{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Headi ng 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "1505 04S Test 3 Take home \+ Maple checks" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 52 "1. geometric ser ies manipulation to get power series" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "f:=x->x/(8-x^3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "plot(f(x),x=-3..3,-5..5);" }}}}{SECT 1 {PARA 3 "" 0 " " {TEXT -1 24 "2. radius of convergence" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "a:=n->(-1)^n*x^n/5^n/n^2;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 67 "simplify((a(n+1)/a(n)));\nlimit(%,n=infinity);\nabs (%)<1;\nsolve(%,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "Sum( subs(x=5,a(n)),n=1..infinity);\nvalue(%);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 45 "Sum(subs(x=-5,a(n)),n=1..infinity);\nvalue(%);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 114 "So the endpoints are also include d in the interval of convergence. As a check we move a bit outside thi s interval:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "Sum(subs(x=- 5.1,a(n)),n=1..infinity);\nvalue(%);\nSum(subs(x=5.1,a(n)),n=1..infini ty);\nvalue(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "Sum(a(n) ,n=1..infinity)\n=sum(a(n),n=1..infinity);" }}}}{SECT 1 {PARA 3 "" 0 " " {TEXT -1 21 "3. limit using series" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "plot((1-cos(x))/(1+x-exp(x)),x=-1..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "Limit((1-cos(x))/(1+x-exp(x)),x=0) \n=limit((1-cos(x))/(1+x-exp(x)),x=0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "taylor(1-cos(x),x=0,5);\ntaylor(1+x-exp(x),x=0,4);\n (1/2*x^2-1/24*x^4)/(-1/2*x^2-1/6*x^3);\nsimplify(%);\nsubs(x=0,%);" }} }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 36 "4. definite integration using s eries" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 132 "after doing handwork we \+ get this series of terms to add; the first 3 are all we need since the 4th is smaller than the target error:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "seq([n,(-1)^n*.5^(2*n+3)/(2*n+3)/n!],n=0..3);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 ".5^3/3-.5^5/5+.5^7/7/2;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "evalf(Int(x^2*exp(-x^2),x=0. .0.5));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "f:=x->x^2*exp(-x^2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "plot((D@@4)(f)(x),x=0..0.5);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "(D@@4)(f)(0);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 136 "The max abs value of the fourth derivati ve of the integrand over the region of integration is clearly 24, for \+ the Simpson error formula:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "24*.5^5/180/2^4;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "S2= 1/3*1/4*(f(0.)+4*f(.25)+f(0.5));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "check on our hand taylor sequence work: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "taylor(f(x),x=0,8);\nT6:=convert(%,polynom);\nIn t(T6,x=0..0.5);\nvalue(%);" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 40 "5 . taylor and gravity at Earth's surface" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "taylor((1+x)^(-2),x=0,4);\nT2:=1-2*x;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "subs(x=h/R,mg*T2);\nexpand(%);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "2*h/R=.01;\nh=solve(%,h);\nR :=6371*km;\n%%;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "h=rhs(%) *5/8*mi/km;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "0 0 0" 38 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }