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{SECT 0 {PARA 256 "" 0 "" {TEXT -1 27 "Tangent Line to a Function " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 18 "Calculus \+
I Project" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 5 "" 0 "" 
{TEXT -1 10 "Objective:" }}{PARA 0 "" 0 "" {TEXT -1 79 "To understand \+
the connection between the derivative of a function at a point,  " }
{TEXT 259 1 "f" }{TEXT 279 4 " ' (" }{TEXT 321 1 "a" }{TEXT 322 1 ")" 
}{TEXT -1 55 ",  and the tangent line to the function at the point  (
" }{TEXT 260 4 "a, f" }{TEXT 287 1 "(" }{TEXT 288 1 "a" }{TEXT 289 1 "
)" }{TEXT -1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 64 "Maple will help wit
h graphing the function and the tangent line." }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 5 "" 0 "
" {TEXT -1 15 "Solved Example:" }}{PARA 0 "" 0 "" {TEXT -1 10 " Consid
er " }{TEXT 261 2 " f" }{TEXT 290 1 "(" }{TEXT 291 1 "x" }{TEXT 292 1 
")" }{TEXT -1 4 "  = " }{XPPEDIT 18 0 "sqrt(x)/(1+x);" "6#*&-%%sqrtG6#
%\"xG\"\"\",&F(F(F'F(!\"\"" }{TEXT -1 4 "   ." }}{PARA 0 "" 0 "" 
{TEXT 271 1 "A" }{TEXT -1 44 ") Find an equation  of  the tangent line
 to " }{TEXT 280 1 "f" }{TEXT -1 1 "(" }{TEXT 281 1 "x" }{TEXT -1 23 "
)  at the point where  " }{TEXT 282 1 "x" }{TEXT -1 5 " = 4." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 272 1 "B" }{TEXT -1 
83 ") Use Maple to plot the graphs of the function and the tangent lin
e in one window. " }}{PARA 0 "" 0 "" {TEXT -1 4 "    " }}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 9 "Solution:" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 
273 1 "A" }{TEXT -1 50 ") The point the tangent line passes through  i
s  (" }{TEXT 262 1 "4" }{TEXT 295 3 ", f" }{TEXT 293 3 "(4)" }{TEXT 
-1 10 "). Since  " }{TEXT 285 1 "f" }{TEXT 263 1 "(" }{TEXT 286 1 "4" 
}{TEXT 283 2 ") " }{TEXT 284 1 "=" }{TEXT 296 1 "0" }{TEXT 297 1 " " }
{TEXT 294 2 ".4" }{TEXT -1 36 ", the coordinates of the point are  " }
{TEXT 264 7 "(4, 0.4" }{TEXT -1 32 ").  We can see this using Maple." 
}}{PARA 0 "" 0 "" {TEXT -1 23 "We define the function." }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "f:=x->sqrt(x)/(x+1);" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&-%%sqrtG6#9$
\"\"\",&F0F1F1F1!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
5 "f(4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"#\"\"&" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"+++++S!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 45 "The slope of the tangent line at the point  (" 
}{TEXT 265 3 "4, " }{TEXT 298 1 "f" }{TEXT 299 3 "(4)" }{TEXT -1 15 ")
 is equal to  " }{TEXT 266 4 "f ' " }{TEXT 300 2 "(4" }{TEXT -1 3 "). \+
" }}{PARA 0 "" 0 "" {TEXT -1 8 "Then    " }{TEXT 267 4 "f ' " }{TEXT 
301 1 "(" }{TEXT 302 1 "x" }{TEXT 303 1 ")" }{TEXT -1 3 " = " }
{XPPEDIT 18 0 "(1-x)/(2*sqrt(x)(1+x)^2);" "6#*&,&\"\"\"F%%\"xG!\"\"F%*
&\"\"#F%*$--%%sqrtG6#F&6#,&F%F%F&F%F)F%F'" }{TEXT -1 15 "  , so that  \+
  " }{TEXT 268 4 "f ' " }{TEXT 304 3 "(4)" }{TEXT 305 3 " = " }{TEXT 
306 10 "-0.03.    " }}{PARA 0 "" 0 "" {TEXT -1 74 "Let us do this usin
g Maple.  We find the derivative at x = 4 using Maple. " }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "fprimex:= diff(f(x),x);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%(fprimexG,&*&\"\"\"F'*(\"\"#F'%\"xG#F'F),&
F*F'F'F'F'!\"\"F'*&F*#F'F)F,!\"#F-" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 28 "fprime4:= subs(x=4,fprimex);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%(fprime4G,$*(\"\"$\"\"\"\"$+#!\"\"\"\"%#F(\"\"#F*" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#$!+++++I!#6" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 51 "So the equation of the tangent line in qu
estion is " }{TEXT 269 7 "   y - " }{TEXT 307 3 "0.4" }{TEXT 308 3 " =
 " }{TEXT 309 6 "-0.03(" }{TEXT 310 4 "x - " }{TEXT 311 2 "4)" }{TEXT 
-1 12 ",   or      " }{TEXT 270 4 "y = " }{TEXT 312 5 "-0.03" }{TEXT 
313 4 "x + " }{TEXT 314 6 "0.52 ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 274 1 "B" }{TEXT -1 28 ") Now, draw the tangent \+
line" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "y:=x->-0.03*x+0.52;" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%\"yGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&*&$\"\"$!\"#\"\"\"9$F1!\"
\"$\"#_F0F1F(F(F(" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"yGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&9$$!\"$!\"
#$\"#_F0\"\"\"F(F(F(" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "plot
(\{f(x),y(x)\},x=-1..10);" }}{PARA 13 "" 1 "" {GLPLOT2D 295 231 231 
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rve 1" "Curve 2" }}}}{PARA 13 "" 1 "" {TEXT -1 0 "" }}{PARA 13 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 101 "Remark:  Notice that a tangent to a curve may intersect \+
the curve at some other point of the curve.  " }}}{PARA 3 "" 0 "" 
{TEXT -1 59 "_________________________________________________________
__" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 10 "A
SSIGNMENT" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
18 "Apply questions  (" }{TEXT 256 1 "A" }{TEXT -1 9 ")  and  (" }
{TEXT 257 1 "B" }{TEXT -1 64 ") of  the solved example to the curves a
t the indicated points. " }{TEXT 277 33 "When plotting, select the int
erva" }{TEXT 258 7 "l of  x" }{TEXT 315 36 " appropriately.  Label, by
 hand, the" }{TEXT 276 1 " " }{TEXT 278 54 "curves, the tangent lines,
 and the points of tangency." }{TEXT 275 1 " " }}{SECT 0 {PARA 4 "" 0 
"" {TEXT -1 11 "Problem 1. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 1 " " }{TEXT 317 1 "y" }{TEXT -1 5 
"  =  " }{XPPEDIT 18 0 "x/(1+x^2);" "6#*&%\"xG\"\"\",&F%F%*$F$\"\"#F%!
\"\"" }{TEXT -1 14 "         at   " }{TEXT 316 1 "x" }{TEXT -1 4 " = 3
" }}{PARA 0 "" 0 "" {TEXT -1 6 "      " }}}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 
10 "Problem 2." }}{PARA 0 "" 0 "" {TEXT -1 5 "     " }{TEXT 318 1 "y" 
}{TEXT -1 3 " = " }{XPPEDIT 18 0 "x^2*(sqrt(2)-sqrt(x));" "6#*&%\"xG\"
\"#,&-%%sqrtG6#F%\"\"\"-F(6#F$!\"\"F*" }{TEXT -1 19 "      at      (i)
  " }{TEXT 319 1 "x" }{TEXT -1 24 " = 2         (ii)       " }{TEXT 
320 2 "x " }{TEXT -1 5 "= 4  " }}}{PARA 0 "" 0 "" {TEXT -1 73 "_______
__________________________________________________________________" }}
{PARA 0 "" 0 "" {TEXT -1 178 "MSIP Grant #P120A80089-98:  \"Three Urba
n Calculus Reform Programs: Adopting the Best,\" 1998-2001; MSEIP Gran
t #P120AA010031:  \"Four Colleges: Calculus + Enhancements\", 2001-200
4 " }}}{MARK "10 0" 59 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
{PAGENUMBERS 0 1 2 33 1 1 }