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SFapFTFW-Feo6$Q#12F'F8-F260Q\"~F'F8F>FAFCFEFGFIFKFMFOFRFTFW-Fjn6%F\\o- Feo6$Q\"2F'F8Fho-F260Q'+F'F8F>FAFCFEFGFIFKF^pF`pFbpFTFW-Feo6$Q\"3 F'F8-F#6'F+-Fjn6%-I(mfencedGF$6$-F#6%F\\oF^q-Feo6$FVF8F8F[qFhoF+-Fjn6% -Fiq6$-F#6&F+-Fjn6%F\\o-F#6#F[qFhoF^q-Feo6$Q\"4F'F8F8F[qFhoF+/%.lineth icknessGQ\"1F'/%+denomalignGQ'centerF'/%)numalignGFas/%)bevelledGF@F+- I'mspaceGF$6&/%'heightGQ&0.0exF'/%&widthGQ&0.3emF'/%&depthGF[t/%*lineb reakGQ%autoF'-F262Q0ⅆF'F5F8F;F>FAFCFEFGFIFK/FNQ'prefixF' FOFRFTFWF\\oF+" }{TEXT 206 1 " " }}}{EXCHG {PARA 207 "" 0 "" {TEXT 208 3 "4) " }{XPPEDIT 18 0 "Int(x/(x^6+1), x);" "6#-%$IntG6$*&%\"xG\" \"\",&*$)F'\"\"'F(F(F(F(!\"\"F'" }{TEXT 206 1 " " }}}{EXCHG {PARA 207 "" 0 "" {TEXT 208 3 "5) " }{XPPEDIT 18 0 "Int((x^3+x)/(1+x^2)^(1/2), x );" "6#-%$IntG6$*&,&*$)%\"xG\"\"$\"\"\"F,F*F,F,),&F,F,*$)F*\"\"#F,F,*& F,F,F1!\"\"F3F*" }{TEXT 206 1 " " }}}{EXCHG {PARA 207 "" 0 "" {TEXT 208 3 "6) " }{XPPEDIT 18 0 "Int(sin(x)^4*cos(x)^2, x);" "6#-%$IntG6$*& )-%$sinG6#%\"xG\"\"%\"\"\")-%$cosGF*\"\"#F-F+" }{TEXT 206 1 " " }}} {EXCHG {PARA 207 "" 0 "" {TEXT 221 0 "" }}}{EXCHG {PARA 203 "" 0 "" {TEXT 216 21 "NUMERICAL INTEGRATION" }}{PARA 207 "" 0 "" {TEXT 221 0 " " }}{PARA 207 "" 0 "" {TEXT 208 79 "To begin we must first load the pa ckage of macros called \"student\" as follows" }}}{EXCHG {PARA 207 "> \+ " 0 "" {MPLTEXT 1 211 14 "with(student);" }}}{EXCHG {PARA 207 "" 0 "" {TEXT 208 32 "The integral to be estimated is:" }{TEXT 208 2 "\n" } {TEXT 208 1 " " }{TEXT 208 2 "\n" }{TEXT 208 57 " \+ " }{TEXT 208 2 "\n" }{TEXT 208 238 "The first method will be the right-hand rule. First, divide [0, 2 ] into n subintervals of equal length, for example n=6, the area under the graph of is approximated by the sum of the areas of 6 rectangles . To visualize Riemann sum, enter" }{TEXT 208 2 "\n" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 211 29 "rightbox(exp(-x^2),x=0..2,6);" }}} {EXCHG {PARA 207 "" 0 "" {TEXT 208 65 "Our first step produces the sig ma notation for the approximation:" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 211 29 "rightsum(exp(-x^2),x=0..2,6);" }}}{EXCHG {PARA 207 "" 0 "" {TEXT 208 30 "Our next step expands the sum:" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 211 9 "value(%);" }}}{EXCHG {PARA 207 "" 0 "" {TEXT 208 47 "Next we get a decimal approximation to the sum:" }}} {EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 211 9 "evalf(%);" }}}{EXCHG {PARA 207 "" 0 "" {TEXT 208 46 "We have an approximate value for the i ntegral." }{TEXT 208 2 "\n" }{TEXT 208 2 "\n" }{TEXT 208 82 "The next \+ mathod will be the left-hand rule. To get a picture of 6 rectangles, t ype" }{TEXT 208 2 "\n" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 211 28 "leftbox(exp(-x^2),x=0..2,6);" }}}{EXCHG {PARA 207 "" 0 "" {TEXT 208 76 "Next we produce the sigma notation, expand the sum, and then a pproximate it." }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 211 28 "lefts um(exp(-x^2),x=0..2,6);" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 211 9 "value(%);" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 211 9 "evalf(%) ;" }}}{EXCHG {PARA 207 "" 0 "" {TEXT 208 65 "The next method is the mi dpoint rule. To view 6 rectangles, enter" }}}{EXCHG {PARA 207 "> " 0 " " {MPLTEXT 1 211 30 "middlebox(exp(-x^2),x=0..2,6);" }}}{EXCHG {PARA 207 "" 0 "" {TEXT 208 68 "Then generate the sigma notation, expand the sum, and approximate it" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 211 30 "middlesum(exp(-x^2),x=0..2,6);" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 211 9 "value(%);" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 211 9 "evalf(%);" }}}{EXCHG {PARA 207 "" 0 "" {TEXT 208 40 "The next m ethod is the Trapezoidal rule:" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 211 30 "trapezoid(exp(-x^2),x=0..2,6);" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 211 9 "value(%);" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 211 9 "evalf(%);" }}}{EXCHG {PARA 207 "" 0 "" {TEXT 208 37 "The nex t method is hthe Simpson rule:" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 211 28 "simpson(exp(-x^2),x=0..2,6);" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 211 9 "value(%);" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 211 9 "evalf(%);" }}}{EXCHG {PARA 207 "" 0 "" {TEXT 221 0 "" }} {PARA 207 "" 0 "" {TEXT 208 44 "Let's use the error bound for Midpoint rule " }}{PARA 207 "" 0 "" {TEXT 208 55 " \+ " }}{PARA 207 "" 0 "" {TEXT 208 88 "to dete rmine a value of n which can be used to approximate the integral withi n 0.00001. " }}{PARA 207 "" 0 "" {TEXT 208 56 "Where , for all . Conse quently, we need to investigate ." }}{PARA 207 "" 0 "" {TEXT 221 0 "" }}{PARA 207 "> " 0 "" {MPLTEXT 1 211 13 "f:=exp(-x^2);" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 211 15 "d1f:=diff(f,x);" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 211 17 "d2f:=diff(d1f,x);" }}}{EXCHG {PARA 207 "" 0 "" {TEXT 221 0 "" }}{PARA 207 "" 0 "" {TEXT 208 52 "To \+ find an upper bound for the second derivative of " }{TEXT 202 4 "f , " }{TEXT 208 24 "find critical values of " }{TEXT 202 7 "f''(x)." }} {PARA 207 "" 0 "" {TEXT 221 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 211 17 "d3f:=diff(d2f,x):" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 211 15 "solve(d3f=0,x);" }}}{EXCHG {PARA 207 "" 0 "" {TEXT 221 0 "" }}{PARA 207 "" 0 "" {TEXT 208 6 "Graph " }{TEXT 210 8 "y=f''( x)" }{TEXT 208 37 " and find an critical value at which " }{TEXT 210 7 "f''(x) " }{TEXT 208 13 "is maximized." }}{PARA 207 "" 0 "" {TEXT 221 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 211 17 "plot(d2f,x=0 ..2);" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 211 29 "K:=evalf(abs(s ubs(x=0,d2f)));" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 211 30 "solv e(2*K/(24*n^2)=0.00001,n);" }}}{EXCHG {PARA 207 "" 0 "" {TEXT 208 2 " \n" }{TEXT 219 43 "Homework for Lab 1 is in the file Hwk1.mws" }{TEXT 208 2 "\n" }{TEXT 208 7 " " }{TEXT 208 2 "\n" }{TEXT 208 2 " " }{TEXT 208 2 "\n" }}}{PARA 208 "" 0 "" {TEXT 222 0 "" }}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }