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-1 9 "         " }{TEXT 256 41 "      Limits, continuity, and derivati
ves" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "sm
121_deriv1.mws" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 161 "NOTE on the sy
ntax: As you'll see almost all commands must end in either a \":\" (to
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{TEXT 257 9 "EXAMPLE 1" }{TEXT -1 96 ": Let's begin by examining the f
unction sin(x)/x. It's plot (NOT drawn to scale) is as follows. " }}}
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LOURG6&%$RGBG$\"#5!\"\"\"\"!F_[l-%+AXESLABELSG6$Q\"x6\"Q\"yFd[l-%(SCAL
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{PARA 0 "" 0 "" {TEXT -1 57 "A table of values of sin(x)/x is easily c
omputed as well." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 166 "print(
`x`.`                 `.`sin(x)/x`);\nfor i from -10 to 10 do\nif i<>0
 then print(evalf(i/100),evalf(sin(i/100)/(i/100)));\nelse print(`unde
fined at x=0`);\nfi;\nod;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%;x~~~~~~
~~~~~~~~~~~sin(x)/xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$!+++++5!#5$\"
+l;M$)**F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$!+++++!*!#6$\"+ma]')**!
#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$!+++++!)!#6$\"+YnL*)**!#5" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6$$!+++++q!#6$\"+P`$=***!#5" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6$$!+++++g!#6$\"+#3,S***!#5" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$$!+++++]!#6$\"+aQ$e***!#5" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6$$!+++++S!#6$\"+[NL(***!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$!+
++++I!#6$\"+m+])***!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$!+++++?!#6$
\"+XLL****!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$!+++++5!#6$\"+ML$)**
**!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%1undefined~at~x=0G" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6$$\"+++++5!#6$\"+ML$)****!#5" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6$$\"+++++?!#6$\"+XLL****!#5" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$$\"+++++I!#6$\"+m+])***!#5" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$$\"+++++S!#6$\"+[NL(***!#5" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$$\"+++++]!#6$\"+aQ$e***!#5" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$$\"+++++g!#6$\"+#3,S***!#5" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$$\"+++++q!#6$\"+P`$=***!#5" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$$\"+++++!)!#6$\"+YnL*)**!#5" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$$\"+++++!*!#6$\"+ma]')**!#5" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$$\"+++++5!#5$\"+l;M$)**F%" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 74 "We define f(x) (in MAPLE's \"arrow\" notation ->) to be s
in(x)/x as follows." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "f:=x
->sin(x)/x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)op
eratorG%&arrowGF(*&-%$sinG6#9$\"\"\"F0!\"\"F(F(F(" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 35 "Limit(f(x), x=0);\nlimit(f(x), x=0);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$*&-%$sinG6#%\"xG\"\"\"F*!\"
\"/F*\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 127 "To test if sin(x)/x is continuous, we must mak
e MAPLE load some commands in the package of programs called \"iscont
\" as follows." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "readlib(i
scont):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "iscont(f(x), x=-
5..5 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 59 "To compute the derivative type the diff command a
s follows." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "diff(f(x),x);
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ollowing command." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Df := \+
unapply(diff(f(x),x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#DfGR6#%
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$$\"1om\"zW?)\\*)FD$!1L;sClv]5FV7$$\"1M$3_!HWb!*FD$!13z+C`'Q2\"FV7$F]]
m$!1%)=1?&)*[3\"FV7$$\"1,+DJ&f@E*FD$!1XISa+H%3\"FV7$$\"1,+++PDj$*FD$!1
(>YfO(*H2\"FV7$$\"1-+voyMk%*FD$!1xCJDLN^5FV7$Fb]m$!1D^9J:y>5FV7$$\"1,+
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ESLABELSG6$Q\"x6\"%!G-%%VIEWG6$;F<Fg]m%(DEFAULTG" 1 2 0 1 0 2 9 1 4 2 
1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT 265 
32 "EXERCISE for Example 1(optional)" }{TEXT -1 66 ": Do all the same \+
commands but replace sin(x)/x by (1-cos(x))/x^2." }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }{TEXT 259 9 "EXAMPLE 2" }{TEXT -1 247 ": Let's d
o some of the same things as we did above but for the simpler function
 sin(x). Since sin(x) is a predefined symbol in MAPLE, to set f(x)=sin
(x) we don't have to use the arrow notation if we don't want to. The f
ollowing command is shorter." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 7 "f:=sin;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG%$sinG" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Df := unapply(diff(f(x),x),x
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#DfG%$cosG" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 119 "plot1 := plot(\{f(x),Df(x)\},x=-1..5):\npl
ot2 := plots[textplot](\{[.2,1.1,`y=D(sin x)`],[3,0.5,`y=sin x`]\},ali
gn = RIGHT):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "plots[displ
ay](\{plot1,plot2\});" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 
{PLOTDATA 2 "6(-%'CURVESG6$7en7$$!\"\"\"\"!$\"1)R\"oeI-.a!#;7$$!1+++]2
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45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 261 9 "EXA
MPLE 4" }{TEXT -1 106 ": Let's do some of the same things as we did ab
ove but for the following piecewise-defined function f(x). " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "f:=x->piecewise(x<0,x^2,x<Pi
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#>%\"fGR6#%\"xG6\"6$%)operatorG%&arrowGF(-%*piecewiseG6'29$\"\"!*$)F0
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e=Fas-%'COLOURG6&%$RGBG$\"#5F)F*F*-%+AXESLABELSG6$Q\"x6\"%!G-%%VIEWG6$
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}}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Df := unapply(diff(f(x),
x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#DfGR6#%\"xG6\"6$%)operato
rG%&arrowGF(-%*piecewiseG6,29$\"\"!,$F0\"\"#/F0F1%*undefinedG2F0%#PiG-
%$cosG6#F0/F0F7F52F7F0\"\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 37 "Limit(f(x), x=Pi);\nlimit(f(x), x=Pi);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#-%&LimitG6$-%*PIECEWISEG6%7$*$)%\"xG\"\"#\"\"\"2
F,\"\"!7$-%$sinG6#F,2F,%#PiG7$,&F,\"\"\"F6!\"\"%*otherwiseG/F,F6" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 105 "Limit(Df(x), x=Pi, right);\nlimit(Df(x), x=Pi, right
);\nLimit(Df(x), x=Pi, left);\nlimit(Df(x), x=Pi, left);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#-%&LimitG6%-%*PIECEWISEG6'7$,$%\"xG\"\"#2F+\"\"!
7$%*undefinedG/F+F.7$-%$cosG6#F+2F+%#PiG7$F0/F+F77$\"\"\"2F7F+F9%&righ
tG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%&LimitG6%-%*PIECEWISEG6'7$,$%\"xG\"\"#2F+\"\"!7$%*und
efinedG/F+F.7$-%$cosG6#F+2F+%#PiG7$F0/F+F77$\"\"\"2F7F+F9%%leftG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"\"" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 60 "Is this function f(x) continuous on the interval -1 < x <
 5?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "iscont(f(x), x=-1..5
 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 24 "iscont(Df(x), x=-1..1 );" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%&falseG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "
iscont(Df(x), x=1..3 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "iscont(Df(x), x=3..5);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 133 "plot1 := plot(\{f(x),Df(x)\},x=-1..5,thickness=3):\n
plot2 := plots[textplot](\{[2.1,-1.2,`y=D(f(x))`],[1.8,1.3,`y=f(x)`]\}
,align = RIGHT):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "plots[d
isplay](\{plot1,plot2\});" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 
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.$\"1cbrMKQbvF07$$!1,+DJJ?B\")F0$\"1#)\\:6Hk)f'F07$F5$\"19;$yTZmq&F07$
$!1++v=>P9pF0$\"1b)zI!R&3y%F07$F:$\"1J%\\?ZXp$RF07$F?$\"1(zE([qJ'[#F07
$FD$\"1+jQ6y6s8F07$FI$\"1&oh#=/+GjFU7$$!1++]iy:+>F0$\"1r0U-**f5OFU7$FN
$\"1,^\"yc=1l\"FU7$$!1.++DTVl'*FU$\"1d9Z#oh?M*Fen7$FS$\"1.\"z?pCK?%Fen
7$$!1+++D@-,LFU$\"1^R(pqu'*3\"Fen7$FY$\"1H:+crs69!#@7$F\\q$\"1PZ'z;l6A
'FU7$Ffq$\"1n))*3p6OD\"F07$F`r$\"1WHI8()4MDF07$Fer$\"1$H<^V^pi$F07$Fjr
$\"19**HQ&e#*z%F07$F_s$\"1)zmzT&f&*eF07$Fds$\"1$eSZ/e#eoF07$Fis$\"1I4+
-XwRwF07$F^t$\"1F'Rpwj)R%)F07$Fct$\"1'z=Zw1w**)F07$Fht$\"1%\\,c@)f(\\*
F07$F]u$\"1LBefZ!**z*F07$$\"1+]Ppj6N9F3$\"1aHjke43**F07$Fbu$\"1^XVX<#[
(**F07$$\"1++]<9Vh:F3$\"1)Hub\\h&****F07$Fgu$\"1et'3:aj)**F07$$\"1+](o
qHto\"F3$\"15>Nim<K**F07$F\\v$\"1&QVvsmp$)*F07$Fav$\"1f3qGNkc&*F07$Ffv
$\"1*H&**=(e_5*F07$F[w$\"1=(*)[\"yO![)F07$F`w$\"1$>#\\3iI:yF07$Few$\"1
YR,q#R5)pF07$Fjw$\"1'z)H]*og+'F07$F_x$\"1Zi&*GEJc\\F07$Fdx$\"1W%H&Hs*f
'QF07$Fix$\"1>]i\"G*>*e#F07$F^y$\"1P#4!fx.,9F07$Fcy$\"1rc'**4z;d(FU7$F
hy$\"1!*H]+1A,6FU7$Fj[l$\"1!o?&Go]oZFU7$F`\\l$\"1m?g^c#Q1\"F07$Ff\\l$
\"1s?59IEZBF07$Fi\\l$\"1l?g^v([b$F07$F\\]l$\"1q?g,F=<[F07$F_]l$\"1m?59
nJ^gF07$Fb]l$\"1m?g^1MVtF07$Fe]l$\"1t?5kKr(e)F07$Fh]l$\"1p?5*)REg)*F07
$F[^l$\"12-;NxA76F37$F^^l$\"12-TcO>G7F37$Fa^l$\"12-\"*)o.6O\"F37$Fd^l$
\"12-T1+)*z9F37$Fg^l$\"12-;bps1;F37$Fj^l$\"12-md*R!G<F37$F]_l$\"12-TYt
Se=F3-F`_l6&Fb_lF*Fc_lF*Fe_l-%%TEXTG6%7$$\"#@F)$!#7F)%*y=D(f(x))G%+ALI
GNRIGHTG-Fj\\m6%7$$\"#=F)$\"#8F)%'y=f(x)GFb]m-%+AXESLABELSG6$Q\"x6\"%!
G-%%VIEWG6$;F(F]_l%(DEFAULTG" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 
45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 267 32 "EX
ERCISE for Example 4(optional)" }{TEXT -1 39 ": Do all the same comman
ds but replace " }}{PARA 0 "" 0 "" {TEXT -1 43 "f:=x->piecewise(x<0,x^
2,x<Pi,sin(x),x-Pi); " }}{PARA 0 "" 0 "" {TEXT -1 24 "by the \"on-off
\" function" }}{PARA 0 "" 0 "" {TEXT -1 64 "f:=x->Heaviside(x)-Heavisi
de(x-1)+Heaviside(x-2)-Heaviside(x-3);" }}{PARA 0 "" 0 "" {TEXT -1 64 
"(In MAPLE, the unit step function u(x) is denoted Heaviside(x).)" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 262 9 "EXAMPLE 5" }{TEXT 
-1 51 ": MAPLE knows the product rule for differentiation." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "diff(F(x)*G(x),x);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#,&*&-%%diffG6$-%\"FG6#%\"xGF+\"\"\"-%\"GGF*F,F,*&F
(F,-F&6$F-F+F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "S:=p->(
1-p+p^2)*sin(p+1):\nS(p);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,(\"\"
\"F%%\"pG!\"\"*$)F&\"\"#\"\"\"F%F%-%$sinG6#,&F&F%F%F%F%" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "diff(S(p),p);\nDS:=unapply(diff(S(p
),p),p);\nS(.75);\nDS(.75);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&,&!
\"\"\"\"\"%\"pG\"\"#F'-%$sinG6#,&F(F'F'F'F'F'*&,(F'F'F(F&*$)F(F)\"\"\"
F'F'-%$cosGF,F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#DSGR6#%\"pG6\"
6$%)operatorG%&arrowGF(,&*&,&!\"\"\"\"\"9$\"\"#F0-%$sinG6#,&F1F0F0F0F0
F0*&,(F0F0F1F/*$)F1F2\"\"\"F0F0-%$cosGF5F0F0F(F(F(" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#$\"+>e)[*z!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+
L0orM!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 268 32 "EXERC
ISE for Example 5(optional)" }{TEXT -1 58 ": Do all the same commands \+
but replace p by t and S(p) by " }}{PARA 0 "" 0 "" {TEXT -1 20 "s:=t->
sin(t)*exp(t);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 263 9 "
EXAMPLE 6" }{TEXT -1 52 ": MAPLE knows the quotient rule for different
iation." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "diff(F(x)/G(x),x
);\nsimplify(diff(F(x)/G(x),x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&
*&-%%diffG6$-%\"FG6#%\"xGF+\"\"\"-%\"GGF*!\"\"\"\"\"*&*&F(F0-F&6$F-F+F
0F,*$)F-\"\"#F,F/!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&*&-%%dif
fG6$-%\"FG6#%\"xGF,\"\"\"-%\"GGF+F-F-*&F)F--F'6$F.F,F-!\"\"\"\"\"*$)F.
\"\"#F4!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "h:=x->(x+1)
/(x^3+x^2+1000*x-3001);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hGR6#%
\"xG6\"6$%)operatorG%&arrowGF(*&,&9$\"\"\"F/F/\"\"\",**$)F.\"\"$F0F/*$
)F.\"\"#F0F/F.\"%+5!%,IF/!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "diff(h(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&
\"\"\"F%,**$)%\"xG\"\"$F%\"\"\"*$)F)\"\"#F%F+F)\"%+5!%,IF+!\"\"F+*&*&,
&F)F+F+F+F+,(F,F*F)F.F/F+F+F%*$)F&\"\"#F%F1!\"\"" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }{TEXT 269 32 "EXERCISE for Example 6(optional)" 
}{TEXT -1 47 ": Do all the same commands but replace h(x) by " }}
{PARA 0 "" 0 "" {TEXT -1 21 "j:=x->sin(t)/(t^2+1);" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 0 "" }{TEXT 264 9 "EXAMPLE 7" }{TEXT -1 49 ": MAPLE k
nows the chain rule for differentiation." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "diff(F(G(x)),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*
&--%\"DG6#%\"FG6#-%\"GG6#%\"xG\"\"\"-%%diffG6$F*F-F." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 22 "k:=t->exp(t^3+1999*t);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"kGR6#%\"tG6\"6$%)operatorG%&arrowGF(-%$expG6#,&*$
)9$\"\"$\"\"\"\"\"\"F2\"%**>F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "diff(k(t),t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&
*$)%\"tG\"\"#\"\"\"\"\"$\"%**>\"\"\"F,-%$expG6#,&*$)F'F*F)F,F'F+F," }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 270 32 "EXERCISE for Exam
ple 7(optional)" }{TEXT -1 47 ": Do all the same commands but replace \+
k(t) by " }}{PARA 0 "" 0 "" {TEXT -1 17 "k:=t->sin(t^2+1);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "62 1 0" 17 }{VIEWOPTS 
1 1 0 1 1 1803 }