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-1 0 }} {SECT 0 {EXCHG {PARA 280 "" 0 "" {TEXT -1 10 "Calculus I" }}{PARA 281 "" 0 "" {TEXT -1 15 "Newton's Method" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 53 "In This lab we will use Maple and New ton's Method to " }}{PARA 258 "" 0 "" {TEXT -1 56 "calculate the roots of non-trivial polynomial functions." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 259 "" 0 "" {TEXT -1 66 "We begin with some theory. The ide a of Newton's Method is to use " }}{PARA 260 "" 0 "" {TEXT -1 68 "a st raight line to approximate the graph of a polynomial function. " }} {PARA 261 "" 0 "" {TEXT -1 69 "Using the straight line we will be able to find a root of the linear " }}{PARA 262 "" 0 "" {TEXT -1 64 "appro ximation provided that the slope of the line is not zero. " }}{PARA 263 "" 0 "" {TEXT -1 71 "The tangent line is the natural chioce for st raight line approximation " }}{PARA 264 "" 0 "" {TEXT -1 22 "and so we begin there." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 272 67 " This command reinitializes the system. We define below a function." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "f := x -> x^7 - 3*x^5 + 2*x^4-5*x^2 + x + 1;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 265 "" 0 "" {TEXT -1 41 "We wish to find the root of the equation " }{TEXT 256 8 "f(x) = 0" }{TEXT -1 2 ". " } }{PARA 266 "" 0 "" {TEXT -1 46 "We begin by computing the tangent line at say " }{TEXT 257 4 "a = " }{TEXT 273 3 "1.0" }{TEXT -1 2 ". " }} {PARA 267 "" 0 "" {TEXT -1 45 "To do this we first compute the derivat ive. " }}{PARA 268 "" 0 "" {TEXT -1 23 "In maple, the operator " } {TEXT 260 4 "D(f)" }{TEXT -1 29 " computes the derivative of " }} {PARA 275 "" 0 "" {TEXT -1 52 "the function and creates a new function , namely the " }}{PARA 276 "" 0 "" {TEXT -1 20 "derivative function " }{TEXT 274 6 "f '(x)" }{TEXT -1 21 ". We denote this by " }{TEXT 275 2 "fp" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "fp := D(f); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 269 "" 0 "" {TEXT -1 6 "Thus " }{TEXT 263 4 "D(f)" }{TEXT -1 34 " is now the derivative function. " }}{PARA 270 "" 0 "" {TEXT -1 31 "To find the slope at the point " }{TEXT 261 2 "x=" }{TEXT 276 3 "1.0" }{TEXT -1 8 " we let " }{TEXT 278 1 "a" }{TEXT 262 3 " = " } {TEXT 277 19 "1.0 denote a fixed " }}{PARA 282 "" 0 "" {TEXT -1 6 "poi nt." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "a := 1.0;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 271 "" 0 "" {TEXT -1 51 "Now we find the tangent line at this fi xed point. " }}{PARA 272 "" 0 "" {TEXT -1 17 "As you can see, " } {TEXT 283 1 "a" }{TEXT -1 43 " is actually a constant in this equatio n. " }}{PARA 273 "" 0 "" {TEXT -1 31 "We define the linear function \+ " }{TEXT 264 10 "tanline(x)" }{TEXT 279 1 " " }{TEXT -1 19 " associate d to the " }}{PARA 274 "" 0 "" {TEXT -1 37 "tangent line so that we ma y graph it." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 " y := fp(a)*(x-a) + f(a);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 " tanline := x -> fp(a)*(x-a) + f(a);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "plot(\{f(x),fp(x),tanline(x)\}, x=- 1.5..1.5, color=[black,blue,red]);" }}}{EXCHG {PARA 283 "" 0 "" {TEXT -1 0 "" }}{PARA 284 "" 0 "" {TEXT -1 49 "Sketch the graph and label th e important aspects." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ":" }}}{EXCHG {PARA 289 "" 0 "" {TEXT -1 0 "" }}{PARA 277 "" 0 "" {TEXT -1 51 "Next, we solve the tangent line fo r the value of " }{TEXT 258 1 "x" }{TEXT -1 27 " at which the tange nt line" }}{PARA 285 "" 0 "" {TEXT -1 12 "crosses the " }{TEXT 259 1 " x" }{TEXT -1 12 "-axis. The " }{TEXT 265 7 "rootval" }{TEXT -1 39 " \+ expression is essentially the formula" }}{PARA 286 "" 0 "" {TEXT -1 8 "used in " }{TEXT 266 15 "Newton's Method" }{TEXT -1 46 ". This comm and calculates the zeros of the " }}{PARA 288 "" 0 "" {TEXT -1 17 "ex pression. The " }{TEXT 267 5 "solve" }{TEXT -1 17 " expression is a \+ " }{TEXT 268 5 "Maple" }{TEXT -1 20 " command for solving" }}{PARA 287 "" 0 "" {TEXT -1 27 "equations. In this case, " }{TEXT 269 1 "y " }{TEXT -1 44 " is set equal to zero and then we solve for " }{TEXT 270 1 "x" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "root val := a -f(a)/fp(a);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "solve(\{y= 0\},\{x\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 280 55 "The following command will plot the polynomial and the " }}{PARA 0 "" 0 "" {TEXT 281 63 "given tangent line. We gather the \+ method into a single set of " }}{PARA 0 "" 0 "" {TEXT 284 63 "commands that can be iterated to obtain the approximate answer." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 12 "a := 0.5690;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "rootval := a - f(a)/fp(a);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "plot(\{f(x),fp(x),tanline(x)\}, x=-1.5..1.5, color=[b lack,blue,red]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ":" }}} {EXCHG {PARA 290 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 73 "Now complete the lab by finding the roots of the following polynom ials. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "1. f := x -> x^5 -6*x^4 + 3*x^2 - 4 *x + 5;" }}{PARA 0 "" 0 "" {TEXT -1 2 "on" }}{PARA 0 "" 0 "" {TEXT -1 9 "(x=-2..7)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 " 2. f := x -> 8*x^3 -4*x^2 -5*x + 1;" }}{PARA 0 "" 0 "" {TEXT -1 2 "on" }}{PARA 0 "" 0 "" {TEXT -1 9 "(x=-1..1)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "3. f := x -> x^(99) - 45* x^(65) + 23*x^(11) + x^3 - x^2 + x -1;" }}{PARA 0 "" 0 "" {TEXT -1 2 " on" }}{PARA 0 "" 0 "" {TEXT -1 33 "(x=-9.8 ..-1.15) and (x=0.7..1.2)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 278 "" 0 "" {TEXT -1 75 "For t his polynomial of degree 99 look for roots between -1.15 and -9.8 and \+ " }}{PARA 279 "" 0 "" {TEXT -1 49 "between 0.7 and 1.2. You should fi nd 5 roots. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 291 "" 0 "" {TEXT -1 67 "Use Newtons method as above in conjunction with plotting \+ methods. " }}{PARA 292 "" 0 "" {TEXT -1 64 "This may be easily accomp lished by copying the given polynomial " }}{PARA 293 "" 0 "" {TEXT -1 69 "into the function definition above. Begin by plotting the functio ns " }}{PARA 294 "" 0 "" {TEXT -1 23 "on the given intervals." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 271 46 "You may use the plot command below to preview \+ " }}{PARA 0 "" 0 "" {TEXT 282 48 "the polynomial functions on various \+ intervals. " }}{PARA 0 "" 0 "" {TEXT 285 66 "Change the function defi nition above or rewrite a definition here." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "plot(f(x),x=-2..2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ":" }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }