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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "Lab 13 Maple Activities fo
r October 24:  (Lesson 10 - The Chain Rule)" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 257 8 "Examples" }{TEXT 258 0 "
" }{TEXT -1 95 "  Use Maple to differentiate the following functions f
rom the lecture notes for the Chain Rule." }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "1.  " }{TEXT 259 1 "y" }
{TEXT -1 4 "  = " }{XPPEDIT 18 0 "x^(2/3);" "6#*$)%\"xG#\"\"#\"\"$\"\"
\"" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 14 "    Using the " }{TEXT 260 4 "diff" }{TEXT -1 9 "
 command:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "diff(x^(2/3), \+
x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 4 "2.  " }{TEXT 261 1 "f" }{TEXT -1 1 "(" }{TEXT 262 1 "x" }{TEXT 
-1 4 ") = " }{XPPEDIT 18 0 "1/(sqrt(x));" "6#*&\"\"\"F$*$-%%sqrtG6#%\"
xGF$!\"\"" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 14 "    Using the " }{TEXT 263 1 "D" }{TEXT 
-1 9 " command:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "f := x -
> 1/sqrt(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "D(f)(x);" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "3.
  " }{TEXT 264 1 "f" }{TEXT -1 1 "(" }{TEXT 265 1 "x" }{TEXT -1 4 ") =
 " }{XPPEDIT 18 0 "(4*x^3+2*x^2-x-3)^3;" "6#*$),**$)%\"xG\"\"$\"\"\"\"
\"%*$)F(\"\"#F*F.F(!\"\"!\"$\"\"\"F)F*" }{TEXT -1 1 " " }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "    Using
 the " }{TEXT 266 1 "D" }{TEXT -1 9 " command:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 41 "f := x -> (4*x^3  +  2*x^2  -  x  - 3)^3;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "D(f)(x); expand(%);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "4. \+
 " }{TEXT 269 1 "g" }{TEXT -1 1 "(" }{TEXT 270 1 "x" }{TEXT -1 6 ")  =
  " }{XPPEDIT 18 0 "(3*x^2-2*x)^(3/5);" "6#*$),&*$)%\"xG\"\"#\"\"\"\"
\"$F(!\"##F+\"\"&F*" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "    Using the " }{TEXT 268 4 "d
iff" }{TEXT -1 9 " command:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
28 "diff((3*x^2 -2*x)^(3/5), x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "5.  " }{TEXT 271 1 "s" }{TEXT -1 1 
"(" }{TEXT 272 1 "t" }{TEXT -1 11 ")  =   5 / " }{XPPEDIT 18 0 "(4*t^5
-3*t^3+t)^4;" "6#*$),(*$)%\"tG\"\"&\"\"\"\"\"%*$)F(\"\"$F*!\"$F(\"\"\"
F+F*" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 14 "    Using the " }{TEXT 273 1 "D" }{TEXT -1 9 " \+
command:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "s := t -> 5/(4*
t^5 - 3*t^3 + t)^4;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "D(s)(
t);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 3 "6. " }{TEXT 275 2 " y" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "(7-6*x)
^3*(8*x^2+9)^5;" "6#*&),&\"\"(\"\"\"%\"xG!\"'\"\"$\"\"\"),&*$)F(\"\"#F
+\"\")\"\"*F'\"\"&F+" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "    Using the " }{TEXT 274 4 "d
iff" }{TEXT -1 9 " command:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
49 "diff( (7 - 6*x)^3 * (8*x^2 + 9)^5, x); factor(%);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 
276 0 "" }{TEXT 277 7 "Example" }{TEXT 278 0 "" }{TEXT -1 21 "  Plot t
he graph of  " }{TEXT 279 1 "y" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "(4*x
^2-8*x+3)^4;" "6#*$),(*$)%\"xG\"\"#\"\"\"\"\"%F(!\")\"\"$\"\"\"F+F*" }
{TEXT -1 45 "  and its tangent line at the point (2, 81). " }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "y :
= x -> (4*x^2 - 8*x + 3)^4;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
43 "plot([y(x), D(y)(2)*(x-2) + y(2)], x=0..4);" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 280 5 "NOTE:
" }{TEXT -1 99 "  This graph does not show the graph of the function n
or the tangent line very well.  Restrict the " }{TEXT 281 1 "y" }
{TEXT -1 8 " values." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 53 "plot([y(x), D(y)(2)*(x-2) + y(2)], x=0..4
, y=0..100);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }{TEXT 283 5 "NOTE:" }{TEXT -1 140 "  Well, at least w
e can see the graph of the function  better.  However, the tangent lin
e does not look very good.  Let's try the following." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "plot([y(x
), D(y)(2)*(x-2) + y(2)], x=0..4, y=0..900);" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 284 5 "NOTE:" }
{TEXT -1 51 "  This shows a better tangent line to the graph of " }
{TEXT 285 1 "y" }{TEXT -1 78 " at the point (2, 81).  Find the equatio
n of the tangent line was using Maple:" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "y = D(y)(2)*(x-2) + y(2
);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }{TEXT 286 0 "" }{TEXT 287 7 "Example" }{TEXT 288 0 "" }{TEXT 
-1 37 "  Find the point(s) on the graph of  " }{TEXT 256 1 "y" }{TEXT 
-1 3 " = " }{XPPEDIT 18 0 "(4*x^2-8*x+3)^4;" "6#*$,(*&\"\"%\"\"\"*$%\"
xG\"\"#F'F'*&\"\")F'F)F'!\"\"\"\"$F'\"\"%" }{TEXT -1 103 "  at which t
he tangent line is horizontal.  Then graph the function and its horizo
ntal tangent line(s)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "y := x -> (4*x^2 - 8*x + 3)^4;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "D(y)(x);" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 21 "solve( D(y)(x) = 0 );" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 290 5 "NOTE:
" }{TEXT -1 215 "  The solutions of 1/2 and 3/2 are solutions of multi
plicity three.  In other words, the factor which produced these soluti
ons was raised to an exponent of 3.  The solutions of 1/2 and 3/2 come
s from the factor of  " }{XPPEDIT 18 0 "(4*x^2-8*x+3)^3;" "6#*$),(*$)%
\"xG\"\"#\"\"\"\"\"%F(!\")\"\"$\"\"\"F-F*" }{TEXT -1 1 "." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "Now, find the " }
{TEXT 301 1 "y" }{TEXT -1 37 "-coordinate of the points which have " }
{TEXT 302 1 "x" }{TEXT -1 33 "-coordinates of 1/2, 1, and 3/2: " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "y(1/2); y(1); y(3/2);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "Th
us, the points on the graph of " }{TEXT 291 1 "y" }{TEXT -1 135 " wher
e the tangent line is horizontal are: (1/2, 0) , (1, 1), and (3/2, 0).
  The equation of the tangent line at the point (1/2, 0) is " }{TEXT 
293 1 "y" }{TEXT -1 3 " = " }{TEXT 294 1 "y" }{TEXT -1 16 "(1/2), whic
h is " }{TEXT 295 1 "y" }{TEXT -1 65 " = 0.  The equation of the tange
nt line at the point (3/2, 0) is " }{TEXT 296 1 "y" }{TEXT -1 3 " = " 
}{TEXT 297 1 "y" }{TEXT -1 21 "(3/2), which is also " }{TEXT 292 1 "y
" }{TEXT -1 63 " = 0.  The equation of the tangent line at the point (
1, 1) is " }{TEXT 298 1 "y" }{TEXT -1 3 " = " }{TEXT 299 1 "y" }{TEXT 
-1 14 "(1), which is " }{TEXT 300 1 "y" }{TEXT -1 92 " = 1.  We can do
 the following to graph the function and its three horizontal tangent \+
lines:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "plot([y(x), y(1/2
), y(1), y(3/2)], x=0..5, y=-10..20);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 303 5 "NOTE:" }
{TEXT -1 35 "  In order to see the tangent line " }{TEXT 304 1 "y" }
{TEXT -1 19 " = 0, which is the " }{TEXT 305 1 "x" }{TEXT -1 71 "-axis
, click around the graph in order to make it active and go to the " }
{TEXT 306 1 "A" }{TEXT 309 3 "xes" }{TEXT -1 31 " menu and choose the \+
option of " }{TEXT 308 1 "N" }{TEXT 310 1 "o" }{TEXT 311 2 "ne" }
{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 50 "In order to obtain a better graph of the function " }
{TEXT 312 1 "y" }{TEXT -1 67 " and the three horizontal tangent lines,
 we could do the following:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
53 "plot(y(x), x=0..5, y=-10..15, axes=none); funct := %:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "plot(y(1/2), x=0.1..0.9, axes=none,
 color=green); tanline1 := %:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 62 "plot(y(1), x=0.6..1.4, axes=none, color=black); tanline2 := %:" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "plot(y(3/2), x=1..2, axes
=none, color=blue); tanline3 := %:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 45 "display(funct, tanline1, tanline2, tanline3);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "1 0 0" 8 }{VIEWOPTS 1 1 0 1 
1 1803 }