{"id":97,"date":"2021-12-15T09:53:29","date_gmt":"2021-12-15T09:53:29","guid":{"rendered":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/numerische-integration\/"},"modified":"2021-12-15T09:53:29","modified_gmt":"2021-12-15T09:53:29","slug":"numerische-integration","status":"publish","type":"chapter","link":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/numerische-integration\/","title":{"raw":"Numerische Integration","rendered":"Numerische Integration"},"content":{"raw":"\n<style>.cmr-5{font-size:50%;}\n.cmr-7{font-size:70%;}\n.cmmi-5{font-size:50%;font-style: italic;}\n.cmmi-7{font-size:70%;font-style: italic;}\n.cmmi-10{font-style: italic;}\n.cmsy-5{font-size:50%;}\n.cmsy-7{font-size:70%;}\n.cmbx-10{ font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-7{font-size:70%;font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-5{font-size:50%;font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmex-7{font-size:70%;}\n.cmex-7x-x-71{font-size:49%;}\n.msam-7{font-size:70%;}\n.msam-5{font-size:50%;}\n.msbm-7{font-size:70%;}\n.msbm-5{font-size:50%;}\n.cmr-17{font-size:170%;}\n.cmr-12{font-size:120%;}\n.cmti-10{ font-style: italic;}\np{margin-top:0;margin-bottom:0}\np.indent{text-indent:0;}\np + p{margin-top:1em;}\np + div, p + pre {margin-top:1em;}\ndiv + p, pre + p {margin-top:1em;}\n@media print {div.crosslinks {visibility:hidden;}}\na img { border-top: 0; border-left: 0; border-right: 0; }\ncenter { margin-top:1em; margin-bottom:1em; }\ntd center { margin-top:0em; margin-bottom:0em; }\n.Canvas { position:relative; }\nmath { text-indent: 0em; }\nli p.indent { text-indent: 0em }\nli p:first-child{ margin-top:0em; }\nli p:last-child, li div:last-child { margin-bottom:0.5em; }\nli p~ul:last-child, li p~ol:last-child{ margin-bottom:0.5em; }\n.enumerate1 {list-style-type:decimal;}\n.enumerate2 {list-style-type:lower-alpha;}\n.enumerate3 {list-style-type:lower-roman;}\n.enumerate4 {list-style-type:upper-alpha;}\n.obeylines-h,.obeylines-v {white-space: nowrap; }\ndiv.obeylines-v p { margin-top:0; margin-bottom:0; }\n.overline{ text-decoration:overline; }\n.overline img{ border-top: 1px solid black; }\ntd.displaylines {text-align:center; white-space:nowrap;}\n.centerline {text-align:center;}\n.rightline {text-align:right;}\npre.verbatim {font-family: monospace,monospace; text-align:left; clear:both; }\n.fbox {padding-left:3.0pt; padding-right:3.0pt; text-indent:0pt; border:solid black 0.4pt; }\ndiv.fbox {display:table}\ndiv.center div.fbox {text-align:center; clear:both; padding-left:3.0pt; padding-right:3.0pt; text-indent:0pt; border:solid black 0.4pt; }\ndiv.minipage{width:100%;}\ndiv.center, div.center div.center {text-align: center; margin-left:1em; margin-right:1em;}\ndiv.center {text-align: left;}\ndiv.flushright, div.flushright div.flushright {text-align: right;}\ndiv.flushright div {text-align: left;}\ndiv.flushleft {text-align: left;}\n.underline{ text-decoration:underline; }\n.underline img{ border-bottom: 1px solid black; margin-bottom:1pt; }\n.framebox-c, .framebox-l, .framebox-r { padding-left:3.0pt; padding-right:3.0pt; text-indent:0pt; border:solid black 0.4pt; }\n.framebox-c {text-align:center;}\n.framebox-l {text-align:left;}\n.framebox-r {text-align:right;}\nspan.thank-mark{ vertical-align: super }\nspan.footnote-mark sup.textsuperscript, span.footnote-mark a sup.textsuperscript{ font-size:80%; }\ndiv.tabular, div.center div.tabular {text-align: center; margin-top:0.5em; margin-bottom:0.5em; }\ntable.tabular td p{margin-top:0em;}\ntable.tabular {margin-left: auto; margin-right: auto;}\ntd p:first-child{ margin-top:0em; }\ntd p:last-child{ margin-bottom:0em; }\ndiv.td00{ margin-left:0pt; margin-right:0pt; }\ndiv.td01{ margin-left:0pt; margin-right:5pt; }\ndiv.td10{ margin-left:5pt; margin-right:0pt; }\ndiv.td11{ margin-left:5pt; margin-right:5pt; }\ntable[rules] {border-left:solid black 0.4pt; border-right:solid black 0.4pt; }\ntd.td00{ padding-left:0pt; padding-right:0pt; }\ntd.td01{ padding-left:0pt; padding-right:5pt; }\ntd.td10{ padding-left:5pt; padding-right:0pt; }\ntd.td11{ padding-left:5pt; padding-right:5pt; }\ntable[rules] {border-left:solid black 0.4pt; border-right:solid black 0.4pt; }\n.hline hr, .cline hr{ height : 0px; margin:0px; }\n.hline td, .cline td{ padding: 0; }\n.hline hr, .cline hr{border:none;border-top:1px solid black;}\n.tabbing-right {text-align:right;}\ndiv.float, div.figure {margin-left: auto; margin-right: auto;}\ndiv.float img {text-align:center;}\ndiv.figure img {text-align:center;}\n.marginpar,.reversemarginpar {width:20%; float:right; text-align:left; margin-left:auto; margin-top:0.5em; font-size:85%; text-decoration:underline;}\n.marginpar p,.reversemarginpar p{margin-top:0.4em; margin-bottom:0.4em;}\n.reversemarginpar{float:left;}\n.equation td{text-align:center; vertical-align:middle; }\ntd.eq-no{ width:5%; }\ntable.equation { width:100%; }\ndiv.math-display, div.par-math-display{text-align:center;}\nmtr.hline mtd{ border-bottom:black solid 1px; padding-top:2px; padding-bottom:0em; }\nmtr.hline mtd mo{ display:none }\nmath .texttt { font-family: monospace; }\nmath .textit { font-style: italic; }\nmath .textsl { font-style: oblique; }\nmath .textsf { font-family: sans-serif; }\nmath .textbf { font-weight: bold; }\nmo.MathClass-op + mi{margin-left:0.3em}\nmi + mo.MathClass-op{margin-left:0.3em}\n math mstyle[mathvariant=\"bold\"] { font-weight: bold; font-style: normal; }\n math mstyle[mathvariant=\"normal\"] { font-weight: normal; font-style: normal; }\n.partToc a, .partToc, .likepartToc a, .likepartToc {line-height: 200%; font-weight:bold; font-size:110%;}\n.index-item, .index-subitem, .index-subsubitem {display:block}\ndiv.caption {text-indent:-2em; margin-left:3em; margin-right:1em; text-align:left;}\ndiv.caption span.id{font-weight: bold; white-space: nowrap; }\nh1.partHead{text-align: center}\np.bibitem { text-indent: -2em; margin-left: 2em; margin-top:0.6em; margin-bottom:0.6em; }\np.bibitem-p { text-indent: 0em; margin-left: 2em; margin-top:0.6em; margin-bottom:0.6em; }\n.paragraphHead, .likeparagraphHead { margin-top:2em; font-weight: bold;}\n.subparagraphHead, .likesubparagraphHead { font-weight: bold;}\n.quote {margin-bottom:0.25em; margin-top:0.25em; margin-left:1em; margin-right:1em; text-align:justify;}\n.verse{white-space:nowrap; margin-left:2em}\ndiv.maketitle {text-align:center;}\nh2.titleHead{text-align:center;}\ndiv.maketitle{ margin-bottom: 2em; }\ndiv.author, div.date {text-align:center;}\ndiv.thanks{text-align:left; margin-left:10%; font-size:85%; font-style:italic; }\ndiv.author{white-space: nowrap;}\n.quotation {margin-bottom:0.25em; margin-top:0.25em; margin-left:1em; }\n.abstract p {margin-left:5%; margin-right:5%;}\ndiv.abstract {width:100%;}\ndiv.tabular, div.center div.tabular {text-align: center; margin-top:0.5em; margin-bottom:0.5em; }\ntable.tabular td p{margin-top:0em;}\ntable.tabular {margin-left: auto; margin-right: auto;}\ntd p:first-child{ margin-top:0em; }\ntd p:last-child{ margin-bottom:0em; }\ndiv.td00{ margin-left:0pt; margin-right:0pt; }\ndiv.td01{ margin-left:0pt; margin-right:5pt; }\ndiv.td10{ margin-left:5pt; margin-right:0pt; }\ndiv.td11{ margin-left:5pt; margin-right:5pt; }\ntable[rules] {border-left:solid black 0.4pt; border-right:solid black 0.4pt; }\ntd.td00{ padding-left:0pt; padding-right:0pt; }\ntd.td01{ padding-left:0pt; padding-right:5pt; }\ntd.td10{ padding-left:5pt; padding-right:0pt; }\ntd.td11{ padding-left:5pt; padding-right:5pt; }\ntable[rules] {border-left:solid black 0.4pt; border-right:solid black 0.4pt; }\n.hline hr, .cline hr{ height : 0px; margin:0px; }\n.hline td, .cline td{ padding: 0; }\n.hline hr, .cline hr{border:none;border-top:1px solid black;}\n.equation-star td{text-align:center; vertical-align:middle; }\ntable.equation-star { width:100%; border-bottom-color: rgb(255,255,255); }\n#content table.equation-star, #content table.equation-star tbody tr td { border: 0px none rgb(255,255,255); }\nmtd.align-odd{margin-left:2em; text-align:right;}\nmtd.align-even{margin-right:2em; text-align:left;}\n.boxed{border: 1px solid black; padding-left:2px; padding-right:2px;}\n.rotatebox{display: inline-block;}\n.item-head{float:left;width:2em;clear:left;}\n.item-content{margin-left:2em;}\n .foreignobject {line-height:100%; font-size:120%; font-family:STIXgeneral,Times,Symbol,cmr10,CMSY10,CMEX10;padding:0; margin:0; text-align:center; }\nmath {vertical-align:baseline; line-height:100%; font-size:100%; font-family:STIXGeneral,Times,Symbol, cmr10,cmsy10,cmex10,cmmi10; font-style: normal; margin:0; padding:0; }\n\n.entry-title{display: none}\n\ndiv.newtheorem { margin-bottom: 2em; margin-top: 2em; border: 1px solid #333; background: #c7e4da; border-color: #4eb79e;}\ndiv.newtheorem h3 { background: #4eb79e; color: white; padding: 0px 15px 0px 15px; margin-top: 12px}\ndiv.newtheorem p { padding: 15px 15px 15px 15px; }\n\ndiv.newtheorem p span.head .ecbx-1095{font-weight: bold}\ndiv.newtheorem p .ecti-1095{font-style: italic}\ndiv.newtheorem div.custom-itemize{font-style: italic}\ndiv.quote{font-style: italic}\ndiv.newtheorem dl, dl.enumerate {display: grid; grid-template-columns: 5% auto; align-items: start; margin-top: 1em}\ndiv.newtheorem dl dd, dl.enumerate dd {margin-bottom: 0.5em}\ndiv.newtheorem dl dt, dl.enumerate dt {font-weight: normal; margin-top: 0px; text-align: right; margin-right: 15%}\ndiv.newtheorem dl dd {font-style: italic}\ndiv.newtheorem dl dt {font-style: italic}\ndiv.proof p span.ecti-1095 {font-style: italic}\ndiv.figure p img { margin-left: auto; margin-right: auto; display: block; }\ndiv.mefigcentered, div.figure { text-align: center }\n\ndl:after {content:\"\";display:table;clear:both;}\ndd {padding:.5em 0;}\ndl {width:100%;}\ndt, dd {display:inline-block; width:125%;}\ndt {text-align:right; font-weight:bold; clear:left; float:left;}\ndd {width:100%; padding-left:1em; padding-top: 0px; clear:right;}\ndd + dd {float:right; clear:both;}\ndd + dt {clear:both;}\ndt + dt {width: 100%; float: none; padding: 0 70% 0 0;}\ndt + dt + dd {margin-top: -2em;}\ndt + dt + dd + dt {margin-top: 2em;}\n<\/style>\n<style>\n\/* CSS Analysis-Skript D-Math ETHZ *\/\n\n\/* Uniform Font, also for headers *\/\nh3 {\n\tfont-family: \"Times New Roman\", serif;\n\tmargin-bottom: 35px;\n}\nh4 {\n\tfont-family: \"Times New Roman\", serif;\n}\nh5 {\n\tfont-family: \"Times New Roman\", serif;\n}\n\n\/* Bold font, e.g. for definitions *\/\n.ecbx-1095 {font-weight: 550 ;}\n\n\n\/* Uniform spacing, indent: larger, noindent, enumerate, itemize *\/\np.indent {\n\tmargin: 25px 0px 0px 0px;\n\ttext-indent: 0px; \n}\np.noindent {\n\tmargin: 15px 0px 0px 0px;\n\ttext-indent: 0px; \n}\ndl.enumerate {\n\tmargin: 0px 0px 0px 0px;\n}\ndl.enumerate dt, dl.enumerate dd {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n}\ndiv.custom-itemize {\n\tmargin: 0px 0px 0px 0px;\n}\ndiv.custom-itemize div.item-head {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n\ttext-align: center;\n}\ndiv.custom-itemize div.item-head:first-of-type {\n\tmargin-top: 0px;\n} \ndiv.custom-itemize div.item-content {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n}\n.MJXc-display {\n\tmargin: 15px 0px 0px 0px;\n}\n\n\n\n\/* green metheorem\/melemma CSS class for more\/medium important latex-theorem-environments *\/\n\/* metheorem box+header *\/\ndiv.metheorem {\n    margin-bottom: 40px;\n    margin-top: 40px;\n\tpadding: 0px 15px 15px 15px;\n    border: 1px solid #333;\n    border-color: #4eb79e;\n    background: #c7e4da;\n}\ndiv.metheorem h4 {\n    background: #4eb79e;\n    color: white;\n\tmargin-top: 12px;\n\tmargin-left: -15px;\n\tmargin-right: -15px;\n\tpadding: 0px 15px 0px 15px;\n}\n\/* melemma box+header *\/\ndiv.melemma {\n    margin-bottom: 40px;\n    margin-top: 40px;\n\tpadding: 0px 15px 15px 15px;\n    border: 1px solid #333;\n    border-color: #4eb79e;\n    background: #F2F2F2;\n}\ndiv.melemma h4 {\n    background: #4eb79e;\n    color: white;\n\tmargin-top: 12px;\n\tmargin-left: -15px;\n\tmargin-right: -15px;\n\tpadding: 0px 15px 0px 15px;\n}\n\/* meexample box+header *\/\ndiv.meexample {\n    margin-bottom: 30px;\n    margin-top: 30px;\n\tpadding: 0px 15px 15px 15px;\n\tborder-color: gainsboro;\n\tborder-style: solid;\n\tborder-width: thin;\n}\ndiv.meexample h4 {\n\tfont-size: inherit;\n\tfont-weight: bold;\n    padding: 15px 0px 0px 0px;\n\tmargin-top: 0px;\n\tmargin-bottom: 5px;\n}\ndiv.meexample h4+p.noindent, div.meexample h4+p.indent {\n\tmargin-top: 5px;\n\ttext-indent: 0px;\n}\n\/* padding and margins for stuff inside these boxes, CSS-selector &gt; doesn't work in WP *\/\ndiv.me details {\n\tmargin: 10px 0px 0px 0px;\n}\ndiv.me dd {\n    width: calc(100% - 30px);\n}\t\n\n\n\/* fixing background of pictures *\/\nimg {\n\tbackground: white;\n}\n\n\/* div-container for centered geoapplet *\/\ndiv.geoapplet {\n\tmargin-left: auto;\n\tmargin-right: auto;\n\tmargin-top: 15px;\n\tmax-width: 100%;\n}\ndiv.geoapplet iframe {\n\tborder-style: none;\n\tmax-height: 110vw;\n}\n\n\/* div-container for centered squeezed tables *\/\ndiv.websqueeze {\n\tmargin-left: auto;\n\tmargin-right: auto;\n}\n\n\/* two containers for squeezing text sizes *\/\ndiv.mesmalltext, div.mesmalltext * {\n\tfont-size: 15px;\n}\nspan.metinytext, span.metinytext * {\n\tfont-size: 12px;\n}\n\n\n\/* removing grid lines in equations *\/\n#content table.equation tr td, #content table.equation tr th {\n    border: none;\n}\n#content table.equation {\n    border: none;\n}\n\n\/* hover\/click-solution for short inline explanations and footnotes *\/\n.hover-text {    \/* hidden part *\/\n    display: none;\n}\n.marginpar {     \/* style for footnote as marginpar *\/\n\ttext-decoration: none;\n\tborder: solid;\n\tborder-width: 1pt;\n\tpadding: 3pt;\t\n\twidth: 30%;\n\tbackground: white;\n}\n.hover-trigger { \/* style for hover\/click-trigger text\/symbol *\/\n\tbackground: none;\n\tborder: none;\n\tpadding: 0;\n\toutline: inherit;\t\n\ttext-transform: none;\n\tfont: inherit;\n\tposition: inherit;\n\tvertical-align: baseline;\n    color: #FF7F00;\n\tcursor: help;\n}\n.hover-trigger:hover +.hover-text{\n    display: inline;\n}\n.hover-trigger:active +.hover-text{\n    display: inline;\n}\n\n\/* simplifying style of details\/summary, removing triangle *\/\ndetails summary {\n  background: none;\n  list-style: none;\n  outline: none;\n  cursor: pointer;\n}\ndetails summary::-webkit-details-marker { \n  display: inline;\n  display: none;\n}\n\n\/* MC-True\/False as inline details\/summary *\/\ndetails.mcquest, div.me details.mcquest {\n\tdisplay: inline;\n\tmargin-top: 0px;\n}\nsummary.mcquest {\n\tdisplay: inline;\n\tcolor: #FF7F00;\n\tcursor: help;\n}\n\n\/* proof style: simple black box with gray background \n                little black square at the end on the right *\/\ndiv.proof {\n\tborder-color: black;\n\tborder-style: solid;\n\tborder-width: thin;\n\tbackground-color: #F2F2F2;\n\tpadding: 15px;\n\tmargin-top: 1em; \n}\ndiv.proof p:first-of-type {\n\tmargin: 0px;\n}\ndiv.qed {\n\tmargin-top: -25px;\n\tmargin-bottom: -7px;\n\ttext-align: right;\n}\ntable.equation+div.qed {\n\tmargin-top: -65px;\n}\n\n\/* The following is making also math-formulas inside the headers of Lemmas, etc., white. *\/\ndiv.melemma h4 span {\n    color: white;\n}\ndiv.metheorem h4 span {\n    color: white;\n}\n\n\/* The following are used to avoid fullstop, period, colon, semicolon, and endquote (broader) to move by itself to the next line after a formula.\n   The math-environment before needs to be wrapped in span.maperiod and the fullstop etc. in a span.period --- together they achieve what we want.  *\/\nspan.maperiod {\n       margin-right: 5px;\n}\nspan.period {\n       display: inline-block;\n       width: 0px;\n       margin-left: -5px;\n       margin-right: 4.9px;\n\t   text-indent: 0px;\n}\nspan.maendquote {\n       margin-right: 8px;\n}\nspan.endquote {\n       display: inline-block;\n       width: 0px;\n       margin-left: -8px;\n       margin-right: 7.9px;\n}\n\n\n\/* The following is removing an extra space left of the equation side in aligned equations *\/\nspan.mjx-mtd {\n    padding-left: 0em !important;\n}\n\n\/* The following fixes the weird problem that math appears smaller if it was rendered while the details tag was closed. *\/\ndetails span.mjx-chtml, details span.MathJax_CHTML {\n font-size: 100% !important;\n}\n\n\/* trying to fix line breaks in verbatim, new lines are missing *\/\npre.verbatim {\n\twhite-space: pre-wrap;\n\tfont-size: small;\n}\n<\/style><h3 id=\"z84d19932b2a2\" class=\"sectionHead\"><span class=\"titlemark\">9.5 <\/span> <a id=\"x1-2790005\"><\/a>Numerische Integration<\/h3> <p class=\"noindent\">Wie bereits erw\u00e4hnt wurde, gibt es Integrale, die sich nicht in Ausdr\u00fccken der \u00fcblichen (den uns bisher bekannten) Funktionen darstellen lassen. Besonderes in diesen F\u00e4llen sind folgende Absch\u00e4tzungen zur approximativen Berechnung von Integralen sehr n\u00fctzlich. Im n\u00e4chsten Abschnitt werden wir weitere Beispiele sehen, die die zugrundeliegende Idee des folgenden Satzes in anderen \u00dcberlegungen gewinnbringend einsetzen. <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"z4675a31b784d\"> <a id=\"x1-279001r56\"><\/a> <span class=\"ecbx-1095\">Satz 9.56.<\/span><\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Seien <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> <span class=\"ecti-1095\">reelle<\/span> <span class=\"ecti-1095\">Zahlen, <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine<\/span> <span class=\"ecti-1095\">Funktion, <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mi>h<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac> <mrow> <mi>b<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mi>a<\/mi><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <\/math> <span class=\"ecti-1095\">die Schrittweite<\/span> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>a<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u2113<\/mi><mi>h<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u2113<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math><\/span><span class=\"period\">.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(a)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">(Rechtecksregel) Falls <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">stetig differenzierbar ist, dann gilt<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>h<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mo>\u2026<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>F<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">wobei der Fehler <\/span><math display=\"inline\"><msub><mrow><mi>F<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">durch <\/span><math display=\"inline\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>F<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo> <mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>2<\/mn><mi>n<\/mi><\/mrow><\/mfrac> <munder class=\"msub\"><mrow><mi class=\"qopname\"> max<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkt ist.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(b)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">(Sehnentrapezregel) Falls <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">zweimal stetig differenzierbar ist, dann gilt<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mi>h<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>F<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>h<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>F<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">wobei der Fehler <\/span><math display=\"inline\"><msub><mrow><mi>F<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">durch <\/span><math display=\"inline\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>F<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo> <mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>6<\/mn><msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <munder class=\"msub\"><mrow><mi class=\"qopname\"> max<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mi class=\"qopname\">\u2033<\/mi><mo>  <\/mo><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkt ist.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(c)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">(Simpson-Regel) Falls <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">viermal<\/span> <span class=\"ecti-1095\">stetig differenzierbar ist und <\/span><math display=\"inline\"><mi>n<\/mi><\/math> <span class=\"ecti-1095\">gerade ist, dann gilt<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>h<\/mi><\/mrow> <mrow><mn>3<\/mn><\/mrow><\/mfrac> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mn>4<\/mn><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-bin\">+<\/mo> <mn>4<\/mn><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>2<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mn>4<\/mn><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> <\/mo><mrow \/><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><msub><mrow><mi>F<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">wobei der Fehler <\/span><math display=\"inline\"><msub><mrow><mi>F<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">durch <\/span><math display=\"inline\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>F<\/mi><\/mrow><mrow><mn>3<\/mn> <\/mrow> <\/msub> <\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo> <mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>5<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>4<\/mn><mn>5<\/mn><msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <munder class=\"msub\"><mrow><mi class=\"qopname\"> max<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mn>4<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkt ist.<\/span><\/p><\/dd><\/dl> <\/div> <p class=\"indent\">Insbesondere verh\u00e4lt sich der Fehler f\u00fcr das Rechtecksverfahren wie <math display=\"inline\"><msub><mrow><mi>O<\/mi><\/mrow><mrow><mi>f<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>n<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u221e<\/mi><\/math><\/span><span class=\"period\">,<\/span> f\u00fcr das Sehnentrapezverfahren wie <math display=\"inline\"><msub><mrow><mi>O<\/mi><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>n<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u221e<\/mi><\/math> und f\u00fcr das Simpson-Verfahren wie <math display=\"inline\"><msub><mrow><mi>O<\/mi><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>n<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>4<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u221e<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\">Wir m\u00f6chten anmerken, dass die Konstanten in obigen Absch\u00e4tzungen nicht optimal sind. Alle drei obigen Approximationsverfahren sind sogenannte Newton-Cotes-Verfahren. Die wesentliche Idee eines solchen Verfahrens ist die folgende: zuerst schreibt man nach Intervalladditivit\u00e4t <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/munderover><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msub><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub> <\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Nun approximiert man jedes obige Integralst\u00fcck <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msub><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><\/mrow><\/msubsup><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> durch einen Ausdruck der Form <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>K<\/mi><\/mrow><\/msubsup><msub><mrow><mi>w<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>z<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> f\u00fcr Gewichte <math display=\"inline\"><msub><mrow><mi>w<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>w<\/mi><\/mrow><mrow><mi>K<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math> mit <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>K<\/mi><\/mrow><\/msubsup><msub><mrow><mi>w<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> und St\u00fctzpunkte <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mi>K<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">.<\/span> Beispielsweise nimmt man f\u00fcr die Sehnentrapezregel zwei St\u00fctzpunkte (<math display=\"inline\"><mi>K<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>2<\/mn><\/math>), n\u00e4mlich die beiden Endpunkte des Intervalls <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">,<\/span> mit den Gewichten <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>w<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>w<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\">Die Summe der Fehler, die auf den St\u00fccken <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msub><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><\/mrow><\/msubsup><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> zustandekommen, ergeben dann den Gesamtfehler der Approximation. Obiger Satz gibt demnach an, wie dieser Fehler kontrolliert werden kann.                                                                                                                                                                           <\/p><p class=\"indent\">Vor dem Beweis des obigen Satzes m\u00f6chten wir kurz erkl\u00e4ren, wie sich die Simpson-Regel als Newton-Cotes-Verfahren auffassen l\u00e4sst. F\u00fcr die \u00e4quidistante Zerlegung des Intervalles <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> mit den Punkten <math display=\"inline\"><msub><mrow><mi>y<\/mi><\/mrow><mrow><mi>\u2113<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>a<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u2113<\/mi><mfrac><mrow><mi>b<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mi>a<\/mi><\/mrow> <mrow><mi>m<\/mi><\/mrow><\/mfrac> <\/math> f\u00fcr <math display=\"inline\"><mi>\u2113<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-punc\">,<\/mo> <mi>m<\/mi><\/math> betrachtet man auf <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mi>\u2113<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">]<\/mo><\/math> die Gewichte <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>w<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>6<\/mn><\/mrow><\/mfrac><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><msub><mrow><mi>w<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mfrac> <mrow> <mn>4<\/mn><\/mrow> <mrow><mn>6<\/mn><\/mrow><\/mfrac><\/math> und <math display=\"inline\"><msub><mrow><mi>w<\/mi><\/mrow><mrow><mn>3<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mfrac> <mrow> <mn>1<\/mn><\/mrow> <mrow><mn>6<\/mn><\/mrow><\/mfrac><\/math> und die St\u00fctzpunkte <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>y<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msub><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mfrac><mrow><msub><mrow><mi>y<\/mi><\/mrow><mrow><mi>\u2113<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">+<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn> <\/mrow> <\/msub> <\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/math> und <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>y<\/mi><\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">.<\/span> Das dazugeh\u00f6rige Newton-Cotes-Verfahren ist genau das Simpson-Verfahren (wieso?). Wir wenden uns nun dem Beweis des obigen Satzes zu. <\/p><p class=\"indent\"> <\/p> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">F\u00fcr (a) verwenden wir den Mittelwertsatz, wonach es zu <math display=\"inline\"><mi>\u2113<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-punc\">,<\/mo> <mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> und <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">]<\/mo><\/math> ein <math display=\"inline\"><msub><mrow><mi>\u03be<\/mi><\/mrow><mrow><mi>x<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> gibt mit <span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>\u03be<\/mi><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Insbesondere gilt <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>\u03be<\/mi><\/mrow><mrow> <mi>x<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><munder class=\"msub\"><mrow><mi class=\"qopname\"> max<\/mi><mo>  <\/mo><\/mrow><mrow><mi>t<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Damit erhalten wir                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msub><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub> <\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mi>h<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">\u2264<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msub><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub> <\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> max<\/mi><mo>  <\/mo><\/mrow><mrow><mi>t<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msub><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub> <\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">\u2264<\/mo> <mfrac><mrow><msup><mrow><mi>h<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <munder class=\"msub\"><mrow><mi class=\"qopname\">max<\/mi><mo>  <\/mo><\/mrow><mrow><mi>t<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><mo class=\"MathClass-punc\">.<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Durch Summation, Intervalladditivit\u00e4t des Integrals und die Dreiecksungleichung erhalten wir <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/munderover><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mi>\u2113<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mi>h<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo> <mfrac><mrow><mi>n<\/mi><msup><mrow><mi>h<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <munder class=\"msub\"><mrow><mi class=\"qopname\">max<\/mi><mo>  <\/mo><\/mrow><mrow><mi>t<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>2<\/mn><mi>n<\/mi><\/mrow><\/mfrac> <munder class=\"msub\"><mrow><mi class=\"qopname\">max<\/mi><mo>  <\/mo><\/mrow><mrow><mi>t<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">F\u00fcr (b) betrachten wir zuerst zu <math display=\"inline\"><mi>\u2113<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> die Endpunkte <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msub><\/math> und <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn> <\/mrow> <\/msub> <\/math> und den Mittelpunkt <math display=\"inline\"><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><mo class=\"MathClass-rel\">=<\/mo> <mfrac> <mrow> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">+<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msub> <\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/math> des Intervalls <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">.<\/span> Des Weiteren definieren wir den Wert <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>M<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> max<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mi class=\"qopname\">\u2033<\/mi><mo>  <\/mo><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><\/math><\/span><span class=\"period\">.<\/span> Nach Korollar <a href=\"..\/..\/chapter\/taylor-approximation#x1-276003r47\">9.47<\/a> gilt f\u00fcr die Approximation durch das erste Taylor-Polynom um <math display=\"inline\"><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><mo class=\"MathClass-close\">)<\/mo><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo> <mfrac><mrow><msub><mrow><mi>M<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow> <mrow><mn>2<\/mn><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <mspace class=\"nbsp\" width=\"0.33em\" \/><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>t<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2264<\/mo> <mfrac><mrow><msub><mrow><mi>M<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mi>h<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><msub><mrow><mi>M<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow> <mrow><mn>8<\/mn><\/mrow><\/mfrac> <msup><mrow><mi>h<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">.<\/span> Wir verwenden dies f\u00fcr die Endpunkte <math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><\/math> und <math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msub> <\/math> des Intervalls <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub><mo class=\"MathClass-close\">]<\/mo><\/math> und erhalten aus der Dreiecksungleichung <\/p><math display=\"block\"> <mtable class=\"multline-star\"> <mtr><mtd class=\"multline-star\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mfrac><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <\/mtd><\/mtr><mtr><mtd class=\"multline-star\"> <mo class=\"MathClass-rel\">=<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><mo class=\"MathClass-bin\">\u2212<\/mo><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo> <mfrac><mrow><msub><mrow><mi>M<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msup><mrow><mi>h<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>8<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">,<\/mo> <\/mtd><\/mtr><\/mtable> <\/math> <p class=\"nopar\"> da sich der lineare Term wegen <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><mo class=\"MathClass-bin\">\u2212<\/mo><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><mo class=\"MathClass-close\">)<\/mo><\/math> aufhebt. <\/p><p class=\"indent\">Aus demselben Grund erhalten wir                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub> <\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mi>h<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><\/mtd><mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub> <\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub> <\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><mspace width=\"2em\" \/><\/mtd><mtd class=\"align-label\" columnalign=\"right\" \/><mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">\u2264<\/mo><mfrac><mrow> <msub><mrow><mi>M<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub> <\/mrow><\/msubsup><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mfrac><mrow> <msub><mrow><mi>M<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mi>h<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><mrow><mfrac><mrow><mi>h<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><\/msubsup><msup><mrow><mi>s<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>s<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><msub><mrow><mi>M<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msup><mrow><mi>h<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>3<\/mn> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mn>8<\/mn><\/mrow><\/mfrac> <mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/><mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Zusammenfassend gilt also f\u00fcr <math display=\"inline\"><mi>\u2113<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> und <math display=\"inline\"><msub><mrow><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover> <\/mrow><mrow><mi>\u2113<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mfrac> <mrow> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msub><mo class=\"MathClass-bin\">+<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/math> <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">\u2264<\/mo> <mfrac><mrow><msub><mrow><mi>M<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msup><mrow><mi>h<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>8<\/mn><\/mrow><\/mfrac> <mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-279005r14\" \/><mstyle class=\"maketag\"><mtext>(9.14)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msub><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub> <\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mi>h<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">\u2264<\/mo> <mfrac><mrow><msub><mrow><mi>M<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msup><mrow><mi>h<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>3<\/mn> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mn>8<\/mn><\/mrow><\/mfrac> <mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-279006r15\" \/><mstyle class=\"maketag\"><mtext>(9.15)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Wir multiplizieren (<a href=\"..\/..\/chapter\/numerische-integration#x1-279005r14\">9.14<\/a>) mit <math display=\"inline\"><mi>h<\/mi><\/math> und summieren sowohl (<a href=\"..\/..\/chapter\/numerische-integration#x1-279005r14\">9.14<\/a>) als auch (<a href=\"..\/..\/chapter\/numerische-integration#x1-279006r15\">9.15<\/a>) \u00fcber <math display=\"inline\"><mi>\u2113<\/mi><\/math> in <span class=\"maperiod\"><math display=\"inline\"><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-punc\">,<\/mo> <mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math><\/span><span class=\"period\">.<\/span> Daraus folgt mit Intervalladditivit\u00e4t des Riemann-Integrals <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>h<\/mi><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/munderover><mfrac><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>n<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><msub><mrow><mi>M<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msup><mrow><mi>h<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>8<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><msub><mrow><mi>M<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msup><mrow><mi>h<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>3<\/mn> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mn>8<\/mn><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><msub><mrow><mi>M<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow> <mrow><mn>6<\/mn><msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">F\u00fcr (c) betrachten wir wieder zuerst zu <math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo> <mfrac><mrow><mi>n<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> die Endpunkte <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><mi>k<\/mi><\/mrow><\/msub><\/math> und <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><mi>k<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>2<\/mn> <\/mrow> <\/msub> <\/math> und den Mittelpunkt <math display=\"inline\"><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><mi>k<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn> <\/mrow> <\/msub> <\/math> des Intervalls <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-close\">]<\/mo><\/math> und verwenden Korollar&nbsp;<a href=\"..\/..\/chapter\/taylor-approximation#x1-276003r47\">9.47<\/a> bei <span class=\"maperiod\"><math display=\"inline\"><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/math><\/span><span class=\"period\">.<\/span> Dies ergibt f\u00fcr <math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub><mo class=\"MathClass-close\">]<\/mo><\/math> <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2033<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2034<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mn>6<\/mn><\/mrow><\/mfrac> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">\u2264<\/mo> <mfrac><mrow><msub><mrow><mi>M<\/mi><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msub><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>4<\/mn><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">\u2264<\/mo> <mfrac><mrow><msub><mrow><mi>M<\/mi><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msub><msup><mrow><mi>h<\/mi><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>4<\/mn><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">,<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-279007r16\" \/><mstyle class=\"maketag\"><mtext>(9.16)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">wobei <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>M<\/mi><\/mrow><mrow><mn>4<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> max<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mn>4<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><\/math><\/span><span class=\"period\">.<\/span> Durch Integration \u00fcber <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub><mo class=\"MathClass-close\">]<\/mo><\/math> erhalten wir <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub> <\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>2<\/mn><mi>h<\/mi><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2033<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>h<\/mi><\/mrow><mrow><mi>h<\/mi><\/mrow><\/msubsup><msup><mrow><mi>s<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>s<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo> <mfrac><mrow><msub><mrow><mi>M<\/mi><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msub><\/mrow> <mrow><mn>4<\/mn><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>h<\/mi><\/mrow><mrow><mi>h<\/mi><\/mrow><\/msubsup><msup><mrow><mi>s<\/mi><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>s<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">oder auch                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub> <\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>2<\/mn><mi>h<\/mi><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2033<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mn>3<\/mn><\/mrow><\/mfrac> <msup><mrow><mi>h<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo> <mfrac><mrow><msub><mrow><mi>M<\/mi><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msub><msup><mrow><mi>h<\/mi><\/mrow><mrow><mn>5<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>6<\/mn><mn>0<\/mn><\/mrow><\/mfrac> <\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Setzen wir <math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><\/math> und <math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msub> <\/math> in Gleichung (<a href=\"..\/..\/chapter\/numerische-integration#x1-279007r16\">9.16<\/a>), so erhalten wir <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mfrac><mrow><mi>h<\/mi><\/mrow> <mrow><mn>3<\/mn><\/mrow><\/mfrac> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mo class=\"MathClass-bin\">\u2212<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>2<\/mn><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2033<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mi>h<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo> <mfrac><mrow><mi>h<\/mi> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mn>2<\/mn> <mo class=\"MathClass-bin\">\u22c5<\/mo> <msub><mrow><mi>M<\/mi><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msub><msup><mrow><mi>h<\/mi><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>3<\/mn> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mn>2<\/mn><mn>4<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><msub><mrow><mi>M<\/mi><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msub><msup><mrow><mi>h<\/mi><\/mrow><mrow><mn>5<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>3<\/mn><mn>6<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">da sich die linearen und kubischen Terme gegenseitig aufheben. Daher gilt                                                                                                                                                                           <\/p><math display=\"block\"> <mtable class=\"multline-star\"> <mtr><mtd class=\"multline-star\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub> <\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow> <mi>h<\/mi><\/mrow> <mrow><mn>3<\/mn><\/mrow><\/mfrac> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mn>4<\/mn><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <\/mtd><\/mtr><mtr><mtd class=\"multline-star\"> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub> <\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow> <mi>h<\/mi><\/mrow> <mrow><mn>3<\/mn><\/mrow><\/mfrac> <mrow><mo class=\"MathClass-open\" fence=\"true\" mathsize=\"1.19em\">(<\/mo><mrow><mn>6<\/mn><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2033<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mi>h<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><mo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow> <mi>h<\/mi><\/mrow> <mrow><mn>3<\/mn><\/mrow><\/mfrac> <mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mn>2<\/mn><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2033<\/mo><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><msup><mrow><mi>h<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow> <mi>h<\/mi><\/mrow> <mrow><mn>3<\/mn><\/mrow><\/mfrac> <mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <\/mtd><\/mtr><mtr><mtd class=\"multline-star\"> <mo class=\"MathClass-rel\">\u2264<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>6<\/mn><mn>0<\/mn><\/mrow><\/mfrac><msub><mrow><mi>M<\/mi><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msub><msup><mrow><mi>h<\/mi><\/mrow><mrow><mn>5<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>3<\/mn><mn>6<\/mn><\/mrow><\/mfrac><msub><mrow><mi>M<\/mi><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msub><msup><mrow><mi>h<\/mi><\/mrow><mrow><mn>5<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo><mfrac><mrow> <mn>2<\/mn><\/mrow> <mrow><mn>4<\/mn><mn>5<\/mn><\/mrow><\/mfrac><msub><mrow><mi>M<\/mi><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msub><msup><mrow><mi>h<\/mi><\/mrow><mrow><mn>5<\/mn><\/mrow><\/msup><mo class=\"MathClass-punc\">.<\/mo> <\/mtd><\/mtr><\/mtable> <\/math> <p class=\"nopar\"> Nach Summation \u00fcber&nbsp;<math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo> <mfrac><mrow><mi>n<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> ergibt sich die Folge&nbsp;<math display=\"inline\"><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>4<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>2<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>4<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>2<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mn>2<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>4<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><\/math> der Gewichte f\u00fcr die Funktionswerte in der Simpson-Regel wie im Satz und die Absch\u00e4tzung genau wie im Beweis von (b) oben. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zdab1fc077dcb\"> <a id=\"x1-279008r57\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 9.57.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Erkl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ren      Sie      unter      Verwendung      der      Simpson-Regel,      wie      man<\/span> <math display=\"inline\"><mi>\u03c0<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>4<\/mn><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msubsup> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>1<\/mn><mo class=\"MathClass-bin\">+<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">auf beliebig viele Dezimalstellen genau bestimmen kann.<\/span> <\/p> <\/div> <a id=\"x1-279009r278\"><\/a> <h4 id=\"z8a28a3ca1010\" class=\"subsectionHead\"><span class=\"titlemark\">9.5.1 <\/span> <a id=\"x1-2800001\"><\/a>Landau-Notation II<\/h4> <p class=\"noindent\">Wie in obigem Beweis der Simpson-Regel ersichtlich wurde, ist das genaue Buchf\u00fchren der Konstanten relativ anstrengend und der eigentliche Wert, den man zum Schluss erh\u00e4lt, steckt nicht so sehr in der konkreten Konstante sondern in den anderen Ausdr\u00fccken. Wir m\u00f6chten nun deswegen ein weiteres St\u00fcck Notation einf\u00fchren, welche uns erlaubt bei Absch\u00e4tzungen unsere Denkleistung auf das Wesentliche zu fokussieren. <\/p><p class=\"indent\">Sei <math display=\"inline\"><mi>X<\/mi><\/math> eine Menge und seien <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>g<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>X<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2102<\/mi><\/math> zwei Funktion. Wir schreiben                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>O<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mspace class=\"quad\" width=\"1em\" \/><mstyle class=\"text\"><mtext>f\u00fcr&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">falls eine Konstante <math display=\"inline\"><mi>C<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> existiert mit <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo> <mi>C<\/mi><mo class=\"MathClass-rel\">|<\/mo><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">|<\/mo><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\">Die obige Notation kann leicht mit der Landau-Notation aus Abschnitt <a href=\"..\/..\/chapter\/landau-notation#x1-1820006\">6.6<\/a> verwirrt werden, weswegen wir uns M\u00fche geben werden, den Zusatz \u201ef\u00fcr <span class=\"maendquote\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math><\/span><span class=\"endquote\">\u201c<\/span> auch immer anzugeben. In einem gewissen Sinne ist die Aussage <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>O<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mstyle class=\"text\"><mtext>f\u00fcr&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> eine \u201eglobale Aussage\u201c, da alle Zahlen <math display=\"inline\"><mi>x<\/mi><\/math> in <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> betroffen sind. Hingegen ist <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>O<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mstyle class=\"text\"><mtext>f\u00fcr&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/math> eine \u201elokale Aussage\u201c , da nur Zahlen <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> betroffen sind, die gross genug sind (das heisst nahe genug bei unendlich sind, siehe Abschnitt <a href=\"..\/..\/chapter\/landau-notation#x1-1820006\">6.6<\/a>). <\/p><p class=\"indent\">In gewissen Situation bedeutet die Notation aber dasselbe, wie folgende \u00dcbung zeigt. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z2f06f7d44fb6\"> <a id=\"x1-280001r58\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 9.58.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">und seien<\/span> <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>g<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">zwei stetige<\/span> <span class=\"ecti-1095\">Funktionen mit <\/span><math display=\"inline\"><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass die Aussagen<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mi>O<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mspace class=\"quad\" width=\"1em\" \/><mstyle class=\"text\"><mtext>f\u00fcr&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mi>O<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mspace class=\"quad\" width=\"1em\" \/><mstyle class=\"text\"><mtext>f\u00fcr&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">quivalent sind.<\/span> <\/p> <\/div> <p class=\"indent\">Die Gross-O-Notation ist per Definition also insbesondere dann n\u00fctzlich, wenn wir Konstanten \u201everstecken\u201c wollen. Beispielsweise ist <span class=\"maperiod\"><math display=\"inline\"><mn>1<\/mn><mn>0<\/mn><mn>0<\/mn><mo class=\"MathClass-punc\">!<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>O<\/mi><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Damit die Notation auch in Rechnungen n\u00fctzlich ist, erlauben wir auch arithmetische Operationen mit ihr (vergleiche auch Abschnitt <a href=\"..\/..\/chapter\/landau-notation#x1-1820006\">6.6<\/a>): Ist <math display=\"inline\"><mi>X<\/mi><\/math> eine Menge und sind <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>g<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>X<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2102<\/mi><\/math> drei Funktionen, dann bedeutet <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>O<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mspace class=\"quad\" width=\"1em\" \/><mstyle class=\"text\"><mtext>f\u00fcr&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">dass <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>O<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> oder intuitiv ausgedr\u00fcckt, dass die Differenz von <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math> und <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <\/math> durch <math display=\"inline\"><mi>g<\/mi><\/math> kontrolliert ist. Des Weiteren folgt aus&nbsp;<math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>O<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>g<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math><\/span><span class=\"period\">,<\/span> auch&nbsp;<math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>O<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>g<\/mi><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\">Die Gross-O-Notation wird auch dazu verwendet, Unabh\u00e4ngigkeiten von gewissen Parametern zum Ausdruck zu bringen. Wir m\u00f6chten dies an einem Beispiel illustrieren.                                                                                                                                                                           <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z353a5690b08f\"> <a id=\"x1-280002r59\"><\/a> <span class=\"ecbx-1095\">Beispiel 9.59 <\/span>(Parameterabh\u00e4ngigkeit bei Landau-Notation)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-punc\">,<\/mo><mn>2<\/mn><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(a)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Nach Taylorapproximation im Sinne von Korollar <\/span><a href=\"..\/..\/chapter\/taylor-approximation#x1-276003r47\"><span class=\"ecti-1095\">9.47<\/span><\/a> <span class=\"ecti-1095\">gilt<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mfrac><mrow><mn>3<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msup><mrow><mi>k<\/mi><\/mrow><mrow><mfrac><mrow><mn>3<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow> <mn>3<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><msup><mrow><mi>k<\/mi><\/mrow><mrow><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>k<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><munder class=\"msub\"><mrow><mi class=\"qopname\"> max<\/mi><mo>  <\/mo><\/mrow><mrow><mi>t<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mo class=\"MathClass-open\">[<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/munder><mrow><mo class=\"MathClass-open\" fence=\"true\" mathsize=\"1.19em\">|<\/mo><mrow><mfrac><mrow><mn>3<\/mn><\/mrow> <mrow><mn>4<\/mn><\/mrow><\/mfrac><msup><mrow><mi>t<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><\/msup><\/mrow><mo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\">|<\/mo><\/mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo><mfrac><mrow> <mn>3<\/mn><\/mrow> <mrow><mn>8<\/mn><\/mrow><\/mfrac><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">.<\/span> <span class=\"ecti-1095\">Insbesondere gilt<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>x<\/mi><\/mrow><mrow><mfrac><mrow><mn>3<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>k<\/mi><\/mrow><mrow><mfrac><mrow><mn>3<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow> <mn>3<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><msup><mrow><mi>k<\/mi><\/mrow><mrow><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>k<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>O<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msup><mrow><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>k<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mspace class=\"quad\" width=\"1em\" \/><mstyle class=\"text\"><mtext>f\u00fcr&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">wobei die in <\/span><math display=\"inline\"><mi>O<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">versteckte implizite Konstante also nicht vom Parameter<\/span> <math display=\"inline\"><mi>k<\/mi><\/math> <span class=\"ecti-1095\">abh<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ngt.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(b)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">H<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ngt die implizite Konstante, nicht wie in (a) oben, von einem Parameter ab, so indizieren wir den Parameter<\/span> <span class=\"ecti-1095\">bei <\/span><math display=\"inline\"><mi>O<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math><span class=\"ecti-1095\">. Ein konkretes<\/span> <span class=\"ecti-1095\">Beispiel: Wenn<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-punc\">,<\/mo><mn>2<\/mn><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">dann gilt<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u03b1<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u03b1<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>O<\/mi><\/mrow><mrow> <mi>\u03b1<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">auf Grund der Taylorapproximation in Korollar <\/span><a href=\"..\/..\/chapter\/taylor-approximation#x1-276003r47\"><span class=\"ecti-1095\">9.47<\/span><\/a><span class=\"ecti-1095\">.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(c)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Auch eine Abh<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ngigkeit des Definitionsbereichs vom Parameter ist denkbar und oft n<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">tzlich. F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle<\/span> <span class=\"ecti-1095\">Zahlen <\/span><math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mi>k<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-punc\">,<\/mo><mi>k<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">gilt<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi class=\"qopname\">log<\/mi><mo>  <\/mo> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi class=\"qopname\">\u2033<\/mi><mo>  <\/mo><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>t<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mi>O<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>k<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">und daher<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\">log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>k<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><\/mrow><\/mfrac> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>k<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>O<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow> <mrow><msup><mrow><mi>k<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>k<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><\/mrow><\/mfrac> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>k<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>O<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>k<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">auf Grund der Taylorapproximation in Korollar <\/span><a href=\"..\/..\/chapter\/taylor-approximation#x1-276003r47\"><span class=\"ecti-1095\">9.47<\/span><\/a><span class=\"ecti-1095\">.<\/span><\/p><\/dd><\/dl> <p class=\"noindent\"><span class=\"ecti-1095\">Im Allgemeinen sollte man die Landau-Notation sorgf<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ltig verwenden, da sich hier oft Fehler<\/span> <span class=\"ecti-1095\">einschleichen insbesondere bei der Frage der Abh<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ngigkeit der impliziten Konstanten.<\/span> <\/p> <\/div> <a id=\"x1-280006r279\"><\/a> \n","rendered":"\n<style scoped=\"scoped\">.cmr-5{font-size:50%;}\n.cmr-7{font-size:70%;}\n.cmmi-5{font-size:50%;font-style: italic;}\n.cmmi-7{font-size:70%;font-style: italic;}\n.cmmi-10{font-style: italic;}\n.cmsy-5{font-size:50%;}\n.cmsy-7{font-size:70%;}\n.cmbx-10{ font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-7{font-size:70%;font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-5{font-size:50%;font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmex-7{font-size:70%;}\n.cmex-7x-x-71{font-size:49%;}\n.msam-7{font-size:70%;}\n.msam-5{font-size:50%;}\n.msbm-7{font-size:70%;}\n.msbm-5{font-size:50%;}\n.cmr-17{font-size:170%;}\n.cmr-12{font-size:120%;}\n.cmti-10{ font-style: italic;}\np{margin-top:0;margin-bottom:0}\np.indent{text-indent:0;}\np + p{margin-top:1em;}\np + div, p + pre {margin-top:1em;}\ndiv + p, pre + p {margin-top:1em;}\n@media print {div.crosslinks {visibility:hidden;}}\na img { border-top: 0; border-left: 0; border-right: 0; }\ncenter { margin-top:1em; margin-bottom:1em; }\ntd center { margin-top:0em; margin-bottom:0em; }\n.Canvas { position:relative; }\nmath { text-indent: 0em; }\nli p.indent { text-indent: 0em }\nli p:first-child{ margin-top:0em; }\nli p:last-child, li div:last-child { margin-bottom:0.5em; }\nli p~ul:last-child, li p~ol:last-child{ margin-bottom:0.5em; }\n.enumerate1 {list-style-type:decimal;}\n.enumerate2 {list-style-type:lower-alpha;}\n.enumerate3 {list-style-type:lower-roman;}\n.enumerate4 {list-style-type:upper-alpha;}\n.obeylines-h,.obeylines-v {white-space: nowrap; }\ndiv.obeylines-v p { margin-top:0; margin-bottom:0; }\n.overline{ text-decoration:overline; }\n.overline img{ border-top: 1px solid black; }\ntd.displaylines {text-align:center; white-space:nowrap;}\n.centerline {text-align:center;}\n.rightline {text-align:right;}\npre.verbatim {font-family: monospace,monospace; text-align:left; clear:both; }\n.fbox {padding-left:3.0pt; padding-right:3.0pt; text-indent:0pt; border:solid black 0.4pt; }\ndiv.fbox {display:table}\ndiv.center div.fbox {text-align:center; clear:both; padding-left:3.0pt; padding-right:3.0pt; text-indent:0pt; border:solid black 0.4pt; }\ndiv.minipage{width:100%;}\ndiv.center, div.center div.center {text-align: center; margin-left:1em; margin-right:1em;}\ndiv.center {text-align: left;}\ndiv.flushright, div.flushright div.flushright {text-align: right;}\ndiv.flushright div {text-align: left;}\ndiv.flushleft {text-align: left;}\n.underline{ text-decoration:underline; }\n.underline img{ border-bottom: 1px solid black; margin-bottom:1pt; }\n.framebox-c, .framebox-l, .framebox-r { padding-left:3.0pt; padding-right:3.0pt; text-indent:0pt; border:solid black 0.4pt; }\n.framebox-c {text-align:center;}\n.framebox-l {text-align:left;}\n.framebox-r {text-align:right;}\nspan.thank-mark{ vertical-align: super }\nspan.footnote-mark sup.textsuperscript, span.footnote-mark a sup.textsuperscript{ font-size:80%; }\ndiv.tabular, div.center div.tabular {text-align: center; margin-top:0.5em; margin-bottom:0.5em; }\ntable.tabular td p{margin-top:0em;}\ntable.tabular {margin-left: auto; margin-right: auto;}\ntd p:first-child{ margin-top:0em; }\ntd p:last-child{ margin-bottom:0em; }\ndiv.td00{ margin-left:0pt; margin-right:0pt; }\ndiv.td01{ margin-left:0pt; margin-right:5pt; }\ndiv.td10{ margin-left:5pt; margin-right:0pt; }\ndiv.td11{ margin-left:5pt; margin-right:5pt; }\ntable[rules] {border-left:solid black 0.4pt; border-right:solid black 0.4pt; }\ntd.td00{ padding-left:0pt; padding-right:0pt; }\ntd.td01{ padding-left:0pt; padding-right:5pt; }\ntd.td10{ padding-left:5pt; padding-right:0pt; }\ntd.td11{ padding-left:5pt; padding-right:5pt; }\ntable[rules] {border-left:solid black 0.4pt; border-right:solid black 0.4pt; }\n.hline hr, .cline hr{ height : 0px; margin:0px; }\n.hline td, .cline td{ padding: 0; }\n.hline hr, .cline hr{border:none;border-top:1px solid black;}\n.tabbing-right {text-align:right;}\ndiv.float, div.figure {margin-left: auto; margin-right: auto;}\ndiv.float img {text-align:center;}\ndiv.figure img {text-align:center;}\n.marginpar,.reversemarginpar {width:20%; float:right; text-align:left; margin-left:auto; margin-top:0.5em; font-size:85%; text-decoration:underline;}\n.marginpar p,.reversemarginpar p{margin-top:0.4em; margin-bottom:0.4em;}\n.reversemarginpar{float:left;}\n.equation td{text-align:center; vertical-align:middle; }\ntd.eq-no{ width:5%; }\ntable.equation { width:100%; }\ndiv.math-display, div.par-math-display{text-align:center;}\nmtr.hline mtd{ border-bottom:black solid 1px; padding-top:2px; padding-bottom:0em; }\nmtr.hline mtd mo{ display:none }\nmath .texttt { font-family: monospace; }\nmath .textit { font-style: italic; }\nmath .textsl { font-style: oblique; }\nmath .textsf { font-family: sans-serif; }\nmath .textbf { font-weight: bold; }\nmo.MathClass-op + mi{margin-left:0.3em}\nmi + mo.MathClass-op{margin-left:0.3em}\n math mstyle[mathvariant=\"bold\"] { font-weight: bold; font-style: normal; }\n math mstyle[mathvariant=\"normal\"] { font-weight: normal; font-style: normal; }\n.partToc a, .partToc, .likepartToc a, .likepartToc {line-height: 200%; font-weight:bold; font-size:110%;}\n.index-item, .index-subitem, .index-subsubitem {display:block}\ndiv.caption {text-indent:-2em; margin-left:3em; margin-right:1em; text-align:left;}\ndiv.caption span.id{font-weight: bold; white-space: nowrap; }\nh1.partHead{text-align: center}\np.bibitem { text-indent: -2em; margin-left: 2em; margin-top:0.6em; margin-bottom:0.6em; }\np.bibitem-p { text-indent: 0em; margin-left: 2em; margin-top:0.6em; margin-bottom:0.6em; }\n.paragraphHead, .likeparagraphHead { margin-top:2em; font-weight: bold;}\n.subparagraphHead, .likesubparagraphHead { font-weight: bold;}\n.quote {margin-bottom:0.25em; margin-top:0.25em; margin-left:1em; margin-right:1em; text-align:justify;}\n.verse{white-space:nowrap; margin-left:2em}\ndiv.maketitle {text-align:center;}\nh2.titleHead{text-align:center;}\ndiv.maketitle{ margin-bottom: 2em; }\ndiv.author, div.date {text-align:center;}\ndiv.thanks{text-align:left; margin-left:10%; font-size:85%; font-style:italic; }\ndiv.author{white-space: nowrap;}\n.quotation {margin-bottom:0.25em; margin-top:0.25em; margin-left:1em; }\n.abstract p {margin-left:5%; margin-right:5%;}\ndiv.abstract {width:100%;}\ndiv.tabular, div.center div.tabular {text-align: center; margin-top:0.5em; margin-bottom:0.5em; }\ntable.tabular td p{margin-top:0em;}\ntable.tabular {margin-left: auto; margin-right: auto;}\ntd p:first-child{ margin-top:0em; }\ntd p:last-child{ margin-bottom:0em; }\ndiv.td00{ margin-left:0pt; margin-right:0pt; }\ndiv.td01{ margin-left:0pt; margin-right:5pt; }\ndiv.td10{ margin-left:5pt; margin-right:0pt; }\ndiv.td11{ margin-left:5pt; margin-right:5pt; }\ntable[rules] {border-left:solid black 0.4pt; border-right:solid black 0.4pt; }\ntd.td00{ padding-left:0pt; padding-right:0pt; }\ntd.td01{ padding-left:0pt; padding-right:5pt; }\ntd.td10{ padding-left:5pt; padding-right:0pt; }\ntd.td11{ padding-left:5pt; padding-right:5pt; }\ntable[rules] {border-left:solid black 0.4pt; border-right:solid black 0.4pt; }\n.hline hr, .cline hr{ height : 0px; margin:0px; }\n.hline td, .cline td{ padding: 0; }\n.hline hr, .cline hr{border:none;border-top:1px solid black;}\n.equation-star td{text-align:center; vertical-align:middle; }\ntable.equation-star { width:100%; border-bottom-color: rgb(255,255,255); }\n#content table.equation-star, #content table.equation-star tbody tr td { border: 0px none rgb(255,255,255); }\nmtd.align-odd{margin-left:2em; text-align:right;}\nmtd.align-even{margin-right:2em; text-align:left;}\n.boxed{border: 1px solid black; padding-left:2px; padding-right:2px;}\n.rotatebox{display: inline-block;}\n.item-head{float:left;width:2em;clear:left;}\n.item-content{margin-left:2em;}\n .foreignobject {line-height:100%; font-size:120%; font-family:STIXgeneral,Times,Symbol,cmr10,CMSY10,CMEX10;padding:0; margin:0; text-align:center; }\nmath {vertical-align:baseline; line-height:100%; font-size:100%; font-family:STIXGeneral,Times,Symbol, cmr10,cmsy10,cmex10,cmmi10; font-style: normal; margin:0; padding:0; }\n\n.entry-title{display: none}\n\ndiv.newtheorem { margin-bottom: 2em; margin-top: 2em; border: 1px solid #333; background: #c7e4da; border-color: #4eb79e;}\ndiv.newtheorem h3 { background: #4eb79e; color: white; padding: 0px 15px 0px 15px; margin-top: 12px}\ndiv.newtheorem p { padding: 15px 15px 15px 15px; }\n\ndiv.newtheorem p span.head .ecbx-1095{font-weight: bold}\ndiv.newtheorem p .ecti-1095{font-style: italic}\ndiv.newtheorem div.custom-itemize{font-style: italic}\ndiv.quote{font-style: italic}\ndiv.newtheorem dl, dl.enumerate {display: grid; grid-template-columns: 5% auto; align-items: start; margin-top: 1em}\ndiv.newtheorem dl dd, dl.enumerate dd {margin-bottom: 0.5em}\ndiv.newtheorem dl dt, dl.enumerate dt {font-weight: normal; margin-top: 0px; text-align: right; margin-right: 15%}\ndiv.newtheorem dl dd {font-style: italic}\ndiv.newtheorem dl dt {font-style: italic}\ndiv.proof p span.ecti-1095 {font-style: italic}\ndiv.figure p img { margin-left: auto; margin-right: auto; display: block; }\ndiv.mefigcentered, div.figure { text-align: center }\n\ndl:after {content:\"\";display:table;clear:both;}\ndd {padding:.5em 0;}\ndl {width:100%;}\ndt, dd {display:inline-block; width:125%;}\ndt {text-align:right; font-weight:bold; clear:left; float:left;}\ndd {width:100%; padding-left:1em; padding-top: 0px; clear:right;}\ndd + dd {float:right; clear:both;}\ndd + dt {clear:both;}\ndt + dt {width: 100%; float: none; padding: 0 70% 0 0;}\ndt + dt + dd {margin-top: -2em;}\ndt + dt + dd + dt {margin-top: 2em;}\n<\/style>\n<style scoped=\"scoped\">\n\/* CSS Analysis-Skript D-Math ETHZ *\/\n\n\/* Uniform Font, also for headers *\/\nh3 {\n\tfont-family: \"Times New Roman\", serif;\n\tmargin-bottom: 35px;\n}\nh4 {\n\tfont-family: \"Times New Roman\", serif;\n}\nh5 {\n\tfont-family: \"Times New Roman\", serif;\n}\n\n\/* Bold font, e.g. for definitions *\/\n.ecbx-1095 {font-weight: 550 ;}\n\n\n\/* Uniform spacing, indent: larger, noindent, enumerate, itemize *\/\np.indent {\n\tmargin: 25px 0px 0px 0px;\n\ttext-indent: 0px; \n}\np.noindent {\n\tmargin: 15px 0px 0px 0px;\n\ttext-indent: 0px; \n}\ndl.enumerate {\n\tmargin: 0px 0px 0px 0px;\n}\ndl.enumerate dt, dl.enumerate dd {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n}\ndiv.custom-itemize {\n\tmargin: 0px 0px 0px 0px;\n}\ndiv.custom-itemize div.item-head {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n\ttext-align: center;\n}\ndiv.custom-itemize div.item-head:first-of-type {\n\tmargin-top: 0px;\n} \ndiv.custom-itemize div.item-content {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n}\n.MJXc-display {\n\tmargin: 15px 0px 0px 0px;\n}\n\n\n\n\/* green metheorem\/melemma CSS class for more\/medium important latex-theorem-environments *\/\n\/* metheorem box+header *\/\ndiv.metheorem {\n    margin-bottom: 40px;\n    margin-top: 40px;\n\tpadding: 0px 15px 15px 15px;\n    border: 1px solid #333;\n    border-color: #4eb79e;\n    background: #c7e4da;\n}\ndiv.metheorem h4 {\n    background: #4eb79e;\n    color: white;\n\tmargin-top: 12px;\n\tmargin-left: -15px;\n\tmargin-right: -15px;\n\tpadding: 0px 15px 0px 15px;\n}\n\/* melemma box+header *\/\ndiv.melemma {\n    margin-bottom: 40px;\n    margin-top: 40px;\n\tpadding: 0px 15px 15px 15px;\n    border: 1px solid #333;\n    border-color: #4eb79e;\n    background: #F2F2F2;\n}\ndiv.melemma h4 {\n    background: #4eb79e;\n    color: white;\n\tmargin-top: 12px;\n\tmargin-left: -15px;\n\tmargin-right: -15px;\n\tpadding: 0px 15px 0px 15px;\n}\n\/* meexample box+header *\/\ndiv.meexample {\n    margin-bottom: 30px;\n    margin-top: 30px;\n\tpadding: 0px 15px 15px 15px;\n\tborder-color: gainsboro;\n\tborder-style: solid;\n\tborder-width: thin;\n}\ndiv.meexample h4 {\n\tfont-size: inherit;\n\tfont-weight: bold;\n    padding: 15px 0px 0px 0px;\n\tmargin-top: 0px;\n\tmargin-bottom: 5px;\n}\ndiv.meexample h4+p.noindent, div.meexample h4+p.indent {\n\tmargin-top: 5px;\n\ttext-indent: 0px;\n}\n\/* padding and margins for stuff inside these boxes, CSS-selector &gt; doesn't work in WP *\/\ndiv.me details {\n\tmargin: 10px 0px 0px 0px;\n}\ndiv.me dd {\n    width: calc(100% - 30px);\n}\t\n\n\n\/* fixing background of pictures *\/\nimg {\n\tbackground: white;\n}\n\n\/* div-container for centered geoapplet *\/\ndiv.geoapplet {\n\tmargin-left: auto;\n\tmargin-right: auto;\n\tmargin-top: 15px;\n\tmax-width: 100%;\n}\ndiv.geoapplet iframe {\n\tborder-style: none;\n\tmax-height: 110vw;\n}\n\n\/* div-container for centered squeezed tables *\/\ndiv.websqueeze {\n\tmargin-left: auto;\n\tmargin-right: auto;\n}\n\n\/* two containers for squeezing text sizes *\/\ndiv.mesmalltext, div.mesmalltext * {\n\tfont-size: 15px;\n}\nspan.metinytext, span.metinytext * {\n\tfont-size: 12px;\n}\n\n\n\/* removing grid lines in equations *\/\n#content table.equation tr td, #content table.equation tr th {\n    border: none;\n}\n#content table.equation {\n    border: none;\n}\n\n\/* hover\/click-solution for short inline explanations and footnotes *\/\n.hover-text {    \/* hidden part *\/\n    display: none;\n}\n.marginpar {     \/* style for footnote as marginpar *\/\n\ttext-decoration: none;\n\tborder: solid;\n\tborder-width: 1pt;\n\tpadding: 3pt;\t\n\twidth: 30%;\n\tbackground: white;\n}\n.hover-trigger { \/* style for hover\/click-trigger text\/symbol *\/\n\tbackground: none;\n\tborder: none;\n\tpadding: 0;\n\toutline: inherit;\t\n\ttext-transform: none;\n\tfont: inherit;\n\tposition: inherit;\n\tvertical-align: baseline;\n    color: #FF7F00;\n\tcursor: help;\n}\n.hover-trigger:hover +.hover-text{\n    display: inline;\n}\n.hover-trigger:active +.hover-text{\n    display: inline;\n}\n\n\/* simplifying style of details\/summary, removing triangle *\/\ndetails summary {\n  background: none;\n  list-style: none;\n  outline: none;\n  cursor: pointer;\n}\ndetails summary::-webkit-details-marker { \n  display: inline;\n  display: none;\n}\n\n\/* MC-True\/False as inline details\/summary *\/\ndetails.mcquest, div.me details.mcquest {\n\tdisplay: inline;\n\tmargin-top: 0px;\n}\nsummary.mcquest {\n\tdisplay: inline;\n\tcolor: #FF7F00;\n\tcursor: help;\n}\n\n\/* proof style: simple black box with gray background \n                little black square at the end on the right *\/\ndiv.proof {\n\tborder-color: black;\n\tborder-style: solid;\n\tborder-width: thin;\n\tbackground-color: #F2F2F2;\n\tpadding: 15px;\n\tmargin-top: 1em; \n}\ndiv.proof p:first-of-type {\n\tmargin: 0px;\n}\ndiv.qed {\n\tmargin-top: -25px;\n\tmargin-bottom: -7px;\n\ttext-align: right;\n}\ntable.equation+div.qed {\n\tmargin-top: -65px;\n}\n\n\/* The following is making also math-formulas inside the headers of Lemmas, etc., white. *\/\ndiv.melemma h4 span {\n    color: white;\n}\ndiv.metheorem h4 span {\n    color: white;\n}\n\n\/* The following are used to avoid fullstop, period, colon, semicolon, and endquote (broader) to move by itself to the next line after a formula.\n   The math-environment before needs to be wrapped in span.maperiod and the fullstop etc. in a span.period --- together they achieve what we want.  *\/\nspan.maperiod {\n       margin-right: 5px;\n}\nspan.period {\n       display: inline-block;\n       width: 0px;\n       margin-left: -5px;\n       margin-right: 4.9px;\n\t   text-indent: 0px;\n}\nspan.maendquote {\n       margin-right: 8px;\n}\nspan.endquote {\n       display: inline-block;\n       width: 0px;\n       margin-left: -8px;\n       margin-right: 7.9px;\n}\n\n\n\/* The following is removing an extra space left of the equation side in aligned equations *\/\nspan.mjx-mtd {\n    padding-left: 0em !important;\n}\n\n\/* The following fixes the weird problem that math appears smaller if it was rendered while the details tag was closed. *\/\ndetails span.mjx-chtml, details span.MathJax_CHTML {\n font-size: 100% !important;\n}\n\n\/* trying to fix line breaks in verbatim, new lines are missing *\/\npre.verbatim {\n\twhite-space: pre-wrap;\n\tfont-size: small;\n}\n<\/style><h3 id=\"z84d19932b2a2\" class=\"sectionHead\"><span class=\"titlemark\">9.5 <\/span> <a id=\"x1-2790005\"><\/a>Numerische Integration<\/h3> <p class=\"noindent\">Wie bereits erw\u00e4hnt wurde, gibt es Integrale, die sich nicht in Ausdr\u00fccken der \u00fcblichen (den uns bisher bekannten) Funktionen darstellen lassen. Besonderes in diesen F\u00e4llen sind folgende Absch\u00e4tzungen zur approximativen Berechnung von Integralen sehr n\u00fctzlich. Im n\u00e4chsten Abschnitt werden wir weitere Beispiele sehen, die die zugrundeliegende Idee des folgenden Satzes in anderen \u00dcberlegungen gewinnbringend einsetzen. <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z4675a31b784d\"> <a id=\"x1-279001r56\"><\/a> <span class=\"ecbx-1095\">Satz 9.56.<\/span><\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Seien <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> <span class=\"ecti-1095\">reelle<\/span> <span class=\"ecti-1095\">Zahlen, <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine<\/span> <span class=\"ecti-1095\">Funktion, <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mi>h<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac> <mrow> <mi>b<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mi>a<\/mi><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <\/math> <span class=\"ecti-1095\">die Schrittweite<\/span> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>a<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u2113<\/mi><mi>h<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u2113<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math><\/span><span class=\"period\">.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(a)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">(Rechtecksregel) Falls <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">stetig differenzierbar ist, dann gilt<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>h<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mo>\u2026<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>F<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">wobei der Fehler <\/span><math display=\"inline\"><msub><mrow><mi>F<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">durch <\/span><math display=\"inline\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>F<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo> <mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>2<\/mn><mi>n<\/mi><\/mrow><\/mfrac> <munder class=\"msub\"><mrow><mi class=\"qopname\"> max<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkt ist.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(b)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">(Sehnentrapezregel) Falls <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">zweimal stetig differenzierbar ist, dann gilt<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mi>h<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>F<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>h<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>F<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">wobei der Fehler <\/span><math display=\"inline\"><msub><mrow><mi>F<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">durch <\/span><math display=\"inline\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>F<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo> <mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>6<\/mn><msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <munder class=\"msub\"><mrow><mi class=\"qopname\"> max<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mi class=\"qopname\">\u2033<\/mi><mo>  <\/mo><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkt ist.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(c)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">(Simpson-Regel) Falls <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">viermal<\/span> <span class=\"ecti-1095\">stetig differenzierbar ist und <\/span><math display=\"inline\"><mi>n<\/mi><\/math> <span class=\"ecti-1095\">gerade ist, dann gilt<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>h<\/mi><\/mrow> <mrow><mn>3<\/mn><\/mrow><\/mfrac> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mn>4<\/mn><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-bin\">+<\/mo> <mn>4<\/mn><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>2<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mn>4<\/mn><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> <\/mo><mrow \/><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><msub><mrow><mi>F<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">wobei der Fehler <\/span><math display=\"inline\"><msub><mrow><mi>F<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">durch <\/span><math display=\"inline\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>F<\/mi><\/mrow><mrow><mn>3<\/mn> <\/mrow> <\/msub> <\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo> <mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>5<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>4<\/mn><mn>5<\/mn><msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <munder class=\"msub\"><mrow><mi class=\"qopname\"> max<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mn>4<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkt ist.<\/span><\/p><\/dd><\/dl> <\/div> <p class=\"indent\">Insbesondere verh\u00e4lt sich der Fehler f\u00fcr das Rechtecksverfahren wie <math display=\"inline\"><msub><mrow><mi>O<\/mi><\/mrow><mrow><mi>f<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>n<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u221e<\/mi><\/math><\/span><span class=\"period\">,<\/span> f\u00fcr das Sehnentrapezverfahren wie <math display=\"inline\"><msub><mrow><mi>O<\/mi><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>n<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u221e<\/mi><\/math> und f\u00fcr das Simpson-Verfahren wie <math display=\"inline\"><msub><mrow><mi>O<\/mi><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>n<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>4<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u221e<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\">Wir m\u00f6chten anmerken, dass die Konstanten in obigen Absch\u00e4tzungen nicht optimal sind. Alle drei obigen Approximationsverfahren sind sogenannte Newton-Cotes-Verfahren. Die wesentliche Idee eines solchen Verfahrens ist die folgende: zuerst schreibt man nach Intervalladditivit\u00e4t <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/munderover><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msub><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub> <\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Nun approximiert man jedes obige Integralst\u00fcck <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msub><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><\/mrow><\/msubsup><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> durch einen Ausdruck der Form <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>K<\/mi><\/mrow><\/msubsup><msub><mrow><mi>w<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>z<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> f\u00fcr Gewichte <math display=\"inline\"><msub><mrow><mi>w<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>w<\/mi><\/mrow><mrow><mi>K<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math> mit <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>K<\/mi><\/mrow><\/msubsup><msub><mrow><mi>w<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> und St\u00fctzpunkte <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mi>K<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">.<\/span> Beispielsweise nimmt man f\u00fcr die Sehnentrapezregel zwei St\u00fctzpunkte (<math display=\"inline\"><mi>K<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>2<\/mn><\/math>), n\u00e4mlich die beiden Endpunkte des Intervalls <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">,<\/span> mit den Gewichten <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>w<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>w<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\">Die Summe der Fehler, die auf den St\u00fccken <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msub><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><\/mrow><\/msubsup><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> zustandekommen, ergeben dann den Gesamtfehler der Approximation. Obiger Satz gibt demnach an, wie dieser Fehler kontrolliert werden kann.                                                                                                                                                                           <\/p><p class=\"indent\">Vor dem Beweis des obigen Satzes m\u00f6chten wir kurz erkl\u00e4ren, wie sich die Simpson-Regel als Newton-Cotes-Verfahren auffassen l\u00e4sst. F\u00fcr die \u00e4quidistante Zerlegung des Intervalles <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> mit den Punkten <math display=\"inline\"><msub><mrow><mi>y<\/mi><\/mrow><mrow><mi>\u2113<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>a<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u2113<\/mi><mfrac><mrow><mi>b<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mi>a<\/mi><\/mrow> <mrow><mi>m<\/mi><\/mrow><\/mfrac> <\/math> f\u00fcr <math display=\"inline\"><mi>\u2113<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-punc\">,<\/mo> <mi>m<\/mi><\/math> betrachtet man auf <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mi>\u2113<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">]<\/mo><\/math> die Gewichte <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>w<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>6<\/mn><\/mrow><\/mfrac><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><msub><mrow><mi>w<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mfrac> <mrow> <mn>4<\/mn><\/mrow> <mrow><mn>6<\/mn><\/mrow><\/mfrac><\/math> und <math display=\"inline\"><msub><mrow><mi>w<\/mi><\/mrow><mrow><mn>3<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mfrac> <mrow> <mn>1<\/mn><\/mrow> <mrow><mn>6<\/mn><\/mrow><\/mfrac><\/math> und die St\u00fctzpunkte <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>y<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msub><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mfrac><mrow><msub><mrow><mi>y<\/mi><\/mrow><mrow><mi>\u2113<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">+<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn> <\/mrow> <\/msub> <\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/math> und <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>y<\/mi><\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">.<\/span> Das dazugeh\u00f6rige Newton-Cotes-Verfahren ist genau das Simpson-Verfahren (wieso?). Wir wenden uns nun dem Beweis des obigen Satzes zu. <\/p><div class=\"wp-nocaption \"><\/div> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open=\"open\"><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">F\u00fcr (a) verwenden wir den Mittelwertsatz, wonach es zu <math display=\"inline\"><mi>\u2113<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-punc\">,<\/mo> <mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> und <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">]<\/mo><\/math> ein <math display=\"inline\"><msub><mrow><mi>\u03be<\/mi><\/mrow><mrow><mi>x<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> gibt mit <span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>\u03be<\/mi><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Insbesondere gilt <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>\u03be<\/mi><\/mrow><mrow> <mi>x<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><munder class=\"msub\"><mrow><mi class=\"qopname\"> max<\/mi><mo>  <\/mo><\/mrow><mrow><mi>t<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Damit erhalten wir                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msub><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub> <\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mi>h<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">\u2264<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msub><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub> <\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> max<\/mi><mo>  <\/mo><\/mrow><mrow><mi>t<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msub><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub> <\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">\u2264<\/mo> <mfrac><mrow><msup><mrow><mi>h<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <munder class=\"msub\"><mrow><mi class=\"qopname\">max<\/mi><mo>  <\/mo><\/mrow><mrow><mi>t<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><mo class=\"MathClass-punc\">.<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Durch Summation, Intervalladditivit\u00e4t des Integrals und die Dreiecksungleichung erhalten wir <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/munderover><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mi>\u2113<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mi>h<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo> <mfrac><mrow><mi>n<\/mi><msup><mrow><mi>h<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <munder class=\"msub\"><mrow><mi class=\"qopname\">max<\/mi><mo>  <\/mo><\/mrow><mrow><mi>t<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>2<\/mn><mi>n<\/mi><\/mrow><\/mfrac> <munder class=\"msub\"><mrow><mi class=\"qopname\">max<\/mi><mo>  <\/mo><\/mrow><mrow><mi>t<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">F\u00fcr (b) betrachten wir zuerst zu <math display=\"inline\"><mi>\u2113<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> die Endpunkte <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msub><\/math> und <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn> <\/mrow> <\/msub> <\/math> und den Mittelpunkt <math display=\"inline\"><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><mo class=\"MathClass-rel\">=<\/mo> <mfrac> <mrow> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">+<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msub> <\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/math> des Intervalls <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">.<\/span> Des Weiteren definieren wir den Wert <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>M<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> max<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mi class=\"qopname\">\u2033<\/mi><mo>  <\/mo><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><\/math><\/span><span class=\"period\">.<\/span> Nach Korollar <a href=\"..\/..\/chapter\/taylor-approximation#x1-276003r47\">9.47<\/a> gilt f\u00fcr die Approximation durch das erste Taylor-Polynom um <math display=\"inline\"><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><mo class=\"MathClass-close\">)<\/mo><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo> <mfrac><mrow><msub><mrow><mi>M<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow> <mrow><mn>2<\/mn><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <mspace class=\"nbsp\" width=\"0.33em\" \/><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>t<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2264<\/mo> <mfrac><mrow><msub><mrow><mi>M<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mi>h<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><msub><mrow><mi>M<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow> <mrow><mn>8<\/mn><\/mrow><\/mfrac> <msup><mrow><mi>h<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">.<\/span> Wir verwenden dies f\u00fcr die Endpunkte <math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><\/math> und <math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msub> <\/math> des Intervalls <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub><mo class=\"MathClass-close\">]<\/mo><\/math> und erhalten aus der Dreiecksungleichung <\/p><math display=\"block\"> <mtable class=\"multline-star\"> <mtr><mtd class=\"multline-star\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mfrac><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <\/mtd><\/mtr><mtr><mtd class=\"multline-star\"> <mo class=\"MathClass-rel\">=<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><mo class=\"MathClass-bin\">\u2212<\/mo><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo> <mfrac><mrow><msub><mrow><mi>M<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msup><mrow><mi>h<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>8<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">,<\/mo> <\/mtd><\/mtr><\/mtable> <\/math> <p class=\"nopar\"> da sich der lineare Term wegen <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><mo class=\"MathClass-bin\">\u2212<\/mo><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><mo class=\"MathClass-close\">)<\/mo><\/math> aufhebt. <\/p><p class=\"indent\">Aus demselben Grund erhalten wir                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub> <\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mi>h<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><\/mtd><mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub> <\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub> <\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><mspace width=\"2em\" \/><\/mtd><mtd class=\"align-label\" columnalign=\"right\" \/><mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">\u2264<\/mo><mfrac><mrow> <msub><mrow><mi>M<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub> <\/mrow><\/msubsup><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mfrac><mrow> <msub><mrow><mi>M<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mi>h<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><mrow><mfrac><mrow><mi>h<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><\/msubsup><msup><mrow><mi>s<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>s<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><msub><mrow><mi>M<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msup><mrow><mi>h<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>3<\/mn> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mn>8<\/mn><\/mrow><\/mfrac> <mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/><mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Zusammenfassend gilt also f\u00fcr <math display=\"inline\"><mi>\u2113<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> und <math display=\"inline\"><msub><mrow><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover> <\/mrow><mrow><mi>\u2113<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mfrac> <mrow> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msub><mo class=\"MathClass-bin\">+<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/math> <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">\u2264<\/mo> <mfrac><mrow><msub><mrow><mi>M<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msup><mrow><mi>h<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>8<\/mn><\/mrow><\/mfrac> <mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-279005r14\" \/><mstyle class=\"maketag\"><mtext>(9.14)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msub><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub> <\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mi>h<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">\u2264<\/mo> <mfrac><mrow><msub><mrow><mi>M<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msup><mrow><mi>h<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>3<\/mn> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mn>8<\/mn><\/mrow><\/mfrac> <mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-279006r15\" \/><mstyle class=\"maketag\"><mtext>(9.15)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Wir multiplizieren (<a href=\"..\/..\/chapter\/numerische-integration#x1-279005r14\">9.14<\/a>) mit <math display=\"inline\"><mi>h<\/mi><\/math> und summieren sowohl (<a href=\"..\/..\/chapter\/numerische-integration#x1-279005r14\">9.14<\/a>) als auch (<a href=\"..\/..\/chapter\/numerische-integration#x1-279006r15\">9.15<\/a>) \u00fcber <math display=\"inline\"><mi>\u2113<\/mi><\/math> in <span class=\"maperiod\"><math display=\"inline\"><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-punc\">,<\/mo> <mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math><\/span><span class=\"period\">.<\/span> Daraus folgt mit Intervalladditivit\u00e4t des Riemann-Integrals <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>h<\/mi><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/munderover><mfrac><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>n<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><msub><mrow><mi>M<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msup><mrow><mi>h<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>8<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><msub><mrow><mi>M<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msup><mrow><mi>h<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>3<\/mn> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mn>8<\/mn><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><msub><mrow><mi>M<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow> <mrow><mn>6<\/mn><msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">F\u00fcr (c) betrachten wir wieder zuerst zu <math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo> <mfrac><mrow><mi>n<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> die Endpunkte <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><mi>k<\/mi><\/mrow><\/msub><\/math> und <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><mi>k<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>2<\/mn> <\/mrow> <\/msub> <\/math> und den Mittelpunkt <math display=\"inline\"><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><mi>k<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn> <\/mrow> <\/msub> <\/math> des Intervalls <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-close\">]<\/mo><\/math> und verwenden Korollar&nbsp;<a href=\"..\/..\/chapter\/taylor-approximation#x1-276003r47\">9.47<\/a> bei <span class=\"maperiod\"><math display=\"inline\"><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/math><\/span><span class=\"period\">.<\/span> Dies ergibt f\u00fcr <math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub><mo class=\"MathClass-close\">]<\/mo><\/math> <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2033<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2034<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mn>6<\/mn><\/mrow><\/mfrac> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">\u2264<\/mo> <mfrac><mrow><msub><mrow><mi>M<\/mi><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msub><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>4<\/mn><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">\u2264<\/mo> <mfrac><mrow><msub><mrow><mi>M<\/mi><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msub><msup><mrow><mi>h<\/mi><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>4<\/mn><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">,<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-279007r16\" \/><mstyle class=\"maketag\"><mtext>(9.16)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">wobei <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>M<\/mi><\/mrow><mrow><mn>4<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> max<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mn>4<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><\/math><\/span><span class=\"period\">.<\/span> Durch Integration \u00fcber <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub><mo class=\"MathClass-close\">]<\/mo><\/math> erhalten wir <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub> <\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>2<\/mn><mi>h<\/mi><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2033<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>h<\/mi><\/mrow><mrow><mi>h<\/mi><\/mrow><\/msubsup><msup><mrow><mi>s<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>s<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo> <mfrac><mrow><msub><mrow><mi>M<\/mi><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msub><\/mrow> <mrow><mn>4<\/mn><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>h<\/mi><\/mrow><mrow><mi>h<\/mi><\/mrow><\/msubsup><msup><mrow><mi>s<\/mi><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>s<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">oder auch                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub> <\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>2<\/mn><mi>h<\/mi><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2033<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mn>3<\/mn><\/mrow><\/mfrac> <msup><mrow><mi>h<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo> <mfrac><mrow><msub><mrow><mi>M<\/mi><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msub><msup><mrow><mi>h<\/mi><\/mrow><mrow><mn>5<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>6<\/mn><mn>0<\/mn><\/mrow><\/mfrac> <\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Setzen wir <math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><\/math> und <math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msub> <\/math> in Gleichung (<a href=\"..\/..\/chapter\/numerische-integration#x1-279007r16\">9.16<\/a>), so erhalten wir <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mfrac><mrow><mi>h<\/mi><\/mrow> <mrow><mn>3<\/mn><\/mrow><\/mfrac> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mo class=\"MathClass-bin\">\u2212<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>2<\/mn><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2033<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mi>h<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo> <mfrac><mrow><mi>h<\/mi> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mn>2<\/mn> <mo class=\"MathClass-bin\">\u22c5<\/mo> <msub><mrow><mi>M<\/mi><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msub><msup><mrow><mi>h<\/mi><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>3<\/mn> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mn>2<\/mn><mn>4<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><msub><mrow><mi>M<\/mi><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msub><msup><mrow><mi>h<\/mi><\/mrow><mrow><mn>5<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>3<\/mn><mn>6<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">da sich die linearen und kubischen Terme gegenseitig aufheben. Daher gilt                                                                                                                                                                           <\/p><math display=\"block\"> <mtable class=\"multline-star\"> <mtr><mtd class=\"multline-star\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub> <\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow> <mi>h<\/mi><\/mrow> <mrow><mn>3<\/mn><\/mrow><\/mfrac> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mn>4<\/mn><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <\/mtd><\/mtr><mtr><mtd class=\"multline-star\"> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub> <\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow> <mi>h<\/mi><\/mrow> <mrow><mn>3<\/mn><\/mrow><\/mfrac> <mrow><mo class=\"MathClass-open\" fence=\"true\" mathsize=\"1.19em\">(<\/mo><mrow><mn>6<\/mn><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2033<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mi>h<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><mo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow> <mi>h<\/mi><\/mrow> <mrow><mn>3<\/mn><\/mrow><\/mfrac> <mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mn>2<\/mn><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2033<\/mo><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mover accent=\"true\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><msup><mrow><mi>h<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow> <mi>h<\/mi><\/mrow> <mrow><mn>3<\/mn><\/mrow><\/mfrac> <mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <\/mtd><\/mtr><mtr><mtd class=\"multline-star\"> <mo class=\"MathClass-rel\">\u2264<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>6<\/mn><mn>0<\/mn><\/mrow><\/mfrac><msub><mrow><mi>M<\/mi><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msub><msup><mrow><mi>h<\/mi><\/mrow><mrow><mn>5<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>3<\/mn><mn>6<\/mn><\/mrow><\/mfrac><msub><mrow><mi>M<\/mi><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msub><msup><mrow><mi>h<\/mi><\/mrow><mrow><mn>5<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo><mfrac><mrow> <mn>2<\/mn><\/mrow> <mrow><mn>4<\/mn><mn>5<\/mn><\/mrow><\/mfrac><msub><mrow><mi>M<\/mi><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msub><msup><mrow><mi>h<\/mi><\/mrow><mrow><mn>5<\/mn><\/mrow><\/msup><mo class=\"MathClass-punc\">.<\/mo> <\/mtd><\/mtr><\/mtable> <\/math> <p class=\"nopar\"> Nach Summation \u00fcber&nbsp;<math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo> <mfrac><mrow><mi>n<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> ergibt sich die Folge&nbsp;<math display=\"inline\"><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>4<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>2<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>4<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>2<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mn>2<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>4<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><\/math> der Gewichte f\u00fcr die Funktionswerte in der Simpson-Regel wie im Satz und die Absch\u00e4tzung genau wie im Beweis von (b) oben. &nbsp;&nbsp;<\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zdab1fc077dcb\"> <a id=\"x1-279008r57\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 9.57.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Erkl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ren      Sie      unter      Verwendung      der      Simpson-Regel,      wie      man<\/span> <math display=\"inline\"><mi>\u03c0<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>4<\/mn><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msubsup> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>1<\/mn><mo class=\"MathClass-bin\">+<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">auf beliebig viele Dezimalstellen genau bestimmen kann.<\/span> <\/p> <\/div> <a id=\"x1-279009r278\"><\/a> <h4 id=\"z8a28a3ca1010\" class=\"subsectionHead\"><span class=\"titlemark\">9.5.1 <\/span> <a id=\"x1-2800001\"><\/a>Landau-Notation II<\/h4> <p class=\"noindent\">Wie in obigem Beweis der Simpson-Regel ersichtlich wurde, ist das genaue Buchf\u00fchren der Konstanten relativ anstrengend und der eigentliche Wert, den man zum Schluss erh\u00e4lt, steckt nicht so sehr in der konkreten Konstante sondern in den anderen Ausdr\u00fccken. Wir m\u00f6chten nun deswegen ein weiteres St\u00fcck Notation einf\u00fchren, welche uns erlaubt bei Absch\u00e4tzungen unsere Denkleistung auf das Wesentliche zu fokussieren. <\/p><p class=\"indent\">Sei <math display=\"inline\"><mi>X<\/mi><\/math> eine Menge und seien <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>g<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>X<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2102<\/mi><\/math> zwei Funktion. Wir schreiben                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>O<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mspace class=\"quad\" width=\"1em\" \/><mstyle class=\"text\"><mtext>f\u00fcr&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">falls eine Konstante <math display=\"inline\"><mi>C<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> existiert mit <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo> <mi>C<\/mi><mo class=\"MathClass-rel\">|<\/mo><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">|<\/mo><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\">Die obige Notation kann leicht mit der Landau-Notation aus Abschnitt <a href=\"..\/..\/chapter\/landau-notation#x1-1820006\">6.6<\/a> verwirrt werden, weswegen wir uns M\u00fche geben werden, den Zusatz \u201ef\u00fcr <span class=\"maendquote\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math><\/span><span class=\"endquote\">\u201c<\/span> auch immer anzugeben. In einem gewissen Sinne ist die Aussage <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>O<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mstyle class=\"text\"><mtext>f\u00fcr&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> eine \u201eglobale Aussage\u201c, da alle Zahlen <math display=\"inline\"><mi>x<\/mi><\/math> in <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> betroffen sind. Hingegen ist <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>O<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mstyle class=\"text\"><mtext>f\u00fcr&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/math> eine \u201elokale Aussage\u201c , da nur Zahlen <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> betroffen sind, die gross genug sind (das heisst nahe genug bei unendlich sind, siehe Abschnitt <a href=\"..\/..\/chapter\/landau-notation#x1-1820006\">6.6<\/a>). <\/p><p class=\"indent\">In gewissen Situation bedeutet die Notation aber dasselbe, wie folgende \u00dcbung zeigt. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z2f06f7d44fb6\"> <a id=\"x1-280001r58\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 9.58.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">und seien<\/span> <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>g<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">zwei stetige<\/span> <span class=\"ecti-1095\">Funktionen mit <\/span><math display=\"inline\"><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass die Aussagen<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mi>O<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mspace class=\"quad\" width=\"1em\" \/><mstyle class=\"text\"><mtext>f\u00fcr&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mi>O<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mspace class=\"quad\" width=\"1em\" \/><mstyle class=\"text\"><mtext>f\u00fcr&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">quivalent sind.<\/span> <\/p> <\/div> <p class=\"indent\">Die Gross-O-Notation ist per Definition also insbesondere dann n\u00fctzlich, wenn wir Konstanten \u201everstecken\u201c wollen. Beispielsweise ist <span class=\"maperiod\"><math display=\"inline\"><mn>1<\/mn><mn>0<\/mn><mn>0<\/mn><mo class=\"MathClass-punc\">!<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>O<\/mi><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Damit die Notation auch in Rechnungen n\u00fctzlich ist, erlauben wir auch arithmetische Operationen mit ihr (vergleiche auch Abschnitt <a href=\"..\/..\/chapter\/landau-notation#x1-1820006\">6.6<\/a>): Ist <math display=\"inline\"><mi>X<\/mi><\/math> eine Menge und sind <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>g<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>X<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2102<\/mi><\/math> drei Funktionen, dann bedeutet <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>O<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mspace class=\"quad\" width=\"1em\" \/><mstyle class=\"text\"><mtext>f\u00fcr&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">dass <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>O<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> oder intuitiv ausgedr\u00fcckt, dass die Differenz von <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math> und <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <\/math> durch <math display=\"inline\"><mi>g<\/mi><\/math> kontrolliert ist. Des Weiteren folgt aus&nbsp;<math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>O<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>g<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math><\/span><span class=\"period\">,<\/span> auch&nbsp;<math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>O<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>g<\/mi><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\">Die Gross-O-Notation wird auch dazu verwendet, Unabh\u00e4ngigkeiten von gewissen Parametern zum Ausdruck zu bringen. Wir m\u00f6chten dies an einem Beispiel illustrieren.                                                                                                                                                                           <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z353a5690b08f\"> <a id=\"x1-280002r59\"><\/a> <span class=\"ecbx-1095\">Beispiel 9.59 <\/span>(Parameterabh\u00e4ngigkeit bei Landau-Notation)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-punc\">,<\/mo><mn>2<\/mn><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(a)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Nach Taylorapproximation im Sinne von Korollar <\/span><a href=\"..\/..\/chapter\/taylor-approximation#x1-276003r47\"><span class=\"ecti-1095\">9.47<\/span><\/a> <span class=\"ecti-1095\">gilt<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mfrac><mrow><mn>3<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msup><mrow><mi>k<\/mi><\/mrow><mrow><mfrac><mrow><mn>3<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow> <mn>3<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><msup><mrow><mi>k<\/mi><\/mrow><mrow><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>k<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><munder class=\"msub\"><mrow><mi class=\"qopname\"> max<\/mi><mo>  <\/mo><\/mrow><mrow><mi>t<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mo class=\"MathClass-open\">[<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/munder><mrow><mo class=\"MathClass-open\" fence=\"true\" mathsize=\"1.19em\">|<\/mo><mrow><mfrac><mrow><mn>3<\/mn><\/mrow> <mrow><mn>4<\/mn><\/mrow><\/mfrac><msup><mrow><mi>t<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><\/msup><\/mrow><mo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\">|<\/mo><\/mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo><mfrac><mrow> <mn>3<\/mn><\/mrow> <mrow><mn>8<\/mn><\/mrow><\/mfrac><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">.<\/span> <span class=\"ecti-1095\">Insbesondere gilt<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>x<\/mi><\/mrow><mrow><mfrac><mrow><mn>3<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>k<\/mi><\/mrow><mrow><mfrac><mrow><mn>3<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow> <mn>3<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><msup><mrow><mi>k<\/mi><\/mrow><mrow><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>k<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>O<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msup><mrow><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>k<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mspace class=\"quad\" width=\"1em\" \/><mstyle class=\"text\"><mtext>f\u00fcr&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">wobei die in <\/span><math display=\"inline\"><mi>O<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">versteckte implizite Konstante also nicht vom Parameter<\/span> <math display=\"inline\"><mi>k<\/mi><\/math> <span class=\"ecti-1095\">abh<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ngt.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(b)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">H<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ngt die implizite Konstante, nicht wie in (a) oben, von einem Parameter ab, so indizieren wir den Parameter<\/span> <span class=\"ecti-1095\">bei <\/span><math display=\"inline\"><mi>O<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math><span class=\"ecti-1095\">. Ein konkretes<\/span> <span class=\"ecti-1095\">Beispiel: Wenn<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-punc\">,<\/mo><mn>2<\/mn><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">dann gilt<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u03b1<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u03b1<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>O<\/mi><\/mrow><mrow> <mi>\u03b1<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">auf Grund der Taylorapproximation in Korollar <\/span><a href=\"..\/..\/chapter\/taylor-approximation#x1-276003r47\"><span class=\"ecti-1095\">9.47<\/span><\/a><span class=\"ecti-1095\">.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(c)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Auch eine Abh<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ngigkeit des Definitionsbereichs vom Parameter ist denkbar und oft n<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">tzlich. F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle<\/span> <span class=\"ecti-1095\">Zahlen <\/span><math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mi>k<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-punc\">,<\/mo><mi>k<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">gilt<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi class=\"qopname\">log<\/mi><mo>  <\/mo> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi class=\"qopname\">\u2033<\/mi><mo>  <\/mo><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>t<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mi>O<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>k<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">und daher<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\">log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>k<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><\/mrow><\/mfrac> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>k<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>O<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow> <mrow><msup><mrow><mi>k<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>k<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><\/mrow><\/mfrac> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>k<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>O<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>k<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">auf Grund der Taylorapproximation in Korollar <\/span><a href=\"..\/..\/chapter\/taylor-approximation#x1-276003r47\"><span class=\"ecti-1095\">9.47<\/span><\/a><span class=\"ecti-1095\">.<\/span><\/p><\/dd><\/dl> <p class=\"noindent\"><span class=\"ecti-1095\">Im Allgemeinen sollte man die Landau-Notation sorgf<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ltig verwenden, da sich hier oft Fehler<\/span> <span class=\"ecti-1095\">einschleichen insbesondere bei der Frage der Abh<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ngigkeit der impliziten Konstanten.<\/span> <\/p> <\/div> <a id=\"x1-280006r279\"><\/a> \n","protected":false},"author":1089,"menu_order":5,"template":"","meta":{"pb_show_title":"","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":92,"_links":{"self":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/97"}],"collection":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/users\/1089"}],"version-history":[{"count":0,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/97\/revisions"}],"part":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/parts\/92"}],"metadata":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/97\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/media?parent=97"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapter-type?post=97"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/contributor?post=97"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/license?post=97"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}