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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 256 13 "Derivati
ves I" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" 
{TEXT -1 0 "" }{TEXT 257 0 "" }{TEXT -1 80 "Maple can be used to find \+
the derivative of any function which is differentiable" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }{TEXT 258 156 "There are two different ways to rep
resent a function in Maple; as an expression or as a function.  Each r
equires a different command to find the derivative." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 0 "" }{TEXT 259 26 
"Derivatives of expressions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 
260 34 "The following commands will input " }{XPPEDIT 291 0 "f(x) = x^
2-3*x+5;" "6#/-%\"fG6#%\"xG,(*$F'\"\"#\"\"\"*&\"\"$F+F'F+!\"\"\"\"&F+
" }{TEXT -1 1 " " }{TEXT 261 93 "as an expression, find its derivative
, and evaluate the function and the derivative at x = 2." }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "f:=x^2-3*x+5;" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 14 "fx:=diff(f,x);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "subs(x=2,f);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 13 "subs(x=2,fx);" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 
{PARA 4 "" 0 "" {TEXT 262 24 "Derivatives of functions" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }{TEXT 263 34 "The following commands will input " 
}{XPPEDIT 289 0 "f(x) = x^2-3*x+5;" "6#/-%\"fG6#%\"xG,(*$F'\"\"#\"\"\"
*&\"\"$F+F'F+!\"\"\"\"&F+" }{TEXT 290 1 " " }{TEXT 264 90 "as a functi
on, find its derivative, and evaluate the function and the derivative \+
at x = 2." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "f:=x->x^2-3*x+5
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "fx:=D(f);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 5 "f(2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "fx(2);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 265 73 "This sequence of c
ommands takes the derivative and makes it a function.  " }}}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restar
t;" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 0 "" }{TEXT 266 9 "Example 1" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 267 19 "Given the function " }
{XPPEDIT 282 0 "f(x) = -x^2+4*x-1;" "6#/-%\"fG6#%\"xG,(*$F'\"\"#!\"\"*
&\"\"%\"\"\"F'F.F.F.F+" }{TEXT 283 1 "," }{TEXT -1 1 " " }{TEXT 268 
146 "find its derivative, find the equation of the tangent line to the
 curve at x = 3, and plot the curve and the tangent line on the same s
et of axes." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 17 "f:=x->-x^2+4*x-1;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 9 "fx:=D(f);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
11 "f(3);fx(3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "eq:=y-f(
3)=fx(3)*(x-3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "tanline:
=solve(eq,y);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "plot(\{f(x
),tanline\},x=-5..5,y=-5..5,thickness=2);" }}}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{SECT 
0 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }{TEXT 269 28 "Find the rate at which both " }{XPPEDIT 284 0 "f(x) \+
= cos(x);" "6#/-%\"fG6#%\"xG-%$cosG6#F'" }{TEXT 285 5 " and " }
{XPPEDIT 286 0 "g(x) = tan(x);" "6#/-%\"gG6#%\"xG-%$tanG6#F'" }{TEXT 
-1 1 " " }{TEXT 270 68 "are changing at their point of intersection in
 the interval 0 < x < " }{XPPEDIT 18 0 "pi;" "6#%#piG" }{TEXT -1 3 ". \+
 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 271 80 "Then plot f and its \+
derivative and g and its derivative in the interval 0 < x < " }
{XPPEDIT 18 0 "pi;" "6#%#piG" }{TEXT -1 1 " " }{TEXT 272 19 "on separa
te graphs." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 273 93 "Use the gra
ph of the derivative to confirm the answer for the rate of change of t
he function." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "f:=x->cos(x)
;g:=x->tan(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "s:=fsolve
(f(x)=g(x),x=0..Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "fx:
=D(f);gx:=D(g);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "evalf(fx
(s));evalf(gx(s));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "plot(
\{f(x),fx(x)\},x=0..Pi,thickness=2);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 47 "plot(\{g(x),gx(x)\},x=0..Pi,y=-5..5,thickness=2);" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{SECT 
0 {PARA 4 "" 0 "" {TEXT -1 0 "" }{TEXT 274 9 "Example 3" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }{TEXT 275 57 "Find the equations of the tangent li
nes x^2 to the curve " }{XPPEDIT 18 0 "y = (x-1)/(x^2-4);" "6#/%\"yG*&
,&%\"xG\"\"\"F(!\"\"F(,&*$F'\"\"#F(\"\"%F)F)" }{TEXT 287 1 " " }{TEXT 
276 102 "which have a slope of -1.  Then plot the tangent lines along \+
with the function to observe the results." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "f:=x->(x-1)/(x^2-4);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "plot(f(x),x=-5..5,y=-5.
.5,discont=true,thickness=2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 9 "fx:=D(f);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "s1:=fsolv
e(-1=fx(x),x=-4..-2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "s2
:=fsolve(-1=fx(x),x=-2..0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
28 "s3:=fsolve(-1=fx(x),x=0..2);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 28 "s4:=fsolve(-1=fx(x),x=2..4);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 36 "eq1:=solve(y-f(s1)=fx(s1)*(x-s1),y);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "eq2:=solve(y-f(s2)=fx(s2)*(x-s2),y)
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "eq3:=solve(y-f(s3)=fx(
s3)*(x-s3),y);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "eq4:=solv
e(y-f(s4)=fx(s4)*(x-s4),y);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
70 "plot(\{f(x),eq1,eq2,eq3,eq4\},x=-5..5,y=-5..5,thickness=2,discont=
true);" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 4 "" 0 "" {TEXT 293 36 "Deri
vatives of functions implicitly;" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
{TEXT 294 34 "The following commands will input " }{TEXT 296 13 "x^2 +
 y^3 = 1" }{TEXT -1 1 " " }{TEXT 295 102 "as an expression, find its d
erivative, and evaluate the function and the derivative at (x,y) = (1,
-2)." }{TEXT -1 1 "\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }
{MPLTEXT 1 0 15 "f := x^2+y^3=1;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "implicitdiff(f,y,x);" }}}{SECT 0 {PARA 4 "" 0 "" 
{TEXT -1 0 "" }{TEXT 277 10 "Exercise 1" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }{TEXT 278 56 "Find the equations of the tangent lines to the funct
ion " }{XPPEDIT 288 0 "f(x) = (x-2)/(x^2-1);" "6#/-%\"fG6#%\"xG*&,&F'
\"\"\"\"\"#!\"\"F*,&*$F'F+F*F*F,F," }{TEXT -1 1 " " }{TEXT 279 95 "whi
ch have a slope of 1.  Then plot the function and the tangent lines on
 the same set of axes." }{TEXT -1 2 "  " }{TEXT 280 49 "Do not show th
e vertical asymptotes in the graph." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 0 "" }{TEXT 281 10 "Exercise 2" }
{TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 292 66 "For each function
 below, Use Maple to differentiate the following\n" }{TEXT 308 1 "1" }
{TEXT 306 1 "." }{TEXT 39 4 "    " }{XPPEDIT 309 1 "y = -2*x^3+5;" "6#
/%\"yG,&*&\"\"#\"\"\"*$%\"xG\"\"$F(!\"\"\"\"&F(" }{XPPEDIT 310 1 "x+sq
rt(x);" "6#,&%\"xG\"\"\"-%%sqrtG6#F$F%" }}{PARA 258 "" 0 "" {TEXT 299 
5 "2.   " }{XPPEDIT 19 1 "y = 5*x/(5*x^2+10*x);" "6#/%\"yG*(\"\"&\"\"
\"%\"xGF',&*&F&F'*$F(\"\"#F'F'*&\"#5F'F(F'F'!\"\"" }}{PARA 258 "" 0 "
" {TEXT 300 5 "3.   " }{XPPEDIT 19 1 "\011y = sin(x^4" "6#/%\"yG-%$sin
G6#*$%\"xG\"\"%" }}{PARA 259 "" 0 "" {TEXT -1 0 "" }{TEXT 301 5 "4.   \+
" }{XPPEDIT 19 1 "\011y = 10tan x - sin(3x" "6#/%\"yG,&*(\"#5\"\"\"%$t
anGF(%\"xGF(F(-%$sinG6#*&\"\"$F(F*F(!\"\"" }}{PARA 259 "" 0 "" {TEXT 
302 5 "5.   " }{XPPEDIT 19 1 "y=sec(x)-4sin(x^2)" "6#/%\"yG,&-%$secG6#
%\"xG\"\"\"*&\"\"%F*-%$sinG6#*$F)\"\"#F*!\"\"" }}{PARA 258 "" 0 "" 
{TEXT 297 5 "6.   " }{XPPEDIT 19 1 "y=(cos x)^4" "6#/%\"yG*$*&%$cosG\"
\"\"%\"xGF(\"\"%" }{TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 0 "" }
{TEXT 303 5 "7.   " }{XPPEDIT 19 1 "y = x^(sin*x)" "6#/%\"yG)%\"xG*&%$
sinG\"\"\"F&F)" }}{PARA 259 "" 0 "" {TEXT 298 5 "8.   " }{XPPEDIT 19 
1 "y=x cos x" "6#/%\"yG*(%\"xG\"\"\"%$cosGF'F&F'" }}{PARA 259 "" 0 "" 
{TEXT -1 0 "" }{TEXT 304 5 "9.   " }{XPPEDIT 19 1 "y=arctan(sin x)" "6
#/%\"yG-%'arctanG6#*&%$sinG\"\"\"%\"xGF*" }}{PARA 0 "" 0 "" {TEXT 307 
0 "" }{TEXT 305 2 "10" }{TEXT 312 2 ". " }{XPPEDIT 311 1 "y= (5x-3x^3+
 ln(x))^10" "6#/%\"yG*$,(*&\"\"&\"\"\"%\"xGF)F)*&\"\"$F)*$F*F,F)!\"\"-
%#lnG6#F*F)\"#5" }}}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 0 "" }{TEXT 
313 10 "Exercise 3" }}{PARA 260 "" 0 "" {TEXT 314 39 "Differentiate th
e following  implicitly" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT 316 
1 "1" }{TEXT 315 1 "." }{TEXT 39 3 "   " }{TEXT -1 1 " " }{XPPEDIT 19 
1 "y ^5 + x^2*y^3= 1-y*exp(x^2)" "6#/,&*$%\"yG\"\"&\"\"\"*&%\"xG\"\"#F
&\"\"$F(,&F(F(*&F&F(-%$expG6#*$F*F+F(!\"\"" }{TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT 322 2 "2." }{TEXT -1 2 "  " }{XPPEDIT 19 1 "x* sin(y) + c
os(2*y) = cos(y) " "6#/,&*&%\"xG\"\"\"-%$sinG6#%\"yGF'F'-%$cosG6#*&\"
\"#F'F+F'F'-F-6#F+" }}{PARA 0 "" 0 "" {TEXT 323 4 "3.  " }{XPPEDIT 19 
1 "x*y =tan(x*y) + exp(y) " "6#/*&%\"xG\"\"\"%\"yGF&,&-%$tanG6#*&F%F&F
'F&F&-%$expG6#F'F&" }{TEXT 325 1 " " }}}}{MARK "2 4 0" 4 }{VIEWOPTS 1 
1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }