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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "with(plots): with(pl
ottools): with(linalg):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "Questi
on 1." }}{PARA 0 "" 0 "" {TEXT -1 58 "We first enter the function and \+
determine its derivatives." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
16 "f:= exp(-x)*x^2;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "f1:= factor
(diff(f,x));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "f2:= factor(diff(f1
, x));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "fderivs:= [x, f, f1, f2];
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG*&-%$expG6#,$%\"xG!\"\"\"\"
\")F*\"\"#F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f1G,$*(%\"xG\"\"\"-
%$expG6#,$F'!\"\"F(,&F'F(\"\"#F-F(F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%#f2G*&-%$expG6#,$%\"xG!\"\"\"\"\",(F*!\"%\"\"#F,*$)F*F/F,F,F," }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(fderivsG7&%\"xG*&-%$expG6#,$F&!\"\"
\"\"\")F&\"\"#F-,$*(F&F-F(F-,&F&F-F/F,F-F,*&F(F-,(F&!\"%F/F-*$F.F-F-F-
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "Now we solve for the critical
 and inflection points." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "
crits:= [solve(f1, x)];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "infls:= \+
[solve(f2, x)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&critsG7$\"\"!\"
\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&inflsG7$,&\"\"#\"\"\"*$-%%sq
rtG6#F'F(F(,&F'F(F)!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "We se
e that f is strictly increasing on [0,2] and strictly decreasing for x
 ³ 2." }}{PARA 0 "" 0 "" {TEXT -1 78 "We also see that the graph of f \+
is concave down on [2 - 2^(1/2), 2 + 2^(1/2)]." }}{PARA 0 "" 0 "" 
{TEXT -1 77 "Also the graph of f is concave up on [0, 2 - 2^(1/2)] and
 for x ³ 2 +2^(1/2)." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "We comput
e [f,f1,f2] at the critical points." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 39 "crit1:= simplify(subs(x = 0, fderivs));" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 39 "crit2:= simplify(subs(x = 2, fderivs));" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "evalf(simplify(subs(x = 2, crit2)))
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&crit1G7&\"\"!F&F&\"\"#" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&crit2G7&\"\"#,$-%$expG6#!\"#\"\"%\"
\"!,$F(F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&$\"\"#\"\"!$\"+G8T8a!#5
$F&F&$!+kcq1FF)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "So [0,0] is a \+
local minimum and [2, 4e^(-2)] is a local maximum." }}{PARA 0 "" 0 "" 
{TEXT -1 94 "In fact we see that [0,0] is the global minimum and [2, 4
e^(-2)] is the unique global maximum." }}{PARA 0 "" 0 "" {TEXT -1 43 "
We also compute f at the inflection points." }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 53 "infl1:= evalf(simplify(subs(x = infls[1], fderivs)
));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "infl2:= evalf(simplify(subs(
x = infls[2], fderivs)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&infl1G
7&$\"+iN@9M!\"*$\"+2*p`$Q!#5$!+JAm)e\"F+$\"\"!F/" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&infl2G7&$\"*Qky&e!\"*$\"+iA=5>!#5$\"+?ze6YF+$\"\"!F/
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "We also see that f(x) goes to
 zero as x goes to infinity, so y = 0 is a horizontal asymptote." }}
{PARA 0 "" 0 "" {TEXT -1 12 "We now plot:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 52 "P1:= plot(f, x = 0..5, color = blue, thickness = 3)
:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "P2:= plot(f1, x = 0..5, color \+
= green, thickness = 3):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "P3:= pl
ot(f2, x = 0..5, color = red, thickness = 3):" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 112 "P4:= pointplot(\{[crit1[1], crit1[2]],[crit2[1], cri
t2[2]]\}, symbol = diamond, symbolsize = 30, color = magenta):" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "P5:= pointplot(\{[infl1[1], infl1[
2]],[infl2[1], infl2[2]]\}, symbol = diamond, symbolsize = 30, color =
 cyan):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "display(P1); " }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 25 "display(P1,P2, P3,P4,P5);" }}{PARA 13 "" 
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1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+
4" "Curve 5" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 154 "Here the blue cu
rve is the graph of the function, the green  curve is the graph of its
 derivative and the red curve is the graph of the second derivative." 
}}{PARA 0 "" 0 "" {TEXT -1 62 "The magenta points are the local and gl
obal minima and maxima." }}{PARA 0 "" 0 "" {TEXT -1 45 "The cyan point
s are the points of inflection." }}{PARA 0 "" 0 "" {TEXT -1 138 "Notic
e that the cyan inflection points for the blue curve occur at the crit
ical points of the green curve and the zeroes of the red curve." }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "Question 2." }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 64 "We enter the function, and construct Newton's iterat
ive formula." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "g:= x^3 - 2
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#F37$$\"3E_GUn!p*fHF0$\"3%G.IS1D%GCF37$$\"3H7^8U\"=3'HF0$\"3ED*Q7AL*HC
F37$$\"3km+3'=g;'HF0$\"3[WNM<$H9V#F37$$\"3cyT,PRViHF0$\"3a(*>N\\W!GV#F
37$$\"3l=C!Hu?L'HF0$\"3'Gs%fZ4QMCF37$$\"3Z]0m7R6kHF0$\"3Q^$ohQ\"zNCF37
$$\"3vze0(ff\\'HF0$\"3NH@lBcHPCF37$$\"3b2\"H\\-pd'HF0$\"3u%ooFxN(QCF37
$$\"3.*ze0znm'HF0$\"3-3<'QLN.W#F37$$\"3qGYbbgWnHF0$\"3!y,)yF3sTCF37$$
\"3$>2'G$RH$oHF0$\"3?=Q6;QHVCF37$$\"3j5JtfW8pHF0$\"3E*Q#*>$ysWCF37$$\"
3k?\\!f2:+(HF0$\"3V`2LloHYCF37$$\"3jwAt)*[xqHF0$\"3XI8o\\5lZCF37$$\"3B
+-C[-krHF0$\"3hB$)[HP>\\CF37$$\"30(GWJ8wC(HF0$\"3xBPj3Vo]CF37$$\"3N()
\\)>Z6L(HF0$\"3#*f7$GLu@X#F37$$\"37/[wPP9uHF0$\"35!ol=HfOX#F37$$\"3\\d
)GYGV\\(HF0$\"3o+v-ki3bCF37$$\"3o-L'fh2e(HF0$\"36ev;!HHmX#F37$$\"3J(*p
R'QJm(HF0$\"3Z8^9B.5eCF37$$\"3SB$))fO)\\xHF0$\"3xS*Qe&*['fCF37$$\"35WF
H\">$GyHF0$\"3'z1h-B^5Y#F37$$\"3&f9v,d]\"zHF0$\"3BpC$fV,EY#F37$$\"3!=L
)*zO\")*zHF0$\"3Pi^3'o'3kCF37$$\"3ljjjV,\"3)HF0$\"3RJ;sL(obY#F37$$\"3!
>]-I!en\")HF0$\"3\\%=j#yr6nCF37$$\"3>M2)o5tC)HF0$\"3tX+)*\\PaoCF37$$\"
3f$f7^l*G$)HF0$\"3BX0*)\\^+qCF37$$\"3y'[%Q@7>%)HF0$\"3Hqktt\">;Z#F37$$
\"3.(\\'z:v+&)HF0$\"3Hs%33'43tCF37$$\"3#46L(HB%e)HF0$\"3g:ySDjduCF37$$
\"3wNL+98p')HF0$\"3)=*Ga,v4wCF37$$\"3K&zadpru)HF0$\"3_43))Rh\\xCF37$$
\"3Z#>Jub.$))HF0$\"3SwnSXu)*yCF37$$\"3iNC#p3H\"*)HF0$\"3DEd[4yY![#F37$
$\"3Z$Rso97+*HF0$\"3fPL`\"z^?[#F37$$\"3s'=C!49z!*HF0$\"3Dkyov*\\M[#F37
$$\"3Q;)zd^!p\"*HF0$\"3Or?mUO1&[#F37$$\"3E!f3.*=]#*HF0$\"3BY,0p-_'[#F3
7$$\"3uhT*Hq/L*HF0$\"3$**3!\\8>'z[#F37$$\"339r`%)z;%*HF0$\"3B8\"))ed7&
*[#F37$$\"3%*f=BeZ.&*HF0$\"3)H404'*p5\\#F37$$\"3c%H`RRBe*HF0$\"3/^tKUt
[#\\#F37$$\"3)>Qe+*3m'*HF0$\"3Q[;+ZH*R\\#F37$$\"3kpOS/#yu*HF0$\"3C9Dwv
EY&\\#F37$$\"3c![w<=k$)*HF0$\"3=V!o*Hj0(\\#F37$$\"3%)4@`)zI\"**HF0$\"3
'\\:6SmN%)\\#F37$$\"\"$F*$\"#DF*-%'COLOURG6&%$RGBG$F*F*Fhbbl$\"*++++\"
!\")-%*THICKNESSG6#Fabbl-%+AXESLABELSG6$Q\"x6\"Q!6\"-%%VIEWG6$;F(F`bbl
%(DEFAULTG" 1 2 0 1 10 3 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "C
urve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 100 "We observe that g is \+
increasing strictly on the interval [2,3] with g' at least 10 on that \+
interval." }}{PARA 0 "" 0 "" {TEXT -1 108 "Since g changes sign on [2,
3] and is continuous, by the intermediate value theorem there is a roo
t in (2,3)." }}{PARA 0 "" 0 "" {TEXT -1 52 "Since g is strictly increa
sing, the root is unique. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "We \+
run Newton's iteration starting at x = 3 and see rapid convergence to \+
the required root." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "x1:= \+
3; for n from 1 to 30 do x1:= evalf(subs(x = x1, h), 30) od;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#x1G\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%#x1G$\"?+++++++++++++gB!#H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#
x1G$\"?V[OA33\\;)e,y'>F@!#H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$
\"?$=A\"Q`]6MO$p.O^4#!#H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\"?
Rn'y;HIxECQn^X4#!#H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\"?Yb?!4
81uMU:[^X4#!#H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\"?#3a'Q#[\"f
EBa\"[^X4#!#H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\"?fSlQ#[\"fEB
a\"[^X4#!#H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\"?fSlQ#[\"fEBa
\"[^X4#!#H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\"?fSlQ#[\"fEBa\"
[^X4#!#H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\"?fSlQ#[\"fEBa\"[^
X4#!#H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\"?fSlQ#[\"fEBa\"[^X4
#!#H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\"?fSlQ#[\"fEBa\"[^X4#!
#H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\"?fSlQ#[\"fEBa\"[^X4#!#H
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\"?fSlQ#[\"fEBa\"[^X4#!#H" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\"?fSlQ#[\"fEBa\"[^X4#!#H" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\"?fSlQ#[\"fEBa\"[^X4#!#H" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\"?fSlQ#[\"fEBa\"[^X4#!#H" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\"?fSlQ#[\"fEBa\"[^X4#!#H" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\"?fSlQ#[\"fEBa\"[^X4#!#H" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\"?fSlQ#[\"fEBa\"[^X4#!#H" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\"?fSlQ#[\"fEBa\"[^X4#!#H" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\"?fSlQ#[\"fEBa\"[^X4#!#H" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\"?fSlQ#[\"fEBa\"[^X4#!#H" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\"?fSlQ#[\"fEBa\"[^X4#!#H" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\"?fSlQ#[\"fEBa\"[^X4#!#H" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\"?fSlQ#[\"fEBa\"[^X4#!#H" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\"?fSlQ#[\"fEBa\"[^X4#!#H" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\"?fSlQ#[\"fEBa\"[^X4#!#H" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\"?fSlQ#[\"fEBa\"[^X4#!#H" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\"?fSlQ#[\"fEBa\"[^X4#!#H" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "We compare with the exact root." }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "rootsg:= simplify([solve(g
, x)]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'rootsgG7%,$*&,&*$),&\"$S
&\"\"\"*&\"#7F--%%sqrtG6#\"%H>F-F-#\"\"#\"\"$F-F-\"#CF-F-*$)F+#F-F6F-!
\"\"#F-\"\"',$*&,**$),&F,F-*(F/F--F16#F6F--F16#\"$V'F-F-F4F-F;F7F;*(^#
F-F-FEF-FBF-F-*&^#!#CF-FEF-F-F-*$)FC#F-F6F-F;#F-F/,$*&,*FAF-F7F-FJF-FL
F-F-*$)FC#F-F6F-F;#F;F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "
evalf(rootsg[1], 35);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"DHIz0a'Q#[
\"fEBa\"[^X4#!#M" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "We see very g
ood agreement." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "We run also sta
rting from x = 0 and from x = 1." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 65 "x1:= 0; for n from 1 to 30 do x1:= evalf(subs(x = x1,
 h), 30) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G\"\"!" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#x1G$!?++++++++++++++D!#H" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%#x1G$!?Y2&)HS>hxW5zT;n:!#H" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#x1G$!?f\")Q)o-/9!o'3XCf-&!#I" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#x1G$!?CoJxDF.L$*pnkq?Q!#H" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#x1G$!?J'34%=D341O\"R$R\\D!#H" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#x1G$!?Ar]L)GZsAG(*\\6\"3;!#H" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#x1G$!?0x(HSOZ'HCgOV+hd!#I" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#x1G$!?a3%fk*)*>>AQ$e4xf%!#H" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#x1G$!?:RQ,j)Q*)oEn9VN3$!#H" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#x1G$!?>j/\"G'[[hVYbU>A?!#H" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#x1G$!?i'[s&o#[aazv5kP7\"!#H" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#x1G$\"?^wrj`#)z^bR7;l37!#H" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#x1G$\"?m:TL:W#QC)*[/!z!e$!#H" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#x1G$\"?+'[mtSY<o4!)>L_l#!#H" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#x1G$\"?%Q&[X%)p_h)QuI1h@#!#H" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#x1G$\"?-m)*)G<Gpy%[:]7-@!#H" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#x1G$\"?(pp(e=;h4txwNe%4#!#H" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#x1G$\"?o`h@oQ[QA7#[^X4#!#H" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#x1G$\"?1B^rr;fEBa\"[^X4#!#H" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#x1G$\"?dSlQ#[\"fEBa\"[^X4#!#H" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#x1G$\"?eSlQ#[\"fEBa\"[^X4#!#H" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#x1G$\"?eSlQ#[\"fEBa\"[^X4#!#H" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#x1G$\"?eSlQ#[\"fEBa\"[^X4#!#H" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#x1G$\"?eSlQ#[\"fEBa\"[^X4#!#H" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#x1G$\"?eSlQ#[\"fEBa\"[^X4#!#H" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#x1G$\"?eSlQ#[\"fEBa\"[^X4#!#H" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#x1G$\"?eSlQ#[\"fEBa\"[^X4#!#H" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#x1G$\"?eSlQ#[\"fEBa\"[^X4#!#H" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#x1G$\"?eSlQ#[\"fEBa\"[^X4#!#H" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#x1G$\"?eSlQ#[\"fEBa\"[^X4#!#H" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 65 "x1:= 1; for n from 1 to 30 do x1:= evalf(subs(
x = x1, h), 30) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G\"\"\"" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\"\"(\"\"!" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%#x1G$\"?!p?'eF[M5$z8C<bw%!#H" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#x1G$\"?6xz+;!))GG![fFq[L!#H" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#x1G$\"?5gO=s*z*4D+T'*fJD!#H" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#x1G$\"?2e<XlL'=BR\\)e\"R<#!#H" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#x1G$\"?>&Q)emL\"RwTko$)y4#!#H" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#x1G$\"?[yL%HS#Gd+&erdX4#!#H" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#x1G$\"?/a0#=zav?k:[^X4#!#H" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#x1G$\"?p\"3cE[\"fEBa\"[^X4#!#H" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#x1G$\"?eSlQ#[\"fEBa\"[^X4#!#H" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#x1G$\"?eSlQ#[\"fEBa\"[^X4#!#H" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#x1G$\"?eSlQ#[\"fEBa\"[^X4#!#H" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#x1G$\"?eSlQ#[\"fEBa\"[^X4#!#H" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#x1G$\"?eSlQ#[\"fEBa\"[^X4#!#H" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#x1G$\"?eSlQ#[\"fEBa\"[^X4#!#H" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#x1G$\"?eSlQ#[\"fEBa\"[^X4#!#H" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#x1G$\"?eSlQ#[\"fEBa\"[^X4#!#H" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#x1G$\"?eSlQ#[\"fEBa\"[^X4#!#H" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#x1G$\"?eSlQ#[\"fEBa\"[^X4#!#H" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#x1G$\"?eSlQ#[\"fEBa\"[^X4#!#H" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#x1G$\"?eSlQ#[\"fEBa\"[^X4#!#H" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#x1G$\"?eSlQ#[\"fEBa\"[^X4#!#H" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#x1G$\"?eSlQ#[\"fEBa\"[^X4#!#H" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#x1G$\"?eSlQ#[\"fEBa\"[^X4#!#H" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#x1G$\"?eSlQ#[\"fEBa\"[^X4#!#H" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#x1G$\"?eSlQ#[\"fEBa\"[^X4#!#H" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#x1G$\"?eSlQ#[\"fEBa\"[^X4#!#H" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#x1G$\"?eSlQ#[\"fEBa\"[^X4#!#H" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#x1G$\"?eSlQ#[\"fEBa\"[^X4#!#H" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#x1G$\"?eSlQ#[\"fEBa\"[^X4#!#H" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 113 "In each case the iteration initally wanders away fr
om the root, but eventually settles down to approach the root." }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "To plot, we enter our previously c
onstructed Newton packages (see quiz 8)." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 52 "Newton1:= proc(h,N,b,a, c) local n, y, pts,PP1,PP2; \+
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "y:= array(1..N): y[1]:= b: " }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "pts:= array(1..N): " }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 77 "for n from 1 to N-1 do y[n+1]:= evalf(y[n] - s
ubs(x = y[n], h/diff(h,x)),a): " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "
pts[n]:=[y[n], evalf(subs(x=y[n], h),a)]: od:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 46 "pts[N]:= [y[N], evalf(subs(x = y[N], h), a)]: " }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "print(y); " }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "print(pts); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "P
P1:= pointplot(convert(pts, set), symbol = cross, symbolsize =30, colo
r=COLOR(RGB, rand()/10^12, rand()/10^12, rand()/10^12)):" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 81 "PP2:= plot(h, x = (b -c)..(b + c), numpoints \+
= 600, thickness = 3,color = green):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 19 "display(PP1, PP2); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "end pro
c;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(Newton1GR6'%\"hG%\"NG%\"bG%\"
aG%\"cG6'%\"nG%\"yG%$ptsG%$PP1G%$PP2G6\"F2C,>8%-%&arrayG6#;\"\"\"9%>&F
56#F:9&>8&F6?(8$F:F:,&F;F:F:!\"\"%%trueGC$>&F56#,&FCF:F:F:-%&evalfG6$,
&&F56#FCF:-%%subsG6$/%\"xGFP*&9$F:-%%diffG6$FXFVFEFE9'>&FAFQ7$FP-FM6$-
FS6$FUFXFfn>&FA6#F;7$&F5F`o-FM6$-FS6$/FVFboFXFfn-%&printG6#F5-Fio6#FA>
8'-%*pointplotG6&-%(convertG6$FA%$setG/%'symbolG%&crossG/%+symbolsizeG
\"#I/%&colorG-%&COLORG6&%$RGBG,$-%%randGF2#F:\".++++++\"FbqFbq>8(-%%pl
otG6'FX/FV;,&F?F:9(FE,&F?F:F_rF:/%*numpointsG\"$+'/%*thicknessG\"\"$/F
]q%&greenG-%(displayG6$F^pFhqF2F2F2" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 75 "Newton2:= proc(h,N,b,a, c) local n, y, PP1, PP2, QQ, \+
RR, s, tline, u, pts; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "y:= array
(1..N): y[1]:= b: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "QQ:= array(1.
.N):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "RR:= array(1..N):  " }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "pts:= array(1..N): " }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 122 "for n from 1 to N-1 do y[n+1]:= evalf(y[n] - \+
subs(x = y[n], h/diff(h,x)),a): pts[n]:=[y[n], evalf(subs(x=y[n], h),a
)]: od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "pts[N]:= [y[N], evalf(su
bs(x = y[N], h), a)]: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "print(y);
 " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "print(pts); " }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 113 "PP1:= pointplot(pts, symbol = cross, symbolsize
 =30, color=COLOR(RGB, rand()/10^12, rand()/10^12, rand()/10^12)):" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "PP2:= plot(h, x = (b -c)..(b + c),
 numpoints = 600, thickness = 3,color = green): display(PP1, PP2); " }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "for s from 1 to N do tline[s]:= su
bs(x = y[s],diff(h, x)*(t - y[s]) + h) od: " }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 197 "for u from 1 to N do RR[u]:= plot(tline[u], t = (y[u
] - c/2)..(y[u] + c/2), color=COLOR(RGB, rand()/10^12, rand()/10^12, r
and()/10^12), thickness = 3) od: display(\{PP1, PP2\}union convert(RR,
 set));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "end proc;" }}{PARA 12 "" 
1 "" {XPPMATH 20 "6#>%(Newton2GR6'%\"hG%\"NG%\"bG%\"aG%\"cG6,%\"nG%\"y
G%$PP1G%$PP2G%#QQG%#RRG%\"sG%&tlineG%\"uG%$ptsG6\"F7C1>8%-%&arrayG6#;
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{PARA 0 "" 0 "" {TEXT -1 11 "Question 3." }}{PARA 0 "" 0 "" {TEXT -1 
46 "We enter the formulas for the volume and cost." }}{PARA 0 "" 0 "" 
{TEXT -1 47 "We write x for the radius and y for the height." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Vol:= Pi*x^2*y;" }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 53 "Cost:= factor(3/100*Pi*(x^2 + 2*x*y) + 5/100
*Pi*x^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$VolG*(%#PiG\"\"\")%\"x
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0 "" {TEXT -1 44 "We eliminate y using the relation Vol = 250." }}
{PARA 0 "" 0 "" {TEXT -1 39 "Then we plot and find the minimum cost." 
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3$)Qt&*>t=_CF-7$$\"3,WP\\#*Qdk&*Ff_l$\"3Ug$4i)))*fX#F-7$$\"3t#)*y(4^,s
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ef^*4SE\\#F-7$$\"3EyHddqa^'*Ff_l$\"3;VLqs'*e'\\#F-7$$\"33I_Fq@3f'*Ff_l
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ntx1(*Ff_l$\"31;%\\uXwD_#F-7$$\"3WZn4@K9:(*Ff_l$\"3UiW))zp_EDF-7$$\"3s
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\"3<D2([EB&Q(*Ff_l$\"3AW$yxO(ePDF-7$$\"33!QdQ)*eku*Ff_l$\"3sV/Bl\"[8a#
F-7$$\"3dPcw3bOa(*Ff_l$\"3^=`Py%)4XDF-7$$\"3YXT(R?h>w*Ff_l$\"3<?j#\\]/
([DF-7$$\"3)pP^;Ns,x*Ff_l$\"3)\\wIk=1Eb#F-7$$\"3:x9xq\")*zx*Ff_l$\"3W8
M*y7Gjb#F-7$$\"3Lu!*)oZMiy*Ff_l$\"3>\"z_J$)[-c#F-7$$\"3wo5G<.p$z*Ff_l$
\"3c+LHV6!Qc#F-7$$\"34')Qm;/$>!)*Ff_l$\"3J%*y2o0tnDF-7$$\"3IaT)f*H#)4)
*Ff_l$\"3T&e@_n(\\rDF-7$$\"3\"\\XXXO'p<)*Ff_l$\"3FbQ6I*e_d#F-7$$\"3RpP
_G,#f#)*Ff_l$\"32rRI76>zDF-7$$\"3%R(pO:X\\L)*Ff_l$\"3wf:G#*f\"Ge#F-7$$
\"3GT'pNs^7%)*Ff_l$\"3#*y(o=aJle#F-7$$\"33FE:.m\")\\)*Ff_l$\"3gujc$oP1
f#F-7$$\"3%Hsm+Srv&)*Ff_l$\"3QG,!z$)eVf#F-7$$\"3?aXYK@]l)*Ff_l$\"3![#=
ZOx;)f#F-7$$\"3S#pJI[nN()*Ff_l$\"3YVkAKZ/-EF-7$$\"33f0n46)4))*Ff_l$\"3
(*ePs3:h0EF-7$$\"3,Ijf&z$)))))*Ff_l$\"3E`rA2oT4EF-7$$\"3gOJwDjs'*)*Ff_
l$\"3fp%RXV'>8EF-7$$\"3))QxG&R:^!**Ff_l$\"3)HJ!*Q4Vsh#F-7$$\"3iv(HdQ=D
\"**Ff_l$\"3fgd;9s\"3i#F-7$$\"3wd#3***)f5#**Ff_l$\"3dV4cRY%\\i#F-7$$\"
3:5;$z&zwG**Ff_l$\"3ges7jEnGEF-7$$\"30OXWyYRO**Ff_l$\"3!)fQ>mWOKEF-7$$
\"3`&e-J&efW**Ff_l$\"3%f4q8!yLOEF-7$$\"3knE?.-$G&**Ff_l$\"3w$f#G<3LSEF
-7$$\"3F)HcDCA.'**Ff_l$\"3[/'35,nRk#F-7$$\"3WGYbb%y#o**Ff_l$\"31o\"\\G
wJyk#F-7$$\"3=i[$=%H/w**Ff_l$\"3)e([bNmg^EF-7$$\"3ejl(osfW)**Ff_l$\"30
\"=\"3mAqbEF-7$$\"3iY]0'eU<***Ff_l$\"3C%*z:r\"\\#fEF-7$$\"#5\"\"!$\"3
\\M=(G7uKm#F--%'COLOURG6&%$RGBG$FbbblFbbbl$\"*++++\"!\")Fibbl-%*THICKN
ESSG6#\"\"$-%+AXESLABELSG6$Q\"x6\"Q!6\"-%%VIEWG6$;$\"\"&!\"\"F`bbl%(DE
FAULTG" 1 2 0 1 10 3 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve
 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "We finish by determining t
he minimum cost." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "factor(
diff(Cost2, x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&*&%#PiG\"\"
\")%\"xG\"\"$F(\"\"%\"$v$!\"\"F(*$)F*\"\"#F(F.#F(\"#D" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "x1:= (375/4/Pi)^(1/3);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%#x1G,$*()\"$v$#\"\"\"\"\"$F*)\"\"%#\"\"#F+F*)
*&F*F*%#PiG!\"\"F)F*#F*F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
90 "mincostderivs:= simplify(subs(x = x1, [x1, y1, Cost2, diff(Cost2, \+
x), diff(Cost2, x,x)]));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "evalf(m
incostderivs);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%.mincostderivsG7',
$*&*&)\"\"$#\"\"\"F*F,)\"\"#F+F,F,*$)%#PiG#F,F*F,!\"\"#\"\"&F.,$F'#\"#
?F*,$*()F*#F.F*F,)F.F<F,)F1F+F,#F*F.\"\"!,$F1#\"#7\"#D" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#7'$\"+bCv,J!\"*$\"+!))R8F)F&$\"+'zjRD(F&$\"\"!F,$
\"+uW'z]\"F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 106 "So the minimum c
ost is 7.25 cents, and the can's dimensions are:  radius = 3.10 cms an
d height = 8.27 cms." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "Question \+
4." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "We first enter the function
 and find the linear approximation about x = 0." }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 13 "f:=arctan(x);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 16 "f1:= diff(f, x);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "flin:= si
mplify(subs(x = 0, f1)*(x - 0) + subs(x = 0, f));" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 29 "linapprox:= subs(x = 1/2, x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"fG-%'arctanG6#%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#f1G*&\"\"\"F&,&F&F&*$)%\"xG\"\"#F&F&!\"\"" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%%flinG%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*l
inapproxG#\"\"\"\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Now we s
tudy the error in the linear approximation." }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 28 "linerror:= 100*(flin/f - 1);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%)linerrorG,&*&%\"xG\"\"\"-%'arctanG6#F'!\"\"\"$+\"F-F
," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "plot(linerror, x = - 1
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$$\"3\\'******\\y))G)Fjq$\"3R/,qSE+'G#F*7$$\"3'*)***\\i_QQ5F*$\"3!=J)e
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\"$\"\"!Fa[lF`[l-%+AXESLABELSG6$Q\"x6\"Q!6\"-%%VIEWG6$;$!+++++]!#5$\"+
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45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "
fsolve(linerror - 2, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+*y=!pC
!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "plot(linerror, x = -
 1/4..1/4);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-
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ff3\"Fap$\"3qQncO<!4$R!#?7$$!3)Grmm\"z%zY#!#@$\"3C')==:cDI?!#B7$$\"3CH
Le*)>px5Fap$\"3$3`dn=!GrQF\\s7$$\"3V(**\\Pf4t.#Fap$\"3K'4Z\"e.R$Q\"Fap
7$$\"3qIL$e*GstIFap$\"3a)pAnWk%[JFap7$$\"3C)*****\\#RW9%Fap$\"3\"4+lay
QGs&Fap7$$\"3#[***\\7j#>>&Fap$\"3XF')fPk\"*y*)Fap7$$\"32'**\\i!RU0iFap
$\"3YB#))GzgAG\"F*7$$\"3c&***\\(=S2L(Fap$\"3OFkE\"=l()y\"F*7$$\"3;jmm;
p)=M)Fap$\"3b6+8U0G:BF*7$$\"3.*****\\(=]@%*Fap$\"3%f8'pA>&=&HF*7$$\"35
L$e*[$z*R5F*$\"3GVfqM2&[f$F*7$$\"3#*****\\iC$p9\"F*$\"3l$Q$3'\\q&pVF*7
$$\"39m;H2qcZ7F*$\"3t')R/,!>n;&F*7$$\"3p**\\7.\"fFN\"F*$\"3RZ<(>os.2'F
*7$$\"3Xmm;/Ogb9F*$\"3]e]+I(QJ-(F*7$$\"3x**\\ilAFj:F*$\"3qALr.&\\O4)F*
7$$\"3YLLL$)*ppm\"F*$\"3-Aje/T(\\>*F*7$$\"3>LL3xe,t<F*$\"3I\"[k>T>#R5F
-7$$\"3em;HdO=y=F*$\"3KA)e.k'*\\;\"F-7$$\"3()*****\\#>#[(>F*$\"38^fr0Q
s'G\"F-7$$\"3hmmT&G!e&3#F*$\"3@B^uNrVL9F-7$$\"3:LLL$)Qk%=#F*$\"3'[k#*y
AL6d\"F-7$$\"37+]iSjE!H#F*$\"3^k^%pbLYs\"F-7$$\"35+]P40O\"R#F*$\"3l*=r
tesz(=F-7$$\"3++++++++DF*F+-%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!Fb[lFa[l-
%+AXESLABELSG6$Q\"x6\"Q!6\"-%%VIEWG6$;$!+++++D!#5$\"+++++DF`\\l%(DEFAU
LTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1
" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "So the error stays below 2 p
ercent for |x| < 0.246 approximately." }}{PARA 0 "" 0 "" {TEXT -1 82 "
We now do L'Hopital's rule for the limit of (arctan(x) - x)/x^3 as x g
oes to zero." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "LHop1:= sim
plify(diff(f - x, x)/diff(x^3, x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%&LHop1G,$*&\"\"\"F',&F'F'*$)%\"xG\"\"#F'F'!\"\"#F-\"\"$" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "subs(x = 0, LHop1);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6##!\"\"\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 24 "cubicapprox:= x - x^3/3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
,cubicapproxG,&%\"xG\"\"\"*&#F'\"\"$F'*$)F&F*F'F'!\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "plot(\{x, cubicapprox, arctan(x)\},
 x = -1.5..1.5, thickness = 3);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 
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{EXCHG {PARA 0 "" 0 "" {TEXT -1 135 "Here the red curve is the linear \+
approximation, the yellow curve is the graph of arctan and the green c
urve is the cubic approximation." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
89 "Finally we determine when the percentage error in the cubic approx
imation is less than 2." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "
cubicerror:= -100*(cubicapprox/arctan(x) - 1);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%+cubicerrorG,&*&,&%\"xG\"\"\"*&#F)\"\"$F)*$)F(F,F)F)!
\"\"F)-%'arctanG6#F(F/!$+\"\"$+\"F)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 33 "plot(cubicerror, x = - 0.7..0.7);" }}{PARA 13 "" 1 "
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;X=:6F-7$$\"3almTSU\"*e_F*$\"3yI?)=GL5R\"F-7$$\"32+++!R,&HbF*$\"3YuFfX
4&\\o\"F-7$$\"3@nm;*zC'ReF*$\"3Lv1kwMvt?F-7$$\"3*)***\\i$QJyfF*$\"31aD
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a@y0FF-7$$\"3*)***\\PvXFT'F*$\"3R\\\\E:[WaHF-7$$\"3V*******exUb'F*$\"3
EM!QKNHs?$F-7$$\"3'*)**\\iU4ep'F*$\"3=Qb\"H7e\\Z$F-7$$\"3E**\\78Z!z%oF
*$\"3>(o1IEv)zPF-7$$\"3a**************pF*F+-%'COLOURG6&%$RGBG$\"#5!\"
\"$\"\"!F`^lF_^l-%+AXESLABELSG6$Q\"x6\"Q!6\"-%%VIEWG6$;$!\"(F^^l$\"\"(
F^^l%(DEFAULTG" 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 "" {MPLTEXT 1 0 30 "fsolve(cubic
error-2, x, 0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+#R1Uy&!#5" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "So the cubic approximation is go
od to within 2 percent on the interval [-0.578, 0.578]." }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 11 "Question 5." }}{PARA 0 "" 0 "" {TEXT -1 
73 "We first find the unit vectors in the direction of each vehicle's \+
motion." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "F1:= [12, -5]: \+
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "F2:= sqrt(F1[1]^2 + F1[2]^2);" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "F3:= evalm(1/F2*F1);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%#F2G\"#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%#F3G-%'vectorG6#7$#\"#7\"#8#!\"&F+" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "H1:= [3, -4]: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "
H2:= sqrt(H1[1]^2 + H1[2]^2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "H3
:= evalm(1/H2*H1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#H2G\"\"&" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#H3G-%'vectorG6#7$#\"\"$\"\"&#!\"%F+
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "Now we can write formulas for
 the position of each vehicle at time t hours:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 29 "F:= evalm([1, 3] + 325*t*F3);" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 28 "H:= evalm([1, 3] + 60*t*H3);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"FG-%'vectorG6#7$,&\"\"\"F**&\"$+$F*%\"tGF*F*,&\"
\"$F**&\"$D\"F*F-F*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"HG-%'v
ectorG6#7$,&\"\"\"F**&\"#OF*%\"tGF*F*,&\"\"$F**&\"#[F*F-F*!\"\"" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "The position vector X of one car r
elative to the other is then given by subtraction:" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 17 "X:= evalm(F - H);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"XG-%'vectorG6#7$,$%\"tG\"$k#,$F*!#x" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 49 "The relative velocity vector V is its der
ivative:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "V:= evalm(map(d
iff, X, t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"VG-%'vectorG6#7$\"
$k#!#x" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "The relative speed in k
ilometers per hour is the length of V:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 33 "relspeed:= sqrt(V[1]^2 + V[2]^2);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%)relspeedG\"$v#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 176 "The area of the triangle formed by the cars and the starting p
oint is given by the following formulas, based on the vectors F4 and H
4 which lie along two sides of the triangle:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 21 "F4:= evalm(325*t*F3);" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "H4:= evalm(60*t*H3);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%#F4G-%'vectorG6#7$,$%\"tG\"$+$,$F*!$D\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#H4G-%'vectorG6#7$,$%\"tG\"#O,$F*!#[" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "FdotF:= factor(F4[1]*F4[1] + F4[2]*
F4[2]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "FdotH:= factor(F4[1]*H4[
1] + F4[2]*H4[2]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "HdotH:= facto
r(H4[1]*H4[1] + H4[2]*H4[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&Fd
otFG,$*$)%\"tG\"\"#\"\"\"\"'Dc5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&
FdotHG,$*$)%\"tG\"\"#\"\"\"\"&+o\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%&HdotHG,$*$)%\"tG\"\"#\"\"\"\"%+O" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 41 "area:= 1/2*sqrt(FdotF*HdotH - (FdotH)^2);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%%areaG,$*$-%%sqrtG6#*$)%\"tG\"\"%\"\"\"F.
\"%]\\" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "So the triangle area is
 4950t^2.\\\\" }}{PARA 0 "" 0 "" {TEXT -1 65 "For illustration, we plo
t the vehicles after one and two minutes." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 30 "F5:= subs(t = 1/60, evalm(F));" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 30 "F6:= subs(t = 1/30, evalm(F));" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 30 "H5:= subs(t = 1/60, evalm(H));" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 30 "H6:= subs(t = 1/30, evalm(H));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#F5G-%'vectorG6#7$\"\"'#\"#6\"#7" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#F6G-%'vectorG6#7$\"#6#!\"(\"\"'" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#H5G-%'vectorG6#7$#\"\")\"\"&#\"#6F+" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%#H6G-%'vectorG6#7$#\"#6\"\"&#\"\"(F+" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "A:= arrow([0,0], [1,3], .2,.4, .1, \+
color = red): " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "B1:= arrow([0,0],
 [6,11/12], .2,.4, .1, color = green): " }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 56 "B2:= arrow([0,0], [11,-7/6], .2,.4, .1, color = green): " }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "C1:= arrow([0,0], [8/5,11/5], .2,.4
, .1, color = blue): " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "C2:= arrow
([0,0], [11/5,7/5], .2,.4, .1, color = blue):" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 58 "D1:= arrow([1,3], [6,11/12], .2,.4, .1, color = magen
ta): " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "D2:= arrow([6,11/12], [11,
-7/6], .2,.4, .1, color = magenta): " }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 56 "E1:= arrow([1,3], [8/5,11/5], .2,.4, .1, color = cyan): " }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "E2:= arrow([8/5,11/5], [11/5,7/5], \+
.2,.4, .1, color = cyan):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "F1:= a
rrow([8/5, 11/5], [6,11/12], .2,.4, .1, color = yellow): " }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 61 "F2:= arrow([11/5,7/5], [11,-7/6], .2,.4, .1,
 color = yellow):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "display(\{A,B1
,B2,C1,C2,D1,D2, E1, E2,F1,F2\});" }}{PARA 13 "" 1 "" {GLPLOT2D 400 
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FC7$$\"++++!)>F5$\"++++g8F57$$\"++++g?F5$\"++++?9F5Fi`l-F$6&7&Fi`lF^al
FcalF^bl7%FfalFCFialFMFa`l-F_q6#7*7$$\"++++s:F5$\"++++/@F57$$\"++++G;F
5$\"++++'H#F57$$\"++++)e&F5$\"++++T6F57$$\"++++;cF5$\"++++P7F5Fbp7$$\"
++++/bF5$\"*+++`)F57$$\"++++KbF5$\"*+++\\*F5Fjbl-F$6&7&FjblF_clFdclFcd
l7%FiclFbpF^dlFM-FR6&FTFWFWFU-F$6&7&7$$\"++++s@F5$\"++++/8F57$$\"++++G
AF5$\"++++'\\\"F57$$\"+++![,\"FY$!*+++9)F57$$\"+++?45FY$!++++15F57%7$$
\"+++g<5FY$!*+++=(F5Fir7$$\"+++S15FY$!++++-6F5FMF\\el-F_q6#7*FaelFfelF
[flFfflFirF[glF`flFael" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 
45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve
 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Cu
rve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve \+
19" "Curve 20" "Curve 21" "Curve 22" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 48 "Here the red vector gives the starting position." }}{PARA 0 "" 
0 "" {TEXT -1 59 "The blue vectors are the two position vecotrs of the
 Honda." }}{PARA 0 "" 0 "" {TEXT -1 62 "The green vectors are the two \+
position vectors of the Ferrari." }}{PARA 0 "" 0 "" {TEXT -1 48 "The c
yan vectors give the movement of the Honda." }}{PARA 0 "" 0 "" {TEXT 
-1 53 "The magenta vectors give the movement of the Ferrari." }}{PARA 
0 "" 0 "" {TEXT -1 94 "The yellow vectors give the relative displaceme
nt vectors of the Ferrari relaive to the Honda." }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 11 "Question 6." }}{PARA 0 "" 0 "" {TEXT -1 138 "We ente
r the formulas for the two volumes and use the fact that r = 100 - 2t \+
and that the total volume is constant to get a formula for s." }}
{PARA 0 "" 0 "" {TEXT -1 37 "Then we differentiate and put t = 20." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 16 "V:= 4/3*Pi*r^3; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "W:= 4/3*P
i*s^3;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "V1:= subs(r = 100 - 2*t, \+
V);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "V0:= subs(r = 100, V);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"VG,$*&%#PiG\"\"\")%\"rG\"\"$F(#\"
\"%F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"WG,$*&%#PiG\"\"\")%\"sG\"
\"$F(#\"\"%F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#V1G,$*&%#PiG\"\"\"
),&\"$+\"F(*&\"\"#F(%\"tGF(!\"\"\"\"$F(#\"\"%F0" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#V0G,$%#PiG#\"(+++%\"\"$" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 29 "s1:= [solve(V1 + W - V0, s)];" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%#s1G7%,$*$),(%\"tG\"%+v*&\"$]\"\"\"\")F*\"\"#F.!\"\"*
$)F*\"\"$F.F.#F.F4F.F0,&F'F1*(^#F.F.-%%sqrtG6#F4F.F(F.F.,&F'F1*(^#F1F.
F9F.F(F.F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "schange:= sub
s(t = 20, diff(s1[1], t));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "evalf
(schange);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(schangeG,$*$)\"&+!)*#
\"\"\"\"\"$F*#\"\"*\"$!\\" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+#H[\"
o%)!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "So at t = 20, s is incr
easing at a rate of 0.85 cms per second, approximately." }}{PARA 0 "" 
0 "" {TEXT -1 36 "We plot the radii on the same graph:" }}}{EXCHG 
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0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Note tha
t s is initially changing at infinite speed!.  " }}{PARA 0 "" 0 "" 
{TEXT -1 43 "We also plot the volumes on the same graph:" }}}{EXCHG 
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