{VERSION 5 0 "IBM INTEL NT" "5.0" }
{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 "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 256 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
257 "" 1 12 0 0 0 0 2 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 
0 0 2 0 0 0 0 0 0 0 0 1 }{CSTYLE "" 26 259 "" 1 18 0 0 0 0 0 0 0 0 0 
0 0 0 0 1 }{CSTYLE "" -1 260 "" 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 261 "" 1 18 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
262 "" 1 18 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 1 12 0 
0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 1 12 0 0 0 0 0 0 0 0 
0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 266 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
267 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 1 12 0 
0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "" 1 12 0 0 0 0 0 2 0 0 
0 0 0 0 0 1 }{CSTYLE "" -1 270 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 271 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
272 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 273 "" 1 12 0 
0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 274 "" 1 12 0 0 0 0 1 2 0 0 
0 0 0 0 0 1 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 276 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
277 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 278 "" 1 12 0 
0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 279 "" 1 12 0 0 0 0 0 0 0 0 
0 0 0 0 0 1 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 281 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
282 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 283 "" 1 12 0 
0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 284 "" 1 12 0 0 0 0 0 2 0 0 
0 0 0 0 0 1 }{CSTYLE "" -1 285 "" 1 12 0 0 0 0 1 2 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 286 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
287 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 288 "" 1 12 0 
0 0 0 1 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 289 "" 1 12 0 0 0 0 0 2 0 0 
0 0 0 0 0 1 }{CSTYLE "" -1 290 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 291 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
292 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 293 "" 0 1 0 0 
0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 294 "" 0 1 0 0 0 0 1 0 0 0 0 0 
0 0 0 1 }{CSTYLE "" -1 295 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 296 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
297 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 298 "" 0 1 0 0 
0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 299 "" 0 1 0 0 0 0 1 0 0 0 0 0 
0 0 0 1 }{CSTYLE "" -1 300 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 301 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
302 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 303 "" 0 1 0 0 
0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 304 "" 0 1 0 0 0 0 1 0 0 0 0 0 
0 0 0 1 }{CSTYLE "" -1 305 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 306 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
307 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 308 "" 0 1 0 0 
0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 309 "" 0 1 0 0 0 0 1 0 0 0 0 0 
0 0 0 1 }{CSTYLE "" -1 310 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 311 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
312 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 313 "" 1 12 0 0 
0 0 1 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 314 "" 0 1 0 0 0 0 1 0 0 0 0 0 
0 0 0 1 }{CSTYLE "" -1 315 "" 1 12 0 0 0 0 1 2 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 316 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
317 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 318 "" 0 1 0 0 
0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 319 "" 0 1 0 0 0 0 1 0 0 0 0 0 
0 0 0 1 }{CSTYLE "" -1 320 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 321 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
322 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 323 "" 0 1 0 0 
0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 324 "" 0 1 0 0 0 0 1 0 0 0 0 0 
0 0 0 1 }{CSTYLE "" -1 325 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 326 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
327 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 328 "" 1 18 0 0 
0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 329 "" 1 12 0 0 0 0 0 0 0 0 0 
0 0 0 0 1 }{CSTYLE "" -1 330 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 331 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
332 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 333 "" 1 12 0 
0 0 0 1 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 334 "" 1 12 0 0 0 0 0 2 0 0 
0 0 0 0 0 1 }{CSTYLE "" -1 335 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 336 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
337 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 338 "" 1 12 0 0 
0 0 1 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 339 "" 1 12 0 0 0 0 0 2 0 0 0 
0 0 0 0 1 }{CSTYLE "" -1 340 "" 1 12 0 0 0 0 1 2 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 341 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal
" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 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 "Maple Output" 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 "R3 Font 0" -1 256 1 
{CSTYLE "" -1 -1 "Helvetica" 1 12 0 0 0 1 2 1 2 0 0 0 0 0 0 1 }0 0 0 
-1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 2" -1 257 1 {CSTYLE "" -1 
-1 "Courier" 1 12 0 0 0 1 2 2 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 
0 0 -1 0 }{PSTYLE "" 0 258 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 "" 0 259 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 "" 0 260 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 "" 4 261 
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 258 "" 0 "" {TEXT 259 32 "NETON'S METHOD FOR FINDING ROO
TS" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 4 "" 0 "" {TEXT -1 90 "Calculus I Project                       \+
                                                 " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "                      " }
}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 1 " " }{TEXT 266 11 "Objectives:" }
{TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 48 "  To solve a word probl
em using Newton's method." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 263 
1 " " }{TEXT 257 70 " To find the roots of a function using graphing a
nd Newton's method.  " }}{PARA 0 "" 0 "" {TEXT -1 37 "  To use a compu
ter algebra system to" }}{PARA 0 "" 0 "" {TEXT -1 59 "       1.  Graph
 and zoom to evaluate a root of a function." }}{PARA 0 "" 0 "" {TEXT 
-1 92 "       2.  Find a set of values that converge to a root of a fu
nction using Newton's method." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "
" }{TEXT 267 24 "Preliminary Information:" }}{PARA 0 "" 0 "" {TEXT 
268 1 " " }{TEXT 282 2 "A " }{TEXT 270 4 "root" }{TEXT 271 16 " of a f
unction  " }{TEXT 313 5 "y = f" }{TEXT 332 1 "(" }{TEXT 333 1 "x" }
{TEXT 334 68 ")  is an x-intercept of the graph of the function.  It i
s a solution" }}{PARA 0 "" 0 "" {TEXT -1 26 "          of the equation
 " }{TEXT 314 3 "  f" }{TEXT -1 1 "(" }{TEXT 336 1 "x" }{TEXT -1 1 ")
" }{TEXT 337 3 " = " }{TEXT -1 1 "0" }}{PARA 0 "" 0 "" {TEXT -1 2 "A \+
" }{TEXT 275 4 "root" }{TEXT -1 14 " can be found " }}{PARA 0 "" 0 "" 
{TEXT -1 64 "            by repeatedly zooming in on the graph on the \+
x-axis " }}{PARA 0 "" 0 "" {TEXT -1 15 "            or " }}{PARA 0 "" 
0 "" {TEXT -1 32 "             by Newton's method." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 276 17 "Newton's Method: " }
{TEXT -1 76 " Newton's method is a numerical method for finding roots \+
of a function to a " }}{PARA 0 "" 0 "" {TEXT -1 96 "very high degree o
f accuracy.   This method is useful in solving complicated equations i
ncluding" }}{PARA 0 "" 0 "" {TEXT -1 97 "equations involving transcend
ental functions, where no direct algebraic methods such as factoring" 
}}{PARA 0 "" 0 "" {TEXT -1 7 "work.  " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 97 "The method is based on the fact that \+
a differentiable function is locally linear.   The curve is " }}{PARA 
0 "" 0 "" {TEXT -1 89 "virtually indistinguishable from its tangent at
 a point on the curve on a small interval." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 269 48 "Newton's metho
d consists of the following steps:" }}{PARA 0 "" 0 "" {TEXT 273 50 "  \+
   a) Start by taking an initial approximation, " }{XPPEDIT 18 0 "x[0]
;" "6#&%\"xG6#\"\"!" }{TEXT 283 13 ", of a root. " }}{PARA 0 "" 0 "" 
{TEXT 272 39 "         Consider the tangent line to  " }{TEXT 274 5 "y
 = f" }{TEXT 335 1 "(" }{TEXT 338 1 "x" }{TEXT 339 1 ")" }{TEXT 340 1 
" " }{TEXT 277 3 "at " }{XPPEDIT 18 0 "x[0];" "6#&%\"xG6#\"\"!" }
{TEXT 285 3 ".  " }}{PARA 0 "" 0 "" {TEXT -1 57 "          The equatio
n of the tangent line is given by   " }{TEXT 286 4 "y - " }{XPPEDIT 
18 0 "y[0];" "6#&%\"yG6#\"\"!" }{TEXT 302 6 " = f '" }{TEXT -1 1 "(" }
{XPPEDIT 18 0 "x[0];" "6#&%\"xG6#\"\"!" }{TEXT -1 2 ")(" }{TEXT 341 4 
"x - " }{XPPEDIT 18 0 "x[0];" "6#&%\"xG6#\"\"!" }{TEXT -1 1 ")" }}
{PARA 0 "" 0 "" {TEXT 284 14 "          Let " }{XPPEDIT 18 0 "x[1];" "
6#&%\"xG6#\"\"\"" }{TEXT 288 1 " " }{TEXT 289 7 "be the " }{TEXT 315 
1 "x" }{TEXT 316 38 "-intercept of this tangent line.  Then" }}{PARA 
0 "" 0 "" {TEXT -1 1 " " }{TEXT 290 33 "                            0 \+
- f" }{TEXT -1 1 "(" }{XPPEDIT 18 0 "x[0];" "6#&%\"xG6#\"\"!" }{TEXT 
-1 1 ")" }{TEXT 303 6 " = f '" }{TEXT -1 1 "(" }{XPPEDIT 18 0 "x[0];" 
"6#&%\"xG6#\"\"!" }{TEXT 304 1 " " }{TEXT -1 2 ")(" }{XPPEDIT 18 0 "x[
1];" "6#&%\"xG6#\"\"\"" }{TEXT 306 3 " - " }{XPPEDIT 18 0 "x[0];" "6#&
%\"xG6#\"\"!" }{TEXT -1 1 ")" }{TEXT 305 2 ", " }{TEXT -1 2 "or" }
{TEXT 310 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 38 "                      \+
                " }{XPPEDIT 18 0 "x[1];" "6#&%\"xG6#\"\"\"" }{TEXT 
291 3 " = " }{XPPEDIT 18 0 "x[0];" "6#&%\"xG6#\"\"!" }{TEXT 308 5 "  -
 f" }{TEXT -1 1 "(" }{XPPEDIT 18 0 "x[0];" "6#&%\"xG6#\"\"!" }{TEXT 
-1 1 ")" }{TEXT 307 4 "/f '" }{TEXT -1 1 "(" }{XPPEDIT 18 0 "x[0];" "6
#&%\"xG6#\"\"!" }{TEXT -1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 95 "      \+
     The assumption behind Newton' method is that the tangent line is \+
close to the curve " }}{PARA 0 "" 0 "" {TEXT -1 17 "         and its \+
" }{TEXT 327 1 "x" }{TEXT -1 12 "-intercept, " }{XPPEDIT 18 0 "x[1];" 
"6#&%\"xG6#\"\"\"" }{TEXT -1 1 "," }{TEXT 311 1 " " }{TEXT -1 18 " is \+
closer to the " }{TEXT 317 1 "x" }{TEXT -1 42 "-intercept, actual root
, of the curve.    " }}{PARA 0 "" 0 "" {TEXT -1 25 "      b) Then we f
ind the" }{TEXT 318 3 " x-" }{TEXT -1 47 "intercept of the tangent lin
e to the curve at (" }{XPPEDIT 18 0 "x[1];" "6#&%\"xG6#\"\"\"" }{TEXT 
300 3 ", f" }{TEXT -1 1 "(" }{XPPEDIT 18 0 "x[1];" "6#&%\"xG6#\"\"\"" 
}{TEXT -1 2 "))" }{TEXT 309 1 " " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 24 "           We call this " }{TEXT 319 1 "x" }{TEXT -1 11 "
-intercept " }{XPPEDIT 18 0 "x[2];" "6#&%\"xG6#\"\"#" }{TEXT -1 1 ":" 
}{TEXT 324 2 "  " }{TEXT -1 1 " " }{XPPEDIT 18 0 "x[2];" "6#&%\"xG6#\"
\"#" }{TEXT 320 3 " = " }{XPPEDIT 18 0 "x[1];" "6#&%\"xG6#\"\"\"" }
{TEXT 321 5 "  - f" }{TEXT -1 1 "(" }{XPPEDIT 18 0 "x[1];" "6#&%\"xG6#
\"\"\"" }{TEXT -1 1 ")" }{TEXT 292 4 "/f '" }{TEXT -1 1 "(" }{XPPEDIT 
18 0 "x[1];" "6#&%\"xG6#\"\"\"" }{TEXT -1 1 ")" }}{PARA 0 "" 0 "" 
{TEXT -1 81 "      c) By repeating the same process several times we c
onstruct a set of points" }}{PARA 0 "" 0 "" {TEXT -1 10 "          " }
{XPPEDIT 18 0 "x[0];" "6#&%\"xG6#\"\"!" }{TEXT 295 2 ", " }{XPPEDIT 
18 0 "x[1];" "6#&%\"xG6#\"\"\"" }{TEXT 293 2 ", " }{XPPEDIT 18 0 "x[2]
;" "6#&%\"xG6#\"\"#" }{TEXT 312 3 " , " }{XPPEDIT 18 0 "x[3];" "6#&%\"
xG6#\"\"$" }{TEXT -1 35 ", ... using the recursive formula: " }
{XPPEDIT 18 0 "x[n+1];" "6#&%\"xG6#,&%\"nG\"\"\"F(F(" }{TEXT 297 3 " =
 " }{XPPEDIT 18 0 "x[n];" "6#&%\"xG6#%\"nG" }{TEXT 298 4 " - f" }
{TEXT -1 1 "(" }{XPPEDIT 18 0 "x[n];" "6#&%\"xG6#%\"nG" }{TEXT -1 1 ")
" }{TEXT 301 4 "/f '" }{TEXT -1 1 "(" }{XPPEDIT 18 0 "x[n];" "6#&%\"xG
6#%\"nG" }{TEXT -1 1 ")" }}{PARA 0 "" 0 "" {TEXT -1 73 "          In m
ost cases the sequence approaches a root of the function.  " }{TEXT 
296 8 "        " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 287 36 "The method is very sensitive to the " }{TEXT -1 30 "choi
ce of the initial value.  " }}{PARA 0 "" 0 "" {TEXT -1 58 "If an inapp
ropriate initial value is chosen, the sequence " }{XPPEDIT 18 0 "x[0];
" "6#&%\"xG6#\"\"!" }{TEXT 325 2 ", " }{XPPEDIT 18 0 "x[1];" "6#&%\"xG
6#\"\"\"" }{TEXT 294 2 ", " }{XPPEDIT 18 0 "x[2];" "6#&%\"xG6#\"\"#" }
{TEXT 326 3 " , " }{XPPEDIT 18 0 "x[3];" "6#&%\"xG6#\"\"$" }{TEXT -1 
17 ", ... may behave " }}{PARA 0 "" 0 "" {TEXT -1 82 "erratically inst
ead of approaching the actual root.   Therefore the initial value " }
{XPPEDIT 18 0 "x[0];" "6#&%\"xG6#\"\"!" }}{PARA 0 "" 0 "" {TEXT -1 25 
"must be chosen carefully." }}}{PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 
264 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 1 " " }{TEXT 258 15 "Solve
d Example:" }}{PARA 0 "" 0 "" {TEXT -1 100 " A car dealer sells a new \+
car for $18,000.  He also offers to sell the same car for monthly paym
ents" }}{PARA 0 "" 0 "" {TEXT -1 75 " of $375.00 for five years.   Wha
t monthly rate is this dealer charging?  (" }{TEXT 299 8 "Calculus" }
{TEXT -1 24 ", Stewart, P.298,   #37)" }}}{SECT 0 {PARA 3 "" 0 "" 
{TEXT -1 1 " " }{TEXT 260 10 "Solution: " }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "f:= x -
> 48*x*(1 + x)^60 - (1 + x)^60 - 1;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,(*(\"#[\"\"\"9$F/),&F0F/F
/F/\"#gF/F/*$F1F/!\"\"F/F5F(F(F(" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 
1 " " }{TEXT 278 19 "Root by Graphing - " }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 22 "plot(f(x), x=0..0.1);\n" }}{PARA 13 "" 1 "" {GLPLOT2D 476 476 
476 {PLOTDATA 2 "6%-%'CURVESG6$7W7$$\"\"!F)$!\"#F)7$$\"3[mmm;arz@!#?$!
3'))\\Z*[.L??!#<7$$\"3mLL$e9ui2%F/$!3zO_f_ZpE?F27$$\"3-nmm\"z_\"4iF/$!
31jO%Hytw,#F27$$\"3[mmmT&phN)F/$!3==,8b<s')>F27$$\"3KLLe*=)H\\5!#>$!3H
iSDWW[G>F27$$\"3omm\"z/3uC\"FE$!3**etF'['>W=F27$$\"3#****\\7LRDX\"FE$!
3\"))3Jy7!H><F27$$\"3emm\"zR'ok;FE$!3Hj%pu%Q7T:F27$$\"3'****\\i5`h(=FE
$!34]>Hb%[LI\"F27$$\"3gLLL3En$4#FE$!3r=Oi=v%z#)*!#=7$$\"3wmm;/RE&G#FE$
!3OqgsjllRiF[o7$$\"3A+++D.&4]#FE$!3S!o$*=*HWv6F[o7$$\"3!)*****\\PAvr#F
E$\"3TTfMf255_F[o7$$\"3)******\\nHi#HFE$\"3yJp%yi&Q$G\"F27$$\"3jmm\"z*
ev:JFE$\"3^z)*oP^?B@F27$$\"31LLL347TLFE$\"3UGd0V7\\PLF27$$\"3cLLLLY.KN
FE$\"3_%fC4154e%F27$$\"3!****\\7o7Tv$FE$\"3?U8<S*o%>jF27$$\"3sKLL$Q*o]
RFE$\"37)QGJuUX;)F27$$\"33++D\"=lj;%FE$\"3w1'o?\"znd5!#;7$$\"33++vV&R<
P%FE$\"3vq(y%)fi8L\"F^r7$$\"3gLL$e9Ege%FE$\"3-\"[`8:)Qq;F^r7$$\"3ILLeR
\"3Gy%FE$\"3zRCbT1eP?F^r7$$\"3smm;/T1&*\\FE$\"3=4,M*H2L]#F^r7$$\"3Smm
\"zRQb@&FE$\"3;5Yu)4Gg2$F^r7$$\"3!****\\(=>Y2aFE$\"3=!*oy-*3,m$F^r7$$
\"3qmm;zXu9cFE$\"3af409oI%R%F^r7$$\"3^******\\y))GeFE$\"3_<Sqhx$GG&F^r
7$$\"3!)****\\i_QQgFE$\"3Y\"R*Q=q***H'F^r7$$\"3j***\\7y%3TiFE$\"3\"=$>
mN:$[W(F^r7$$\"3o****\\P![hY'FE$\"3;2[pE5)*H*)F^r7$$\"33KLL$Qx$omFE$\"
3k\")\\oDee[5!#:7$$\"3k+++v.I%)oFE$\"3%*Gi(zwD9C\"F[v7$$\"3Amm\"zpe*zq
FE$\"3\"paIuA#[V9F[v7$$\"37+++D\\'QH(FE$\"3))RV5hl])p\"F[v7$$\"3GKLe9S
8&\\(FE$\"3mx\"p$*e*ov>F[v7$$\"3]++D1#=bq(FE$\"3f58'=Fc'4BF[v7$$\"3>LL
L3s?6zFE$\"3'p9?SW_ho#F[v7$$\"3)*)**\\7`Wl7)FE$\"3W;LH=/+TJF[v7$$\"3[n
mmm*RRL)FE$\"33TDl@%[ik$F[v7$$\"3Smm;a<.Y&)FE$\"3:$HySj55C%F[v7$$\"3-M
Le9tOc()FE$\"3nSnW'>\"**>\\F[v7$$\"3u******\\Qk\\*)FE$\"3ys5=\\d/LcF[v
7$$\"3!QLL3dg6<*FE$\"3]5Y54@1qlF[v7$$\"3A***\\(oTAq#*FE$\"3UBQ')*z$=Nq
F[v7$$\"3-mmmmxGp$*FE$\"3qs1dM#y8`(F[v7$$\"3sK$eRA5\\Z*FE$\"3IlGH!\\))
o4)F[v7$$\"3!3+]7oK0e*FE$\"3:%*)o\">*QDq)F[v7$$\"3o******\\oi\"o*FE$\"
36!z1T%[LA$*F[v7$$\"3'****\\(=5s#y*FE$\"3G(*QK(4VR)**F[v7$$\"3c**\\P40
O\"*)*FE$\"3%**fAvAjW2\"!#97$$\"3/+++++++5F[o$\"3O%Rd-BIg:\"F[\\l-%'CO
LOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q\"x6\"Q!F\\]l-%%VIEWG6$;F(
$\"\"\"Fg\\l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 
45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "
plot(f(x), x=0.02..0.05);\n" }}{PARA 13 "" 1 "" {GLPLOT2D 259 187 187 
{PLOTDATA 2 "6%-%'CURVESG6$7S7$$\"3/+++++++?!#>$!3`;Y`J7CJ6!#<7$$\"3A+
+]i9Rl?F*$!3X.n8wUOH5F-7$$\"38+]PC#)GA@F*$!3110TiAxS$*!#=7$$\"3%****\\
Peui=#F*$!37^#3gaC7>)F87$$\"3/++D'3&o]AF*$!3)pqn^Moh%pF87$$\"32+](oX*y
9BF*$!3O.2'3^VXh&F87$$\"33+]P9CAuBF*$!3WGf0q&HIH%F87$$\"30+]P*zhdV#F*$
!3%H,]%e(H=$GF87$$\"37+]P>fS*\\#F*$!3YOli$4Ag@\"F87$$\"39+](=$f%Gc#F*$
\"3y&>L@;_=1&F*7$$\"3E++]#y,\"GEF*$\"3YfMZtZE+CF87$$\"3/++Dr\"zbo#F*$
\"3wA%fXvQp<%F87$$\"3)*****\\(4&G]FF*$\"3(R]1&\\r)[I'F87$$\"3)*****\\7
nD:GF*$\"35=U$*)f?\\e)F87$$\"3=++]-*oy(GF*$\"3SD?*=;KD4\"F-7$$\"3#)**
\\PpnsMHF*$\"3/6Hk`i&yJ\"F-7$$\"3C++]siL-IF*$\"3@lok#f^Cg\"F-7$$\"37++
+!R5'fIF*$\"3gpyz6sSe=F-7$$\"3V+]P/QBEJF*$\"3)36%4<Y>u@F-7$$\"3e*****
\\\"o?&=$F*$\"3?teSFyrqCF-7$$\"3?+]Pa&4*\\KF*$\"3.PO)\\tc`\"GF-7$$\"3'
)**\\7j=_6LF*$\"3#3(='ydVK;$F-7$$\"3a++vVy!eP$F*$\"3O6JoRqwZNF-7$$\"3C
+](=WU[V$F*$\"3*)=()fzr?@RF-7$$\"3k***\\7B>&)\\$F*$\"3]5sDH/\"pM%F-7$$
\"3=+]P>:mkNF*$\"3E<`6`;^:[F-7$$\"3H+]iv&QAi$F*$\"3]9')f>@UY_F-7$$\"3E
++vtLU%o$F*$\"3-.Jp0D-PdF-7$$\"3/+++bjm[PF*$\"3;m6$)*)omsiF-7$$\"30++v
yb^6QF*$\"3Uul#yF6k#oF-7$$\"3%***\\PMaKsQF*$\"3#)He%))oD:R(F-7$$\"3,++
D6W%)RRF*$\"3aw%*H)RfW0)F-7$$\"3t*****\\@80+%F*$\"3Ej9$=P:No)F-7$$\"3=
++]7,HlSF*$\"3mw#)[mjw\"R*F-7$$\"3D+]P4w)R7%F*$\"3%HU(GU?x15!#;7$$\"3I
++]x%f\")=%F*$\"3e'=bqYmX3\"Fcv7$$\"3_**\\P/-a[UF*$\"3k]SRm>lh6Fcv7$$
\"3r**\\(=Yb;J%F*$\"3!*)R_#zwWY7Fcv7$$\"3m****\\i@OtVF*$\"3ig4hWUrL8Fc
v7$$\"3`**\\PfL'zV%F*$\"3F/e60f\"*H9Fcv7$$\"3N+++!*>=+XF*$\"3@Bv*\\!>Z
F:Fcv7$$\"3w***\\i_4Qc%F*$\"32I=rExWK;Fcv7$$\"3=+]P%>5pi%F*$\"3'>B*RU?
+U<Fcv7$$\"3/+++bJ*[o%F*$\"3s([0EOsw%=Fcv7$$\"3)****\\7<[8v%F*$\"3=JLL
Zd%\\(>Fcv7$$\"3y******Hjy5[F*$\"3=>J7f&4Y4#Fcv7$$\"33+]P/)fT([F*$\"3%
e[j.@S&GAFcv7$$\"3^+]i0j\"[$\\F*$\"3q5T$o8OJO#Fcv7$$\"3G+++++++]F*$\"3
S>s<Dg3:DFcv-%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!Fb[lFa[l-%+AXESLABELSG6$
Q\"x6\"Q!Fg[l-%%VIEWG6$;$\"\"#!\"#$\"\"&F_\\l%(DEFAULTG" 1 2 0 1 10 0 
2 9 1 4 2 1.000000 47.000000 52.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 28 "plot(f(x), x=0.025..0.026);\n" }}{PARA 
13 "" 1 "" {GLPLOT2D 222 195 195 {PLOTDATA 2 "6%-%'CURVESG6$7S7$$\"39+
++++++D!#>$!3oZ*pB]?/?\"!#=7$$\"3ummT:(z@]#F*$!3i[^$yj$4V6F-7$$\"3MLe9
ui2/DF*$!3[?;+@m5$4\"F-7$$\"3pm;z_\"4i]#F*$!3W1hM%frn.\"F-7$$\"3%pmT&p
hN3DF*$!3mK`'ej_$*z*F*7$$\"3KLe*=)H\\5DF*$!3*)\\5#>^9CB*F*7$$\"3pm\"z/
3uC^#F*$!3AbFR&*4l0()F*7$$\"3-+DJ$RDX^#F*$!3GZJ;A21f\")F*7$$\"3%o;zR'o
k;DF*$!3+hIdod`#f(F*7$$\"39+D1J:w=DF*$!33k,Xu,dEqF*7$$\"3SLL3En$4_#F*$
!3k4u'yx#4VkF*7$$\"3*omT!RE&G_#F*$!35nQFB91GfF*7$$\"3&****\\K]4]_#F*$!
3ep:sC&=qM&F*7$$\"3/++vB_<FDF*$!3yz'pg^mAw%F*7$$\"35++v'Hi#HDF*$!3_b&4
0t#\\(>%F*7$$\"3jm\"z*ev:JDF*$!3U9)GFV`No$F*7$$\"3HLL347TLDF*$!3s*\\U,
K,62$F*7$$\"3[LLLY.KNDF*$!3s@k64H9^DF*7$$\"3>+D\"o7Tv`#F*$!3a&oTC(***
\\%>F*7$$\"3XLL$Q*o]RDF*$!3O91e%Q&G29F*7$$\"32+D\"=lj;a#F*$!3'GM\\<s^0
;)!#?7$$\"3'**\\PaR<Pa#F*$!3\\/qro!\\#=DF\\r7$$\"3EL$e9Egea#F*$\"3yps)
G[7=Q$F\\r7$$\"3QLeR\"3Gya#F*$\"3mwo,FQa6))F\\r7$$\"3wm;/T1&*\\DF*$\"3
7)y_K*\\2o9F*7$$\"3em\"zRQb@b#F*$\"3q!p'>bt4z?F*7$$\"3;+v=>Y2aDF*$\"3H
QFN\"GQ@h#F*7$$\"3sm;zXu9cDF*$\"3;gZcJ`.*=$F*7$$\"31++]y))GeDF*$\"3I(z
\"GaaL'y$F*7$$\"3))**\\i_QQgDF*$\"3okUx(om>P%F*7$$\"35+D\"y%3TiDF*$\"3
jb5Bf6\")R\\F*7$$\"3))**\\P![hYc#F*$\"3q$>$[-%4<d&F*7$$\"3MLL$Qx$omDF*
$\"3O$=/SBg29'F*7$$\"39++v.I%)oDF*$\"3y1*o/fk'\\nF*7$$\"3om\"zpe*zqDF*
$\"3&pR;Dw-EI(F*7$$\"3#****\\#\\'QHd#F*$\"3\\su%=`'R3zF*7$$\"3OLe9S8&
\\d#F*$\"3KrLc4-jz%)F*7$$\"3y*\\i?=bqd#F*$\"3!*=Bp-*4!y!*F*7$$\"39LL3s
?6zDF*$\"3y/2a'R)Hk'*F*7$$\"3-+DJXaE\"e#F*$\"3e27_RG%z-\"F-7$$\"3cmmm*
RRLe#F*$\"3o&3(z,qJ(3\"F-7$$\"3Um;a<.Y&e#F*$\"3Od2oTw;[6F-7$$\"3?Le9tO
c(e#F*$\"3w!plRWY'37F-7$$\"3-++]Qk\\*e#F*$\"3IQTP-jLk7F-7$$\"3;L$3dg6<
f#F*$\"3%\\zVP<+$G8F-7$$\"3mmmmxGp$f#F*$\"3QScX2Wj&Q\"F-7$$\"3)**\\7oK
0ef#F*$\"3i!*\\')[S*oW\"F-7$$\"3#)*\\(=5s#yf#F*$\"3;z6!Ri_c]\"F-7$$\"3
))*************f#F*$\"3g)36:dL*o:F--%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!F
a[lF`[l-%+AXESLABELSG6$Q\"x6\"Q!Ff[l-%%VIEWG6$;$\"#D!\"$$\"#EF^\\l%(DE
FAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 41.000000 40.000000 0 0 "Curve
 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "plot(f(x), x=0.02544
..0.02545);\n" }}{PARA 13 "" 1 "" {GLPLOT2D 180 225 225 {PLOTDATA 2 "6
%-%'CURVESG6$7S7$$\"34++++++WD!#>$!3D%*ped[*3u\"!#?7$$\"3xmT:(z@Sa#F*$
!3musat$G4o\"F-7$$\"3Qe9ui2/WDF*$!3;#*)oiu](G;F-7$$\"3\"p\"z_\"4iSa#F*
$!3Q5)fJJq+d\"F-7$$\"3o;aphN3WDF*$!3n`VX)e**4^\"F-7$$\"3Xe*=)H\\5WDF*$
!3I'Qh?I3AX\"F-7$$\"3'=z/3uCTa#F*$!3;e=?#3+xR\"F-7$$\"3(\\7LRDXTa#F*$!
3;C!e)*ye7M\"F-7$$\"3q\"zR'ok;WDF*$!3!oxGK&e)GG\"F-7$$\"3@D1J:w=WDF*$!
3G%QIG$))pC7F-7$$\"3EL3En$4Ua#F*$!3E7F6!*\\%[;\"F-7$$\"3z;/RE&GUa#F*$!
3\"**R=rcC@6\"F-7$$\"3\"**\\K]4]Ua#F*$!3+m9)ffsF0\"F-7$$\"36+vB_<FWDF*
$!3+!RSPvb<$**!#@7$$\"3C+v'Hi#HWDF*$!376OK\"G:uN*Fio7$$\"3g\"z*ev:JWDF
*$!3tPA!)o#[e$))Fio7$$\"3LL347TLWDF*$!3WwBy87k:#)Fio7$$\"3BLLY.KNWDF*$
!39_W>qIB!p(Fio7$$\"3BD\"o7TvVa#F*$!3ShGN?`/zqFio7$$\"3^L$Q*o]RWDF*$!3
B*GW'y(G!QlFio7$$\"31D\"=lj;Wa#F*$!3o%=\\S6NW%fFio7$$\"3'\\PaR<PWa#F*$
!3OH<1^6=z`Fio7$$\"3Q$e9EgeWa#F*$!3oCGz@]Q*y%Fio7$$\"3deR\"3GyWa#F*$!3
3@j`SkvZUFio7$$\"3!oT5k]*\\WDF*$!3mD12sV_jOFio7$$\"3o\"zRQb@Xa#F*$!3K(
o7Fudm0$Fio7$$\"3'\\(=>Y2aWDF*$!3SVKR(=n$GDFio7$$\"3w;zXu9cWDF*$!3WIF#
p-(yd>Fio7$$\"3@+]y))GeWDF*$!3A(pgZ#*3$o8Fio7$$\"3(*\\i_QQgWDF*$!3M2f(
paog\"z!#A7$$\"3%\\7y%3TiWDF*$!36?%GjWfgL#Fjt7$$\"3**\\P![hYYa#F*$\"3w
'[wAhF(fQFjt7$$\"3^L$Qx$omWDF*$\"3V<?v-?/F%*Fjt7$$\"3A+v.I%)oWDF*$\"3#
)\\?%e\"o9P:Fio7$$\"3s\"zpe*zqWDF*$\"3W^%3uu6e2#Fio7$$\"3!**\\#\\'QHZa
#F*$\"3#\\6CaQFZm#Fio7$$\"3Ze9S8&\\Za#F*$\"3hO++7J')=KFio7$$\"3)\\i?=b
qZa#F*$\"3#plkr02\")z$Fio7$$\"3SL3s?6zWDF*$\"37mb%o=OWO%Fio7$$\"3:DJXa
E\"[a#F*$\"3am6PGLMd\\Fio7$$\"3rmm*RRL[a#F*$\"38PsYsfRGbFio7$$\"3y;a<.
Y&[a#F*$\"3tcnbkDR7hFio7$$\"3Xe9tOc([a#F*$\"3_N'3y8m:p'Fio7$$\"3'***\\
Qk\\*[a#F*$\"3+5H+6xxBsFio7$$\"3T$3dg6<\\a#F*$\"3(H'*)*zVlP$yFio7$$\"3
xmmxGp$\\a#F*$\"3kF+x!od$z$)Fio7$$\"3$\\7oK0e\\a#F*$\"3/kLhM_3h*)Fio7$
$\"3*[(=5s#y\\a#F*$\"3M8'3Lu%)y^*Fio7$$\"30++++++XDF*$\"3=\"[6b\\D;,\"
F--%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!Fb[lFa[l-%+AXESLABELSG6$Q\"x6\"Q!F
g[l-%%VIEWG6$;$\"%WD!\"&$\"%XDF_\\l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 
1.000000 45.000000 46.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "The interest rate
 is the root of the function.  The root turns out to be 2.554452% per \+
month." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" 
{TEXT -1 0 "" }{TEXT 279 17 " Newton's Method:" }}{PARA 0 "" 0 "" 
{TEXT -1 62 "Assume the intitial value of the zero (interest rate) is \+
.02. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "diff(f(x),x);\n" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&\"#[\"\"\"),&%\"xGF&F&F&\"#gF&F&*(
\"%!)GF&F)F&)F(\"#fF&F&*&F*F&F-F&!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 56 "fprime := x -> 48*(1+x)^60+2880*x*(1+x)^59-60*(1+x)^5
9;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'fprimeGf*6#%\"xG6\"6$%)oper
atorG%&arrowGF(,(*&\"#[\"\"\"),&9$F/F/F/\"#gF/F/*(\"%!)GF/F2F/)F1\"#fF
/F/*&F3F/F6F/!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "x:= n -> x(n-1)- f(x(n-
1))/fprime(x(n-1));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xGf*6#%\"
nG6\"6$%)operatorG%&arrowGF(,&-F$6#,&9$\"\"\"F1!\"\"F1*&-%\"fG6#F-F1-%
'fprimeGF6F2F2F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "x(0
):= .02;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"xG6#\"\"!$\"\"#!\"#
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 31 "L:= k -> [k, evalf(x(k),8)];  \n" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%\"LGf*6#%\"kG6\"6$%)operatorG%&arrowGF(7$9$-%&
evalfG6$-%\"xG6#F-\"\")F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 44 "for k from 1 by 1 to 10 do print (L(k)) od;\n" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#7$\"\"\"$\")>KbF!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#7$\"\"#$\")(=cc#!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$\"\"$$\")
*f[a#!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$\"\"%$\")EjWD!\"*" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#7$\"\"&$\")JjWD!\"*" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#7$\"\"'$\")KjWD!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#7$\"\"($\")LjWD!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$\"\")$\")Kj
WD!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$\"\"*$\")LjWD!\"*" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#7$\"#5$\")KjWD!\"*" }}}{PARA 0 "" 0 "
" {TEXT -1 29 "The monthly rate is 2.534463%" }}}}{PARA 0 "" 0 "" 
{TEXT -1 78 "_________________________________________________________
_____________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 
0 "" {TEXT 328 10 "ASSIGNMENT" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 1 " \+
" }{TEXT 256 10 "Problem 1:" }}{PARA 0 "" 0 "" {TEXT -1 90 "    Jeremy
 decides to deposit $2,000 in his IRA (Individual Retirement Account).
  At what " }}{PARA 0 "" 0 "" {TEXT -1 98 "    minimum interest rate s
hould he deposit $2,000 per year for 30 years so that his accumulation
 " }}{PARA 0 "" 0 "" {TEXT -1 68 "    reaches a minimum of $300,000 af
ter 30 years?  Do the problem by" }}{PARA 0 "" 0 "" {TEXT -1 48 "    (
a) Graphing a function and finding its zero" }}{PARA 0 "" 0 "" {TEXT 
-1 25 "    (b) Newton's method. " }}{PARA 0 "" 0 "" {TEXT -1 65 "     \+
 The formula to be used in this case is (the future value): " }{TEXT 
280 3 "F =" }{TEXT -1 1 " " }{XPPEDIT 18 0 "R*((1+i)^(n+1)-1)/i;" "6#*
(%\"RG\"\"\",&),&F%F%%\"iGF%,&%\"nGF%F%F%F%F%!\"\"F%F)F," }}{PARA 0 "
" 0 "" {TEXT -1 11 "     where " }{TEXT 281 83 "n = number of years, i
 = interest rate, R = annual payment, F = total value after n" }}
{PARA 0 "" 0 "" {TEXT 322 12 "      years." }}{PARA 0 "" 0 "" {TEXT 
-1 90 "     Decide the initial value from the first approximation of t
he zero given by the graph." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 
0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 329 10 "Problem 2:" }}{PARA 0 "
" 0 "" {TEXT -1 115 "     a) Define an appropriate function to find th
e square root of a given positive number, a, by          graphing." }}
{PARA 0 "" 0 "" {TEXT -1 26 "     b) Find the value of " }{XPPEDIT 18 
0 "sqrt(10003);" "6#-%%sqrtG6#\"&.+\"" }{TEXT -1 87 "  correct to four
 decimal places using the graph of the function          defined in a)
" }}{PARA 0 "" 0 "" {TEXT -1 27 "     c) Find the value of  " }
{XPPEDIT 18 0 "sqrt(10003)" "6#-%%sqrtG6#\"&.+\"" }{TEXT -1 65 " corre
ct to four decimal places using the function defined in a) " }}{PARA 
0 "" 0 "" {TEXT -1 36 "         and Newton's method ( with " }
{XPPEDIT 18 0 "x[0];" "6#&%\"xG6#\"\"!" }{TEXT -1 31 "  =   100).     \+
               " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "
" 0 "" {TEXT -1 0 "" }{TEXT 330 10 "Problem 3:" }}{PARA 0 "" 0 "" 
{TEXT -1 54 "  Find the smallest positive root of the equation  sin" }
{TEXT 323 2 " x" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "2*x/3;" "6#*(\"\"#
\"\"\"%\"xGF%\"\"$!\"\"" }{TEXT -1 7 "   by: " }}{PARA 0 "" 0 "" 
{TEXT -1 19 "      a)  graphing." }}{PARA 0 "" 0 "" {TEXT -1 8 "      \+
b)" }{TEXT 331 2 "  " }{TEXT -1 41 "Newton's method using the initial \+
value  " }{XPPEDIT 18 0 "x[0];" "6#&%\"xG6#\"\"!" }{TEXT -1 8 " = 1.2.
 " }}{PARA 0 "" 0 "" {TEXT -1 50 "      c)  Newton's method using the \+
initial value " }{XPPEDIT 18 0 "x[0];" "6#&%\"xG6#\"\"!" }{TEXT -1 9 "
 = 0.903." }}}{PARA 3 "" 0 "" {TEXT 265 77 "__________________________
___________________________________________________" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 "MSIP Grant #P120A80089-
98:  \"Three Urban Calculus Reform Programs: Adopting the Best\" 1998-
2001    " }}{PARA 259 "" 0 "" {TEXT 261 1 " " }}{PARA 260 "" 0 "" 
{TEXT 262 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 1 " " }}}{MARK "10 4 10 0" 1 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 
1 }{PAGENUMBERS 0 1 2 33 1 1 }