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}}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "c)  When f(x) has a max or min
 at x = a, what is D(f)(a)? " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "d
)  When f(x) is increasing (decreasing) what is the sign of D(f)? " }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "When f is increasing D(f) >0, and
 when f is decreasing D(f) <0. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT 259 10 "Example 4\n" }{TEXT -1 11 "Let f(x)
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\"F%" }{TEXT -1 19 " if x > 0 and 0 if " }{XPPEDIT 18 0 "x <= 0" "6#1%
\"xG\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 63 "Plot f(x). \+
 Use the graph to determine whether D(f)(0) exists. " }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 42 "f:= x -> piecewise( x > 0, x*sin(1/x), 0)
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*[Vqf]_<a(F_]l7$$\"3gBJq&=dj7)F_]l$\"3uwK4h/`gwF_]l7$$\"3_LLmgLxL$)F_]
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8&HCc()F_]l$\"3UkQ!\\!pmjzF_]l7$$\"3m**\\Q9)Q&\\*)F_]l$\"3;fY/(yC_/)F_
]l7$$\"3[;/*oox5<*F_]l$\"3;7?V?T&H8)F_]l7$$\"3OLLW&pC#p$*F_]l$\"3aLEs+
Bi1#)F_]l7$$\"3[C\"=Xt!\\!e*F_]l$\"3%>.!QSpc!G)F_]l7$$\"31v=&3H*p#y*F_
]l$\"3)GVTKGDsM)F_]l7$$\"\"\"F,$\"30l*y![)4ZT)F_]l-%'COLOURG6&%$RGBG$
\"#5!\"\"F+F+-%+AXESLABELSG6$Q\"x6\"Q!6\"-%%VIEWG6$;$Fj_m!\"%F`_m%(DEF
AULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve \+
1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "Based on the plots, we conj
ecture that f is not differentiable when x = 0. " }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 10 "Example 5\n" }
{TEXT -1 11 "Let f(x) = " }{XPPEDIT 18 0 "x^2 *sin(1/x)" "6#*&%\"xG\"
\"#-%$sinG6#*&\"\"\"F*F$!\"\"F*" }{TEXT -1 29 " for x > 0 and 0 for x \+
<= 0. " }}{PARA 0 "" 0 "" {TEXT -1 58 "Plot f(x).  Use the graph to de
termine if D(f)(0) exits.  " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
43 "f:= x -> piecewise(x>0, x^2 * sin(1/x), 0);" }{TEXT -1 0 "" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&arrowG
F(-%*piecewiseG6%2\"\"!9$*&)F1\"\"#\"\"\"-%$sinG6#*&F5F5F1!\"\"F5F0F(F
(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "plot(f(x), x = -.000
01...00001);" }{TEXT -1 0 "" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 
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}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "Based on the plot, we conjectur
e that D(f)(0) = 0.  " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 261 10 "Example 6\n" }{TEXT -1 12 "Let f(x) =  \+
" }{XPPEDIT 18 0 "3 - 2*x^2" "6#,&\"\"$\"\"\"*&\"\"#F%*$%\"xGF'F%!\"\"
" }{TEXT -1 47 ".  Plot f(x), the tangent line when x = .5 and " }}
{PARA 0 "" 0 "" {TEXT -1 46 "the normal line when x =.5 on the same ax
es.  " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "restart:  " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "f:=x ->  3 - 2*x^2;" }{TEXT 
-1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operato
rG%&arrowGF(,&\"\"$\"\"\"*&\"\"#F.)9$F0F.!\"\"F(F(F(" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 9 "D(f)(.5);" }{TEXT -1 0 "" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#$!#?!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
6 "f(.5);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$]#!\"#
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "t:= x -> -2*(x - .5) + \+
2.5;" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"tGR6#%\"xG6
\"6$%)operatorG%&arrowGF(,&9$!\"#$\"#N!\"\"\"\"\"F(F(F(" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "n:= x -> (1/2)*(x - .5) + 2.5;" }
{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nGR6#%\"xG6\"6$%)o
peratorG%&arrowGF(,&9$#\"\"\"\"\"#$\"++++]A!\"*F/F(F(F(" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "plot(\{f(x), t(x), n(x)\}, x = -1..
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