{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 "2D Output" 2 20 "" 0 1 0 0 255 1 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 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{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 "Maple Out put" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Heading 2 " -1 256 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 } 3 1 0 0 8 2 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 10 "Calculus I" }}{PARA 256 " " 0 "" {TEXT -1 35 "Lesson 13: Quadratic Approximation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "We want to a pproximate a given function f(x) at x=a with a second degree pol ynomial." }}{PARA 0 "" 0 "" {TEXT -1 71 "Express the second degree pol ynomial as P(x) = A + B (x - a) + C " }{XPPEDIT 18 0 "(x-a)^2;" "6#*$,&%\"xG\"\"\"%\"aG!\"\"\"\"#" }}{PARA 0 "" 0 "" {TEXT -1 9 "We wa nt: " }}{PARA 0 "" 0 "" {TEXT -1 47 " P(a ) = f (a) " }}{PARA 0 "" 0 "" {TEXT -1 49 " \+ P'(a) = f ' (a) " }}{PARA 0 "" 0 "" {TEXT -1 51 " \+ P''(a) = f '' (a) " }}{PARA 0 "" 0 "" {TEXT -1 16 "Hence, we want: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 41 " A = f (a) " }}{PARA 0 " " 0 "" {TEXT -1 43 " B = f ' (a) " }} {PARA 0 "" 0 "" {TEXT -1 45 " 2 C = f '' (a ) " }}{PARA 0 "" 0 "" {TEXT -1 6 "Thus, " }}{PARA 0 "" 0 "" {TEXT -1 67 " P(x) = f (a) + f ' (a) (x-a) + [ f '' (a) / \+ 2 ] " }{XPPEDIT 18 0 "(x-a)^2" "6#*$,&%\"xG\"\"\"%\"aG!\"\"\"\"#" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "Conclusio n: The quadratic approximation to f(x) near x = a is given by: \+ " }}{PARA 0 "" 0 "" {TEXT -1 79 " P(x) = f \+ (a) + f ' (a) (x-a) + [ f '' (a) / 2 ] " }{XPPEDIT 18 0 "(x-a)^2" " 6#*$,&%\"xG\"\"\"%\"aG!\"\"\"\"#" }}{PARA 0 "" 0 "" {TEXT -1 55 " \+ " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 7 "Example" }{TEXT -1 1 " " }{TEXT 260 1 "1" }{TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 67 "Find the quadratic approximation to f(x) = sec (x) for a = 0. " }}{PARA 0 "" 0 "" {TEXT -1 54 "Plot both f(x) and P(x) on the sam e axes near 0. 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" }}{PARA 0 "" 0 "" {TEXT -1 52 "Plot both f (x) and P (x) on the same axes near 1. " }}{PARA 0 "" 0 "" {TEXT -1 103 "Plot both f (x) and P (x) on the sam e axes for a larger domain; is P(x) a good estimate away from 1? " }} {PARA 0 "" 0 "" {TEXT -1 94 "Determine the values of x for which the q uadratic approximation is accurate to within 0.01. " }}{PARA 0 "" 0 "" {TEXT -1 28 "Can you use a plot to help? 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" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "We are to find x values for which | f (x) - P (x) | < 0.01, i.e., " }}{PARA 0 "" 0 "" {TEXT -1 9 "we wa nt: " }}{PARA 0 "" 0 "" {TEXT -1 75 " \+ -0.01 < f (x) - P (x) < 0.01" }}{PARA 0 "" 0 "" {TEXT -1 2 "OR" }}{PARA 0 "" 0 "" {TEXT -1 88 " \+ f (x) - 0.01 < P (x) < f (x) + 0.01. 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Our answer is a numerical estimate. " }}{PARA 0 "" 0 "" {TEXT -1 26 " " }}}}{MARK "35 1" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }