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{SECT 0 {EXCHG {PARA 3 "" 0 "" {TEXT -1 19 "Homework Problem 61" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "restart:  #this is always a \+
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{MPLTEXT 1 0 160 "with(plots); with(plottools);  #These loads the grap
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mmmA\"F-$!3`-v=nmmmUFdel7$7$$\"3Xuozmmmm7F-FijlF[\\[l7$Fa\\[l7$$\"3;P%
)RLLL88F-$!3myV)RLLL`$Fdel7$7$F`t$!3v`(ozmmm1$FdelFe\\[l7$7$F`t$\"3v`(
ozmmm1$Fdel7$$\"3Mvozmmmm7F-Feam7$Fb][lFg[[l7$F[][l7$$\"39#)33\\DP\"R
\"F-$!3jF)33\\DPr#Fdel7$7$$\"3u**\\Pf#f#>9F-F_`oFg][l7$F]^[l7$$\"32QlL
5kD]9F-$!3K*QlL5kDq\"Fdel7$7$$\"356h[q.P!\\\"F-F]^mFa^[l7$Fg^[l7$Fh^[l
$!3#3=6'[q.P]F_^m7$7$Fh^[lFc_mF[_[l7$F__[l7$$\"3e+](ommmY\"F-$\"3/()*
\\7LLLL\"Fdel7$7$$\"3>+]Pf#f#>9F-Fc`mFa_[l7$Fg_[lF_][l-F_[l6&Fa[lFb[lF
*F*-%+AXESLABELSG6$%\"xG%\"yG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 
45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve
 5" "Curve 6" "Curve 7" "Curve 8" }}}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 19 "Homework Prob
lem 62" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 165 "We \+
will use Maple to evaluate the directional derivatives analytically.  \+
We will just use the definition for directional derivatives (we could \+
do it other ways too)." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 236 "d:=(u1,u
2,h,x,y)->(f(x+h*u1,y+h*u2) - f(x,y))/(h*sqrt(u1^2+u2^2)); #The differ
ence quotient.  Note that I have \"unitized\" the vector in the defini
tion of the diffence quotient.  I don't need to worry whether I have a
 unit vector or not." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"dGf*6'%#u1
G%#u2G%\"hG%\"xG%\"yG6\"6$%)operatorG%&arrowGF,*(,&-%\"fG6$,&9'\"\"\"*
&9&F79$F7F7,&9(F7*&F9F79%F7F7F7-F36$F6F<!\"\"F7F9FA-%%sqrtG6#,&*$)F:\"
\"#F7F7*$)F>FHF7F7FAF,F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
98 "dirder:=(u1,u2,x,y)->limit(d(u1,u2,h,x,y),h=0); #Take the limit to
 get the directional derivative." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
'dirderGf*6&%#u1G%#u2G%\"xG%\"yG6\"6$%)operatorG%&arrowGF+-%&limitG6$-
%\"dG6'9$9%%\"hG9&9'/F7\"\"!F+F+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
259 3 "(a)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dirder(1,0,0,1);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 256 "" 0 "" 
{TEXT -1 3 "(b)" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
16 "dirder(0,1,0,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}}
{EXCHG {PARA 257 "" 0 "" {TEXT -1 3 "(c)" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "dirder(1,-1,1,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
,$*&\"\"$\"\"\"\"\"##F&F'!\"\"" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 
3 "(d)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dirder(1,1,1,1);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"&\"\"\"\"\"##F&F'F&" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 3 "" 0 "" 
{TEXT -1 19 "Homework Problem 63" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 72 "restart:  #Let's clear the memory so we can use the s
ame names as above." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "f:=(
x,y,z) -> x^2*y + y^2*z;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6%
%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF*,&*&)9$\"\"#\"\"\"9%F3F3*&)F4
F2F39&F3F3F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "d:=(u1,
u2,u3,h,x,y,z)->(f(x+h*u1,y+h*u2,z+h*u3) - f(x,y,z))/(h*sqrt(u1^2+u2^2
+u3^2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"dGf*6)%#u1G%#u2G%#u3G%
\"hG%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF.*(,&-%\"fG6%,&9(\"\"\"*&9
'F99$F9F9,&9)F9*&F;F99%F9F9,&9*F9*&F;F99&F9F9F9-F56%F8F>FB!\"\"F9F;FG-
%%sqrtG6#,(*$)F<\"\"#F9F9*$)F@FNF9F9*$)FDFNF9F9FGF.F.F." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "dirder:=(u1,u2,u3,x,y,z)->limit(d(u
1,u2,u3,h,x,y,z),h=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'dirderGf*
6(%#u1G%#u2G%#u3G%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF--%&limitG6$-
%\"dG6)9$9%9&%\"hG9'9(9)/F:\"\"!F-F-F-" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 258 3 "(a)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "dirder(0,-1,1,1
,2,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"*\"\"\"\"\"#!\"\"F'#
F&F'F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 257 3 "(b)" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 20 "dirder(2,3,5,1,2,3);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,$*(\"#n\"\"\"\"#Q!\"\"F'#F&\"\"#F&" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}}{MARK "11 0 0" 3 }{VIEWOPTS 1 1 0 1 1 1803 1 
1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }