{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 } {PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 3" 4 5 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 4" 5 20 1 {CSTYLE "" -1 -1 "" 1 10 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 4 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 20 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 4 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 3 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 261 "" 0 "" {TEXT -1 25 "Best Taylor Approximation" }} {PARA 260 "" 0 "" {TEXT -1 18 "Jim Herod, Retired" }}{PARA 256 "" 0 " " {TEXT -1 21 "School of Mathematics" }}{PARA 257 "" 0 "" {TEXT -1 12 "Georgia Tech" }}{PARA 258 "" 0 "" {TEXT -1 22 "Atlanta, Ga 30332-0160 " }}{PARA 259 "" 0 "" {TEXT -1 21 "herod@math.gatech.edu" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 102 "In this worksheet, we ask which Taylor quadratic is be st in a sense we describe. We take the function" }}{PARA 0 "" 0 "" {TEXT -1 44 " f(x) = " }{XPPEDIT 18 0 "5*x*exp(x/10)" "6#*(\"\"&\"\"\"%\"xGF%-%$expG6#*&F&F%\"#5!\"\"F% " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 31 "and make a Taylor fit at x = a." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "f:=x->5*x*exp( -5*x/10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "plot(f(x),x=0. .3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "taylor(f(x),x=a,3); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "convert(%,polynom);" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "P3:=unapply(%,(a,x));" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "By compar ing several of these, we see how close the various fits are." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "plot([f(x),P3(3/2,x),P3(1/2, x)],x=0..3,color=[BLACK,RED,GREEN]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 153 "The choice for which a to use is det ermined by requiring that the area between the curves be as small as p ossible, as measured by the following integral." }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 29 "int((f(x)-P3(a,x))^2,x=0..3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "diff(%,a);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "fsolve(%=0,a);" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 29 "Here is a plot of our answer." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "plot(\{f(x),P3(%,x)\},x=0..3 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 9 "Exercise:" }{TEXT -1 98 " Give the Taylor series expansion for x exp(-x) about x = 0, x = 1/2, and x = 1. Graph all these." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {MARK "0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }