{VERSION 4 0 "IBM INTEL NT" "4.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 
0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 256 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 
0 0 1 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Tim
es" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }
{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 
2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 
4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 
0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 
-1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 
0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 
1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Title" 
-1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 
1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Heading 2" -1 256 1 {CSTYLE "" 
-1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 8 2 1 0 1 0 2 
2 0 1 }}
{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 11 "Calculus II" }}{PARA 256 
"" 0 "" {TEXT -1 59 "Lesson 21:  Approximating Sine and Cosine with Po
wer Series" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 31 "Approximating the \+
sine function" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Using " }{TEXT 
256 7 "Maple's" }{TEXT -1 50 " ability to compute limits, it is easy t
o see how " }{XPPEDIT 18 0 "sin(x);" "6#-%$sinG6#%\"xG" }{TEXT -1 29 "
 behaves for small values of " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT 
-1 17 " .  We know that " }{XPPEDIT 18 0 "sin(0) = 0;" "6#/-%$sinG6#\"
\"!F'" }{TEXT -1 15 ", and the limit" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 40 "Limit(sin(x), x=0) = limit(sin(x), x=0);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#/-%&LimitG6$-%$sinG6#%\"xG/F*\"\"!F," }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "shows that the sine function is co
ntinuous at 0.  (This should be clear to you from the graph of \n" }
{XPPEDIT 18 0 "y = sin(x);" "6#/%\"yG-%$sinG6#%\"xG" }{TEXT -1 48 " .)
  The interesting question is: how fast does " }{XPPEDIT 18 0 "sin(x);
" "6#-%$sinG6#%\"xG" }{TEXT -1 12 " go to 0 as " }{XPPEDIT 18 0 "proc \+
(x) options operator, arrow; 0 end;" "6#R6#%\"xG7\"6$%)operatorG%&arro
wG6\"\"\"!F*F*F*" }{TEXT -1 37 ".  We can discover this by comparing \+
" }{XPPEDIT 18 0 "sin(x);" "6#-%$sinG6#%\"xG" }{TEXT -1 24 " with vari
ous powers of " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 2 " :" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "limit(sin(x)/x, x=0);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 34 "The meaning of this limit is that " }{XPPEDIT 18 0 "sin(x
)/x;" "6#*&-%$sinG6#%\"xG\"\"\"F'!\"\"" }{TEXT -1 33 " is approximatel
y equal to 1, or " }{XPPEDIT 18 0 "sin(x);" "6#-%$sinG6#%\"xG" }{TEXT 
-1 27 " is approximately equal to " }{XPPEDIT 18 0 "x;" "6#%\"xG" }
{TEXT -1 7 ", when " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 47 " is c
lose to 0.  We can check this graphically:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 34 "plot( \{sin(x),x\}, x=-Pi/2..Pi/2 );" }}{PARA 13 "
" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7S7$$!3+++lBjz
q:!#<F(7$$!3WNzQW&=B]\"F*F,7$$!3.42![ROFW\"F*F/7$$!3?xt3O+tv8F*F27$$!3
xxIt:&z#38F*F57$$!3\")zz#fd\\6C\"F*F87$$!3W:#)\\G:\"*y6F*F;7$$!3,%4``j
nW6\"F*F>7$$!3)yYu)o'>y/\"F*FA7$$!3EWfpKW&Q\")*!#=FD7$$!3(fe1Vw'\\I\"*
FFFH7$$!333*H!e\\fG&)FFFK7$$!35`u@%3'*4&yFFFN7$$!3%ewEV#\\hqrFFFQ7$$!3
SI*3_gT\\^'FFFT7$$!3B!RBoTF&>fFFFW7$$!3!=qRrNA:@&FFFZ7$$!3Z[5T-#\\<h%F
FFgn7$$!3#4J\\*R/29RFFFjn7$$!3wFQ0=l]'H$FFF]o7$$!3)eE=r,T*=EFFF`o7$$!3
eSV%*H%QP(>FFFco7$$!3)*RWU;s`+8FFFfo7$$!3.gSQ<NGBo!#>Fio7$$!3y]Py0ul]:
!#?F]p7$$\"3O=#e5YQ8x'F[pFap7$$\"3I^(**zOz+G\"FFFdp7$$\"3dNv()\\qFJ>FF
Fgp7$$\"3y0j*\\(z-/EFFFjp7$$\"3_\"oMd]$=iKFFF]q7$$\"3SH4XBG)*)*QFFF`q7
$$\"3=DLY%*)Rgg%FFFcq7$$\"3U6;S?@OT_FFFfq7$$\"3KMYS.Uq>fFFFiq7$$\"3g!o
i?&HQMlFFF\\r7$$\"3Jr$zA=*Q1sFFF_r7$$\"3gHJCuYpQyFFFbr7$$\"3)fG\"*Q5O'
*\\)FFFer7$$\"3#=wm/;Fe9*FFFhr7$$\"37E5$3JHB#)*FFF[s7$$\"31h>eG\")QZ5F
*F^s7$$\"3PK#)fG(=S6\"F*Fas7$$\"330`g$f(4!=\"F*Fds7$$\"3&oOhy?<3C\"F*F
gs7$$\"3XnP+Q(3/J\"F*Fjs7$$\"3/%yp@B_EP\"F*F]t7$$\"3cNK@zn,R9F*F`t7$$
\"3*p@f'=h`-:F*Fct7$$\"3+++lBjzq:F*Fft-%'COLOURG6&%$RGBG$\"#5!\"\"$\"
\"!F`uF_u-F$6$7S7$F($F^uF`u7$F,$!3m`D4FJcw**FF7$F/$!3U3<el_6=**FF7$F2$
!3\"4/LZA[.\")*FF7$F5$!3'y#3A\\*)Rd'*FF7$F8$!37e%*=Cvch%*FF7$F;$!3%pr9
&*Q3>C*FF7$F>$!3'>>2jDjn(*)FF7$FA$!3;GB9]HOj')FF7$FD$!3\"p*GLjHo7$)FF7
$FH$!3'yIzL\"zr8zFF7$FK$!3$4mG5]X;`(FF7$FN$!3#)**zF%Rc*oqFF7$FQ$!3CPT#
\\TE<d'FF7$FT$!3qviKn?vjgFF7$FW$!3E(yNitD)zbFF7$FZ$!3'[fQ.S(zy\\FF7$Fg
n$!3[\"[%=0f+]WFF7$Fjn$!3:A+*3E%*[\"QFF7$F]o$!3d#oIW5DrB$FF7$F`o$!3[/&
ebI0\"*e#FF7$Fco$!3!e^J'*R[4'>FF7$Ffo$!3Y97m2T(oH\"FF7$Fio$!3%p\"pQ+-*
z\"oF[p7$F]p$!3W[zM%yc1b\"F_p7$Fap$\"3w$p_/5lhw'F[p7$Fdp$\"3!)4c_Fjew7
FF7$Fgp$\"3-&4Sv&QH>>FF7$Fjp$\"3CoO$\\!zpuDFF7$F]q$\"3!4\"))*Q>JY?$FF7
$F`q$\"3>l%fdLV4!QFF7$Fcq$\"3&[Dw?K#*[W%FF7$Ffq$\"3\"*eoz8Ol/]FF7$Fiq$
\"3U?'z@Ws*zbFF7$F\\r$\"3Y(GhB%**>zgFF7$F_r$\"3!ogrtz[')f'FF7$Fbr$\"3U
]!e4/]-1(FF7$Fer$\"3=L,8\")Qc7vFF7$Fhr$\"3>!)Qd?13BzFF7$F[s$\"3G\"fd*>
4R<$)FF7$F^s$\"3))[j\\nn?h')FF7$Fas$\"3_3O')>Uyu*)FF7$Fds$\"3o5rJQ=VY#
*FF7$Fgs$\"35a:'o4\"\\g%*FF7$Fjs$\"3qln3SE!Hm*FF7$F]t$\"3MurX7gL/)*FF7
$F`t$\"3?l,yOjH8**FF7$Fct$\"3c3edk<rw**FF7$Fft$\"\"\"F`u-Fit6&F[uF_uF
\\uF_u-%+AXESLABELSG6$Q\"x6\"Q!6\"-%%VIEWG6$;$!+Fjzq:!\"*$\"+Fjzq:Fe_l
%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "C
urve 1" "Curve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "The graph ce
rtainly seems to confirm that the line " }{XPPEDIT 18 0 "y = x;" "6#/%
\"yG%\"xG" }{TEXT -1 28 " is tangent to the graph of " }{XPPEDIT 18 0 
"sin(x);" "6#-%$sinG6#%\"xG" }{TEXT -1 42 " at the origin, so the diff
erence between " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "sin(x);" "6#-%$sinG6#%\"xG" }{TEXT -1 20 " is very smal
l when " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 31 " is close to 0.  \+
Of course, as " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 163 " gets lar
ger, the sine graph curves away from the tangent line, so the linear a
pproximation is no longer good.  Can we find a better approximation?\n
\nSince we know " }{XPPEDIT 18 0 "sin(x);" "6#-%$sinG6#%\"xG" }{TEXT 
-1 5 " and " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 21 " are close fo
r small " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 44 ", let's see how \+
their difference behaves as " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 
11 " goes to 0." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "limit(si
n(x) - x, x=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 116 "Unfortunately, this limit doesn't tell u
s anything we didn't already know.  Once again, what we need to find o
ut is " }{TEXT 257 8 "how fast" }{TEXT -1 2 "  " }{XPPEDIT 18 0 "sin(x
)-x;" "6#,&-%$sinG6#%\"xG\"\"\"F'!\"\"" }{TEXT -1 48 "  goes to 0: how
 big (or small) is it for small " }{XPPEDIT 18 0 "x;" "6#%\"xG" }
{TEXT -1 30 " ?  We compare with powers of " }{XPPEDIT 18 0 "x;" "6#%
\"xG" }{TEXT -1 40 " in the same way as before: by dividing." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "limit( (sin(x) - x)/x, x=0 )
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 98 "(If you think about it, you can probably see that this li
mit still isn't telling us anything new.)" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 31 "limit( (sin(x) - x)/x^2, x=0 );" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "(This l
imit is giving new information:  " }{XPPEDIT 18 0 "x;" "6#%\"xG" }
{TEXT -1 44 " is apparently a very good approximation to " }{XPPEDIT 
18 0 "sin(x);" "6#-%$sinG6#%\"xG" }{TEXT -1 11 " for small " }
{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 3 " .)" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 31 "limit( (sin(x) - x)/x^3, x=0 );" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6##!\"\"\"\"'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
81 "Aha!  The non-zero value of this last limit shows that we have fou
nd how quickly " }{XPPEDIT 18 0 "sin(x)-x;" "6#,&-%$sinG6#%\"xG\"\"\"F
'!\"\"" }{TEXT -1 14 " goes to 0 as " }{XPPEDIT 18 0 "x;" "6#%\"xG" }
{TEXT -1 38 " goes to 0:  it goes about as fast as " }{XPPEDIT 18 0 "x
^3;" "6#*$%\"xG\"\"$" }{TEXT -1 29 ".  In fact, we now know that " }
{XPPEDIT 18 0 "sin(x)-x;" "6#,&-%$sinG6#%\"xG\"\"\"F'!\"\"" }{TEXT -1 
27 " is approximately equal to " }{XPPEDIT 18 0 "(-1/6)*x^3;" "6#*&,$*
&\"\"\"F&\"\"'!\"\"F(F&*$%\"xG\"\"$F&" }{TEXT -1 11 " for small " }
{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 11 " , or that " }{XPPEDIT 18 
0 "sin(x);" "6#-%$sinG6#%\"xG" }{TEXT -1 27 " is approximately equal t
o " }{XPPEDIT 18 0 "x-x^3/6;" "6#,&%\"xG\"\"\"*&F$\"\"$\"\"'!\"\"F)" }
{TEXT -1 53 " .  We can check this computation again with a graph." }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "p3 := x -> x - x^3/6;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p3GR6#%\"xG6\"6$%)operatorG%&arrowG
F(,&9$\"\"\"*&#F.\"\"'F.*$)F-\"\"$F.F.!\"\"F(F(F(" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 38 "plot( \{sin(x),p3(x)\}, x=-Pi/2..Pi/2 );" }}
{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7S7$
$!3+++lBjzq:!#<$!3wWhB+BK[#*!#=7$$!3WNzQW&=B]\"F*$!3q#p\"Q\"[h?P*F-7$$
!3.42![ROFW\"F*$!31[p**f!*HA%*F-7$$!3?xt3O+tv8F*$!3eC/7#G8xT*F-7$$!3xx
It:&z#38F*$!3mWYIf+s]$*F-7$$!3\")zz#fd\\6C\"F*$!3Q')H*o&H%\\A*F-7$$!3W
:#)\\G:\"*y6F*$!3?')f\"*o&*He!*F-7$$!3,%4``jnW6\"F*$!3#eE<1cdw$))F-7$$
!3)yYu)o'>y/\"F*$!3-f!pO#e\"3c)F-7$$!3EWfpKW&Q\")*F-$!3Q-*H5zQ&Q#)F-7$
$!3(fe1Vw'\\I\"*F-$!3#yI6xav='yF-7$$!333*H!e\\fG&)F-$!3+w?ws&)o%\\(F-7
$$!35`u@%3'*4&yF-$!3M43%4VhW/(F-7$$!3%ewEV#\\hqrF-$!3;PU&G\\?hb'F-7$$!
3SI*3_gT\\^'F-$!3IZ(RlfpS0'F-7$$!3B!RBoTF&>fF-$!3'=RT_2>Qd&F-7$$!3!=qR
rNA:@&F-$!3)zdbLT9c(\\F-7$$!3Z[5T-#\\<h%F-$!3o,(>CHw#[WF-7$$!3#4J\\*R/
29RF-$!3S'[\"z<:89QF-7$$!3wFQ0=l]'H$F-$!3j!*R%o`,oB$F-7$$!3)eE=r,T*=EF
-$!3G>J<-G+*e#F-7$$!3eSV%*H%QP(>F-$!3%o'y\\hM#4'>F-7$$!3)*RWU;s`+8F-$!
3q=GS35(oH\"F-7$$!3.gSQ<NGBo!#>$!3T=m-o*))z\"oFir7$$!3y]Py0ul]:!#?$!3'
Q(yM%yc1b\"F_s7$$\"3O=#e5YQ8x'Fir$\"3aP3H9R;mnFir7$$\"3I^(**zOz+G\"F-$
\"3!zWVWY$ew7F-7$$\"3dNv()\\qFJ>F-$\"3Y*=a!)[r#>>F-7$$\"3y0j*\\(z-/EF-
$\"3U.'z]G)fuDF-7$$\"3_\"oMd]$=iKF-$\"3]-Ga2TK/KF-7$$\"3SH4XBG)*)*QF-$
\"3a,Obf^>+QF-7$$\"3=DLY%*)Rgg%F-$\"3'>tZvOtJW%F-7$$\"3U6;S?@OT_F-$\"3
;Bwm&py8+&F-7$$\"3KMYS.Uq>fF-$\"3mRr<()['Rd&F-7$$\"3g!oi?&HQMlF-$\"3m%
pq7ts$pgF-7$$\"3Jr$zA=*Q1sF-$\"3=c%=ij^Ee'F-7$$\"3gHJCuYpQyF-$\"3nkC%R
EXf.(F-7$$\"3)fG\"*Q5O'*\\)F-$\"3'f5m!*)eAwuF-7$$\"3#=wm/;Fe9*F-$\"3%y
)4vT]!3(yF-7$$\"37E5$3JHB#)*F-$\"3yfId#**GHC)F-7$$\"31h>eG\")QZ5F*$\"3
[u$y=Wo)e&)F-7$$\"3PK#)fG(=S6\"F*$\"3[]Wym^&f$))F-7$$\"330`g$f(4!=\"F*
$\"3=\"35lw4>1*F-7$$\"3&oOhy?<3C\"F*$\"3k_8Yw&yTA*F-7$$\"3XnP+Q(3/J\"F
*$\"3aa>'exgPN*F-7$$\"3/%yp@B_EP\"F*$\"3aHYF**e*fT*F-7$$\"3cNK@zn,R9F*
$\"3_osUm[rB%*F-7$$\"3*p@f'=h`-:F*$\"3c;`D&e\"yr$*F-7$$\"3+++lBjzq:F*$
\"3wWhB+BK[#*F--%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!Fb[lFa[l-F$6$7S7$F($F
`[lFb[l7$F/$!3m`D4FJcw**F-7$F4$!3U3<el_6=**F-7$F9$!3\"4/LZA[.\")*F-7$F
>$!3'y#3A\\*)Rd'*F-7$FC$!37e%*=Cvch%*F-7$FH$!3%pr9&*Q3>C*F-7$FM$!3'>>2
jDjn(*)F-7$FR$!3;GB9]HOj')F-7$FW$!3\"p*GLjHo7$)F-7$Ffn$!3'yIzL\"zr8zF-
7$F[o$!3$4mG5]X;`(F-7$F`o$!3#)**zF%Rc*oqF-7$Feo$!3CPT#\\TE<d'F-7$Fjo$!
3qviKn?vjgF-7$F_p$!3E(yNitD)zbF-7$Fdp$!3'[fQ.S(zy\\F-7$Fip$!3[\"[%=0f+
]WF-7$F^q$!3:A+*3E%*[\"QF-7$Fcq$!3d#oIW5DrB$F-7$Fhq$!3[/&ebI0\"*e#F-7$
F]r$!3!e^J'*R[4'>F-7$Fbr$!3Y97m2T(oH\"F-7$Fgr$!3%p\"pQ+-*z\"oFir7$F]s$
!3W[zM%yc1b\"F_s7$Fcs$\"3w$p_/5lhw'Fir7$Fhs$\"3!)4c_Fjew7F-7$F]t$\"3-&
4Sv&QH>>F-7$Fbt$\"3CoO$\\!zpuDF-7$Fgt$\"3!4\"))*Q>JY?$F-7$F\\u$\"3>l%f
dLV4!QF-7$Fau$\"3&[Dw?K#*[W%F-7$Ffu$\"3\"*eoz8Ol/]F-7$F[v$\"3U?'z@Ws*z
bF-7$F`v$\"3Y(GhB%**>zgF-7$Fev$\"3!ogrtz[')f'F-7$Fjv$\"3U]!e4/]-1(F-7$
F_w$\"3=L,8\")Qc7vF-7$Fdw$\"3>!)Qd?13BzF-7$Fiw$\"3G\"fd*>4R<$)F-7$F^x$
\"3))[j\\nn?h')F-7$Fcx$\"3_3O')>Uyu*)F-7$Fhx$\"3o5rJQ=VY#*F-7$F]y$\"35
a:'o4\"\\g%*F-7$Fby$\"3qln3SE!Hm*F-7$Fgy$\"3MurX7gL/)*F-7$F\\z$\"3?l,y
OjH8**F-7$Faz$\"3c3edk<rw**F-7$Ffz$\"\"\"Fb[l-F[[l6&F][lFa[lF^[lFa[l-%
+AXESLABELSG6$Q\"x6\"Q!6\"-%%VIEWG6$;$!+Fjzq:!\"*$\"+Fjzq:Fgel%(DEFAUL
TG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" 
"Curve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 100 "This looks promisin
g: if you compare it to our earlier graph, you can see that the cubic \+
polynomial " }{XPPEDIT 18 0 "p3;" "6#%#p3G" }{TEXT -1 170 " stays clos
er to the graph of sine for longer than the tangent line did.  Can we \+
repeat the process and get a better approximation still?  We will look
 at the difference " }{XPPEDIT 18 0 "sin(x)-p3(x);" "6#,&-%$sinG6#%\"x
G\"\"\"-%#p3G6#F'!\"\"" }{TEXT -1 87 ", and see how fast it goes to 0.
  Hopefully, it goes at least as fast (or faster) than " }{XPPEDIT 18 
0 "x^3;" "6#*$%\"xG\"\"$" }{TEXT -1 32 ", so we'll start by dividing b
y " }{XPPEDIT 18 0 "x^3;" "6#*$%\"xG\"\"$" }{TEXT -1 30 " and work up \+
to higher powers." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "limit(
 (sin(x) - (x - x^3/6))/x^3, x=0 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "limit( (sin(x) - \+
(x - x^3/6))/x^4, x=0 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "limit( (sin(x) - (x - x^3/6)
)/x^5, x=0 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\"\"$?\"" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "We see that " }{XPPEDIT 18 0 "sin(
x)-p3(x);" "6#,&-%$sinG6#%\"xG\"\"\"-%#p3G6#F'!\"\"" }{TEXT -1 14 " be
haves like " }{XPPEDIT 18 0 "x^5/120;" "6#*&%\"xG\"\"&\"$?\"!\"\"" }
{TEXT -1 4 " as " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 15 " goes to
 0, so " }{XPPEDIT 18 0 "sin(x);" "6#-%$sinG6#%\"xG" }{TEXT -1 26 " is
 approximately equal to" }{XPPEDIT 18 0 "p3(x)+x^5/120 = x-x^3/6+x^5/1
20;" "6#/,&-%#p3G6#%\"xG\"\"\"*&F(\"\"&\"$?\"!\"\"F),(F(F)*&F(\"\"$\"
\"'F-F-*&F(F+F,F-F)" }{TEXT -1 2 " ." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 31 "p5 := x -> x - x^3/6 + x^5/120;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#p5GR6#%\"xG6\"6$%)operatorG%&arrowGF(,(9$\"\"\"*&#F.
\"\"'F.*$)F-\"\"$F.F.!\"\"*&#F.\"$?\"F.)F-\"\"&F.F.F(F(F(" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "plot( \{sin(x),p5(x)\}, x=-Pi/2..Pi
/2 );" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURV
ESG6$7S7$$!3+++lBjzq:!#<$!3?<?Zb[_/5F*7$$!3WNzQW&=B]\"F*$!3/oZL!)z(4+
\"F*7$$!3.42![ROFW\"F*$!3m<bZh#)>V**!#=7$$!3?xt3O+tv8F*$!3$3o%Q'fv$G)*
F77$$!3xxIt:&z#38F*$!3i_,`n,6q'*F77$$!3\")zz#fd\\6C\"F*$!39x3w\"ez.Z*F
77$$!3W:#)\\G:\"*y6F*$!3Q3\"eePo![#*F77$$!3,%4``jnW6\"F*$!3U(H5P#y#4)*
)F77$$!3)yYu)o'>y/\"F*$!3;]*eW%H2m')F77$$!3EWfpKW&Q\")*F7$!3[VVN&[*R9$
)F77$$!3(fe1Vw'\\I\"*F7$!3aU5fnav9zF77$$!333*H!e\\fG&)F7$!3%*G-'>;!HKv
F77$$!35`u@%3'*4&yF7$!3+S<oy!=$pqF77$$!3%ewEV#\\hqrF7$!3o!**pOW=>d'F77
$$!3SI*3_gT\\^'F7$!3W<vDH.&Q1'F77$$!3B!RBoTF&>fF7$!3]/c]Cg()zbF77$$!3!
=qRrNA:@&F7$!3k#oO\"R!=)y\\F77$$!3Z[5T-#\\<h%F7$!35N;?#o9+X%F77$$!3#4J
\\*R/29RF7$!3vlsIZq*[\"QF77$$!3wFQ0=l]'H$F7$!381F_Uf7PKF77$$!3)eE=r,T*
=EF7$!3K6m1ta5*e#F77$$!3eSV%*H%QP(>F7$!3iE9xA%[4'>F77$$!3)*RWU;s`+8F7$
!3GP&4*3T(oH\"F77$$!3.gSQ<NGBo!#>$!3GJN_+-*z\"oFir7$$!3y]Py0ul]:!#?$!3
m[zM%yc1b\"F_s7$$\"3O=#e5YQ8x'Fir$\"3c\">#e+^;mnFir7$$\"3I^(**zOz+G\"F
7$\"3u/GkGjew7F77$$\"3dNv()\\qFJ>F7$\"3Y!y7u(QH>>F77$$\"3y0j*\\(z-/EF7
$\"3e2'ze1)puDF77$$\"3_\"oMd]$=iKF7$\"3'3F2G(>j/KF77$$\"3SH4XBG)*)*QF7
$\"3G:\"**y/Y4!QF77$$\"3=DLY%*)Rgg%F7$\"33GxPB5!\\W%F77$$\"3U6;S?@OT_F
7$\"33^g@$4vY+&F77$$\"3KMYS.Uq>fF7$\"3#zy[4uA+e&F77$$\"3g!oi?&HQMlF7$
\"3a\"Q1=F+$zgF77$$\"3Jr$zA=*Q1sF7$\"3'=1)Q>w%))f'F77$$\"3gHJCuYpQyF7$
\"3!*4Lc'y211(F77$$\"3)fG\"*Q5O'*\\)F7$\"38qph<M>8vF77$$\"3#=wm/;Fe9*F
7$\"3$o1&G!RIT#zF77$$\"37E5$3JHB#)*F7$\"3'o-@a!y6>$)F77$$\"31h>eG\")QZ
5F*$\"3g,:`#**3Rm)F77$$\"3PK#)fG(=S6\"F*$\"3CaeW;r$*y*)F77$$\"330`g$f(
4!=\"F*$\"3Y4&[x4NED*F77$$\"3&oOhy?<3C\"F*$\"3WuaI\\nGp%*F77$$\"3XnP+Q
(3/J\"F*$\"3AjK\\\"Redn*F77$$\"3/%yp@B_EP\"F*$\"3,@$zA-&3A)*F77$$\"3cN
K@zn,R9F*$\"3Vw\"zlxLz$**F77$$\"3*p@f'=h`-:F*$\"3lyv3$='*4+\"F*7$$\"3+
++lBjzq:F*$\"3?<?Zb[_/5F*-%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!Fb[lFa[l-F$
6$7S7$F($F`[lFb[l7$F.$!3m`D4FJcw**F77$F3$!3U3<el_6=**F77$F9$!3\"4/LZA[
.\")*F77$F>$!3'y#3A\\*)Rd'*F77$FC$!37e%*=Cvch%*F77$FH$!3%pr9&*Q3>C*F77
$FM$!3'>>2jDjn(*)F77$FR$!3;GB9]HOj')F77$FW$!3\"p*GLjHo7$)F77$Ffn$!3'yI
zL\"zr8zF77$F[o$!3$4mG5]X;`(F77$F`o$!3#)**zF%Rc*oqF77$Feo$!3CPT#\\TE<d
'F77$Fjo$!3qviKn?vjgF77$F_p$!3E(yNitD)zbF77$Fdp$!3'[fQ.S(zy\\F77$Fip$!
3[\"[%=0f+]WF77$F^q$!3:A+*3E%*[\"QF77$Fcq$!3d#oIW5DrB$F77$Fhq$!3[/&ebI
0\"*e#F77$F]r$!3!e^J'*R[4'>F77$Fbr$!3Y97m2T(oH\"F77$Fgr$!3%p\"pQ+-*z\"
oFir7$F]s$!3W[zM%yc1b\"F_s7$Fcs$\"3w$p_/5lhw'Fir7$Fhs$\"3!)4c_Fjew7F77
$F]t$\"3-&4Sv&QH>>F77$Fbt$\"3CoO$\\!zpuDF77$Fgt$\"3!4\"))*Q>JY?$F77$F
\\u$\"3>l%fdLV4!QF77$Fau$\"3&[Dw?K#*[W%F77$Ffu$\"3\"*eoz8Ol/]F77$F[v$
\"3U?'z@Ws*zbF77$F`v$\"3Y(GhB%**>zgF77$Fev$\"3!ogrtz[')f'F77$Fjv$\"3U]
!e4/]-1(F77$F_w$\"3=L,8\")Qc7vF77$Fdw$\"3>!)Qd?13BzF77$Fiw$\"3G\"fd*>4
R<$)F77$F^x$\"3))[j\\nn?h')F77$Fcx$\"3_3O')>Uyu*)F77$Fhx$\"3o5rJQ=VY#*
F77$F]y$\"35a:'o4\"\\g%*F77$Fby$\"3qln3SE!Hm*F77$Fgy$\"3MurX7gL/)*F77$
F\\z$\"3?l,yOjH8**F77$Faz$\"3c3edk<rw**F77$Ffz$\"\"\"Fb[l-F[[l6&F][lFa
[lF^[lFa[l-%+AXESLABELSG6$Q\"x6\"Q!6\"-%%VIEWG6$;$!+Fjzq:!\"*$\"+Fjzq:
Fgel%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 
0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "The appr
oximation is now almost perfect over the interval " }{XPPEDIT 18 0 "[-
Pi/2, Pi/2];" "6#7$,$*&%#PiG\"\"\"\"\"#!\"\"F)*&F&F'F(F)" }{TEXT -1 
46 " , so let's look at it over a longer interval." }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 34 "plot( \{sin(x),p5(x)\}, x=-Pi..Pi );" }}
{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7]o7
$$!3*****4tk#fTJ!#<$!3a,/RE\"R/C&!#=7$$!3E[<mdPN2JF*$!31WqmM%G'4_F-7$$
!35'\\8!o[6tIF*$!3sKmH&o:5?&F-7$$!3#RCl$yf()QIF*$!3c$=!=*o&38_F-7$$!3w
\"*pr)3PY+$F*$!3!eZe2\\lVC&F-7$$!3)**z*4R\\0XHF*$!3.9A_fsOS`F-7$$!3?3E
[*ysa)GF*$!3?\"\\t&RK)H[&F-7$$!3gX&))>2g9v#F*$!3D<m\"*[\")4RfF-7$$!3)4
577.flh#F*$!3W#>j[*HZHlF-7$$!3u\"QM::*H#[#F*$!3s))>%o4TX=(F-7$$!3y\\Ah
cI#yN#F*$!3gvEZlmI/yF-7$$!3&e=d-FN*GAF*$!3O+`x%y]yT)F-7$$!3u%*GBP$Rc4#
F*$!3sh\\=CVd&)*)F-7$$!3mCj&f)3xi>F*$!3GW*RA&fs_%*F-7$$!3(or4AN*4E=F*$
!3Cq;LL7=/)*F-7$$!3ix<br\"4fw\"F*$!3gr'QsZQ?\"**F-7$$!3QQQ*3**=dq\"F*$
!3yQY)*)o*=*)**F-7$$!3f9&yM5fzj\"F*$!3uul@&G$z.5F*7$$!3e!>jg@*>q:F*$!3
wUr(pw7X+\"F*7$$!3b.*R+5h@]\"F*$!3%y;=0yk4+\"F*7$$!3`;m,%)H7M9F*$!3Hh@
s@)33$**F-7$$!39d=2_cbo8F*$!3(H`fR&Qd8)*F-7$$!3v(4F,K))HI\"F*$!3i$H_be
!*el*F-7$$!3S*4!R#[0R=\"F*$!35P?tt/@n#*F-7$$!3pY@QqWIU5F*$!3)GF5P=(HQ'
)F-7$$!3Wp3w$R)\\B#*F-$!39jMA\"fV8(zF-7$$!3EJc8o39GyF-$!3K'3)zQk7`qF-7
$$!3'=[BP-8If'F-$!3%zvd$eTvDhF-7$$!3CB3<@?)yB&F-$!3aMF&>xh;+&F-7$$!3_*
)R<YoZZRF-$!3w(y5/'evXQF-7$$!38TqX=W2,EF-$!33oJ)4(R%=d#F-7$$!3qTJY)ocY
O\"F-$!3U!*o5$*\\Ug8F-7$$!38,z7VKJ,J!#?$!3q=3lr#385$Ffu7$$\"39SFf3xEa8
F-$\"3s\"=hr'=8]8F-7$$\"3'R2()Hve,c#F-$\"3q'Qi$yJGKDF-7$$\"3IcwR<TbiQF
-$\"39%3')43Esw$F-7$$\"3*4*=Jof03_F-$\"3C6,:K9\"e(\\F-7$$\"3q]1XIqOClF
-$\"3i+?[HNMrgF-7$$\"3%HoBlmlzz(F-$\"3XkztwFpJqF-7$$\"3^u'f#4)z?@*F-$
\"3o&QxCdKW'zF-7$$\"39Hv<ECF[5F*$\"3oGa)\\dT$o')F-7$$\"3Olj%G%3%R=\"F*
$\"3%*oD$f?XtE*F-7$$\"3\\-)REfwoI\"F*$\"3-Z6*3Wljm*F-7$$\"3_o_p:s2u8F*
$\"3k+*[kc3]#)*F-7$$\"3cM2vQyFT9F*$\"3MA\"Q2Y\\6%**F-7$$\"3=6*yzQ3X]\"
F*$\"3r!e>R;f6+\"F*7$$\"3!y32s$*Qxc\"F*$\"3Zl&[!z%fW+\"F*7$$\"3&R90-3L
Qj\"F*$\"3]0:Mgp&R+\"F*7$$\"35+K?Bs#**p\"F*$\"3[[#e/S^\\***F-7$$\"39YK
*)Gjak<F*$\"3*e9e$3(HT\"**F-7$$\"3<#H$eMa;H=F*$\"3C#pc\">&))yz*F-7$$\"
3%>>CZ'eYk>F*$\"3w&RG)y!)\\Z%*F-7$$\"3kYuyfix%4#F*$\"3E:?tIu&*))*)F-7$
$\"3T4q))fu.GAF*$\"3JVnH&**R>U)F-7$$\"3y?w'**=&>gBF*$\"3c'))==5LEz(F-7
$$\"3!*>3a=Wj\"[#F*$\"31%zfpldy=(F-7$$\"3A?c*)yu\"3i#F*$\"3z=\"RLh&[4l
F-7$$\"3-i-HnWIXFF*$\"3AE*4$)=&ojfF-7$$\"3u=RWhN.yGF*$\"3;p7&\\H<P]&F-
7$$\"3tX=#4!HbTHF*$\"3Gn\"G=?HvM&F-7$$\"3ss(*RSA20IF*$\"3$*y'G(=M&QC&F
-7$$\"3;HB\"HM-#RIF*$\"3tYpQ]w(G@&F-7$$\"3/')[UXCLtIF*$\"3\\7rrNN+,_F-
7$$\"3#HWPzaiu5$F*$\"3*)3&o[O!p4_F-7$$\"3!)***\\/l#fTJF*$\"3pt6GI\"R/C
&F--%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!Fa`lF``l-F$6$7en7$F($!3=5KT_Kzzi!
#E7$F>$!3UWkbU#y_O\"F-7$FH$!34`kI;!*GLDF-7$FM$!3mJ^0x.6.QF-7$FR$!3G?))
Q')4U7]F-7$FW$!3`Zq;6deDhF-7$Ffn$!37nDKOFafqF-7$F[o$!3-uO$)y!>8\"zF-7$
F`o$!3+YUphh-a')F-7$Feo$!32R0rFbcT#*F-7$Fjo$!3ufg#\\9oen*F-7$F_p$!3sNz
lxzD5)*F-7$Fdp$!3@6I/pt64**F-7$Fip$!3uM'fSGau(**F-7$F^q$!2Q#RH<#)*****
*F*7$Fcq$!3'\\s#Gt_Xw**F-7$Fhq$!3)3![&GGZn!**F-7$F]r$!3k-]%fk*='z*F-7$
Fbr$!3HAO\\SD`V'*F-7$Fgr$!3))pm;Cl'3E*F-7$F\\s$!3$*e_me]oN')F-7$Fas$!3
ik]JA+BqzF-7$Ffs$!3I`45V?x_qF-7$F[t$!3rZ9lmtkDhF-7$F`t$!35W$e?RS;+&F-7
$Fet$!3%\\2/L!HvXQF-7$Fjt$!3'>R58\"Q%=d#F-7$F_u$!3/q&e8*\\Ug8F-Fcu7$Fj
u$\"37AR]l=8]8F-7$F_v$\"3+F!fa.$GKDF-7$Fdv$\"33j2BTNAnPF-7$Fiv$\"3I+8:
*)3zv\\F-7$F^w$\"3iGkzpUCrgF-7$Fcw$\"3E#Ry7\"yMJqF-7$Fhw$\"3[$yTdeGL'z
F-7$F]x$\"3eLa()3Mil')F-7$Fbx$\"37!GJs$***4E*F-7$Fgx$\"3E:8W$p[Pl*F-7$
F\\y$\"3g&\\o%Q682)*F-7$Fay$\"3ZfD`\"*>C;**F-7$Ffy$\"3i]On*eP!y**F-7$F
[z$\"3Y\\4)=E`*****F-7$F`z$\"3'>2!z;%Q,)**F-7$Fez$\"3nG)er%=u;**F-7$Fj
z$\"3%z'z236*G\")*F-7$F_[l$\"3#)Q>7M'z!o'*F-7$Fd[l$\"3-'f-+?x]B*F-7$Fi
[l$\"33yU$=iZ$e')F-7$F^\\l$\"3Jjs%pI2o\"zF-7$Fc\\l$\"3ycCmz?sUqF-7$Fh
\\l$\"3B,&=XdQ38'F-7$F]]l$\"3[6)p&**p_v\\F-7$Fb]l$\"3_^@G2'o*fQF-7$Fg]
l$\"3H4EwtQ=0EF-7$Fa^l$\"3%eLwJMn4O\"F-7$Fe_l$\"3pawpOMzRJFh`l-Fj_l6&F
\\`lF``lF]`lF``l-%+AXESLABELSG6$Q\"x6\"Q!6\"-%%VIEWG6$;$!+aEfTJ!\"*$\"
+aEfTJF]\\m%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 
45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
64 "We can now see that the approximation is reasonably good out to " 
}{XPPEDIT 18 0 "x = 2;" "6#/%\"xG\"\"#" }{TEXT -1 52 " (or -2), but if
 we wanted to approximate as far as " }{XPPEDIT 18 0 "x = Pi;" "6#/%\"
xG%#PiG" }{TEXT -1 32 " we would still need more terms." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 
-1 10 "Question 1" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 94 "By repeating \+
the procedure illustrated above, find the next two terms in the approx
imation of " }{XPPEDIT 18 0 "sin(x);" "6#-%$sinG6#%\"xG" }{TEXT -1 1 "
." }}}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 3 
"" 0 "" {TEXT -1 33 "Approximating the cosine function" }}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 39 "This time, we can start by noting that " 
}{XPPEDIT 18 0 "cos(0) = 1;" "6#/-%$cosG6#\"\"!\"\"\"" }{TEXT -1 68 ".
  This already gives us the linear approximation (tangent line) to " }
{XPPEDIT 18 0 "cos(x);" "6#-%$cosG6#%\"xG" }{TEXT -1 4 " at " }
{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" }{TEXT -1 1 ":" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "plot( \{cos(x),1\}, x=-Pi/2..Pi/2 )
;" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6
$7S7$$!3+++lBjzq:!#<$\"\"\"\"\"!7$$!3WNzQW&=B]\"F*F+7$$!3.42![ROFW\"F*
F+7$$!3?xt3O+tv8F*F+7$$!3xxIt:&z#38F*F+7$$!3\")zz#fd\\6C\"F*F+7$$!3W:#
)\\G:\"*y6F*F+7$$!3,%4``jnW6\"F*F+7$$!3)yYu)o'>y/\"F*F+7$$!3EWfpKW&Q\"
)*!#=F+7$$!3(fe1Vw'\\I\"*FIF+7$$!333*H!e\\fG&)FIF+7$$!35`u@%3'*4&yFIF+
7$$!3%ewEV#\\hqrFIF+7$$!3SI*3_gT\\^'FIF+7$$!3B!RBoTF&>fFIF+7$$!3!=qRrN
A:@&FIF+7$$!3Z[5T-#\\<h%FIF+7$$!3#4J\\*R/29RFIF+7$$!3wFQ0=l]'H$FIF+7$$
!3)eE=r,T*=EFIF+7$$!3eSV%*H%QP(>FIF+7$$!3)*RWU;s`+8FIF+7$$!3.gSQ<NGBo!
#>F+7$$!3y]Py0ul]:!#?F+7$$\"3O=#e5YQ8x'F^pF+7$$\"3I^(**zOz+G\"FIF+7$$
\"3dNv()\\qFJ>FIF+7$$\"3y0j*\\(z-/EFIF+7$$\"3_\"oMd]$=iKFIF+7$$\"3SH4X
BG)*)*QFIF+7$$\"3=DLY%*)Rgg%FIF+7$$\"3U6;S?@OT_FIF+7$$\"3KMYS.Uq>fFIF+
7$$\"3g!oi?&HQMlFIF+7$$\"3Jr$zA=*Q1sFIF+7$$\"3gHJCuYpQyFIF+7$$\"3)fG\"
*Q5O'*\\)FIF+7$$\"3#=wm/;Fe9*FIF+7$$\"37E5$3JHB#)*FIF+7$$\"31h>eG\")QZ
5F*F+7$$\"3PK#)fG(=S6\"F*F+7$$\"330`g$f(4!=\"F*F+7$$\"3&oOhy?<3C\"F*F+
7$$\"3XnP+Q(3/J\"F*F+7$$\"3/%yp@B_EP\"F*F+7$$\"3cNK@zn,R9F*F+7$$\"3*p@
f'=h`-:F*F+7$$\"3+++lBjzq:F*F+-%'COLOURG6&%$RGBG$\"#5!\"\"$F-F-Fbu-F$6
$7S7$F($\"3&pI?mi'*[9$!#E7$F/$\"3/+Gv?xUUoF^p7$F2$\"3gb6eB?5x7FI7$F5$
\"3m&\\2))o:$Q>FI7$F8$\"3?'*G%4c>^f#FI7$F;$\"314EY&>)3PKFI7$F>$\"3%zp)
3e+J>QFI7$FA$\"3-PRxsVb1WFI7$FD$\"3mj(RcL6Y*\\FI7$FG$\"3#39Fb09(ebFI7$
FK$\"3`Z6(>o^L6'FI7$FN$\"3ugk,u;KylFI7$FQ$\"3!p!G<*fyJ2(FI7$FT$\"3xM,Z
t3SPvFI7$FW$\"3#)=n:Ymy^zFI7$FZ$\"3#>F(fhm_)H)FI7$Fgn$\"3gh=&o!3Ys')FI
7$Fjn$\"33=YcbFIb*)FI7$F]o$\"34?j_E?tV#*FI7$F`o$\"3))3cV'*[bh%*FI7$Fco
$\"3u.9#>38!f'*FI7$Ffo$\"39eW&[M\\e!)*FI7$Fio$\"3p=!\\7G\\b\"**FI7$F\\
p$\"3sT?<J/tw**FI7$F`p$\"2#e/Jxz)*****F*7$Fdp$\"3'e4OgC$3x**FI7$Fgp$\"
3V4EMb;==**FI7$Fjp$\"3!oa$Hzt39)*FI7$F]q$\"3Q&QuL[jGm*FI7$F`q$\"3CKyKJ
*4EZ*FI7$Fcq$\"3A%yk%psZ\\#*FI7$Ffq$\"3=Y2HT>%y&*)FI7$Fiq$\"35X&f]glvl
)FI7$F\\r$\"3q,*ok,G%)H)FI7$F_r$\"3WFkMvG)*RzFI7$Fbr$\"3temLGG%Q^(FI7$
Fer$\"3i\"H<$e!p=3(FI7$Fhr$\"31%)f\\U[5+mFI7$F[s$\"3#3&>#)fC@,hFI7$F^s
$\"3i[%GE;n;b&FI7$Fas$\"3MF&R1Y\\$)*\\FI7$Fds$\"3YPFIeOe5WFI7$Fgs$\"3[
es=peM3QFI7$Fjs$\"3![m6&RFBSKFI7$F]t$\"3S'*='\\%4buDFI7$F`t$\"35*fH')y
+&o>FI7$Fct$\"3WGV'*zY)RJ\"FI7$Fft$\"3@%*fWC6s?oF^p7$FitFgu-F\\u6&F^uF
buF_uFbu-%+AXESLABELSG6$Q\"x6\"Q!6\"-%%VIEWG6$;$!+Fjzq:!\"*$\"+Fjzq:Fg
_l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 
"Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "Now we fol
low the same procedure we did for the sine function: find how fast the
 difference " }{XPPEDIT 18 0 "cos(x)-1;" "6#,&-%$cosG6#%\"xG\"\"\"F(!
\"\"" }{TEXT -1 11 " goes to 0." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 25 "limit( cos(x) - 1, x=0 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "limit( (cos(x) - 1
)/x, x=0 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 31 "limit( (cos(x) - 1)/x^2, x=0 );" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6##!\"\"\"\"#" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 62 "We have found a non-zero limit, and so the next approxima
tion:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "p2 := x -> 1 - x^2
/2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p2GR6#%\"xG6\"6$%)operatorG%
&arrowGF(,&\"\"\"F-*&#F-\"\"#F-*$)9$F0F-F-!\"\"F(F(F(" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "plot( \{cos(x),p2(x)\}, x=-Pi/2..Pi
/2 );" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURV
ESG6$7S7$$!3+++lBjzq:!#<$\"3&pI?mi'*[9$!#E7$$!3WNzQW&=B]\"F*$\"3/+Gv?x
UUo!#>7$$!3.42![ROFW\"F*$\"3gb6eB?5x7!#=7$$!3?xt3O+tv8F*$\"3m&\\2))o:$
Q>F97$$!3xxIt:&z#38F*$\"3?'*G%4c>^f#F97$$!3\")zz#fd\\6C\"F*$\"314EY&>)
3PKF97$$!3W:#)\\G:\"*y6F*$\"3%zp)3e+J>QF97$$!3,%4``jnW6\"F*$\"3-PRxsVb
1WF97$$!3)yYu)o'>y/\"F*$\"3mj(RcL6Y*\\F97$$!3EWfpKW&Q\")*F9$\"3#39Fb09
(ebF97$$!3(fe1Vw'\\I\"*F9$\"3`Z6(>o^L6'F97$$!333*H!e\\fG&)F9$\"3ugk,u;
KylF97$$!35`u@%3'*4&yF9$\"3!p!G<*fyJ2(F97$$!3%ewEV#\\hqrF9$\"3xM,Zt3SP
vF97$$!3SI*3_gT\\^'F9$\"3#)=n:Ymy^zF97$$!3B!RBoTF&>fF9$\"3#>F(fhm_)H)F
97$$!3!=qRrNA:@&F9$\"3gh=&o!3Ys')F97$$!3Z[5T-#\\<h%F9$\"33=YcbFIb*)F97
$$!3#4J\\*R/29RF9$\"34?j_E?tV#*F97$$!3wFQ0=l]'H$F9$\"3))3cV'*[bh%*F97$
$!3)eE=r,T*=EF9$\"3u.9#>38!f'*F97$$!3eSV%*H%QP(>F9$\"39eW&[M\\e!)*F97$
$!3)*RWU;s`+8F9$\"3p=!\\7G\\b\"**F97$$!3.gSQ<NGBoF3$\"3sT?<J/tw**F97$$
!3y]Py0ul]:!#?$\"2#e/Jxz)*****F*7$$\"3O=#e5YQ8x'F3$\"3'e4OgC$3x**F97$$
\"3I^(**zOz+G\"F9$\"3V4EMb;==**F97$$\"3dNv()\\qFJ>F9$\"3!oa$Hzt39)*F97
$$\"3y0j*\\(z-/EF9$\"3Q&QuL[jGm*F97$$\"3_\"oMd]$=iKF9$\"3CKyKJ*4EZ*F97
$$\"3SH4XBG)*)*QF9$\"3A%yk%psZ\\#*F97$$\"3=DLY%*)Rgg%F9$\"3=Y2HT>%y&*)
F97$$\"3U6;S?@OT_F9$\"35X&f]glvl)F97$$\"3KMYS.Uq>fF9$\"3q,*ok,G%)H)F97
$$\"3g!oi?&HQMlF9$\"3WFkMvG)*RzF97$$\"3Jr$zA=*Q1sF9$\"3temLGG%Q^(F97$$
\"3gHJCuYpQyF9$\"3i\"H<$e!p=3(F97$$\"3)fG\"*Q5O'*\\)F9$\"31%)f\\U[5+mF
97$$\"3#=wm/;Fe9*F9$\"3#3&>#)fC@,hF97$$\"37E5$3JHB#)*F9$\"3i[%GE;n;b&F
97$$\"31h>eG\")QZ5F*$\"3MF&R1Y\\$)*\\F97$$\"3PK#)fG(=S6\"F*$\"3YPFIeOe
5WF97$$\"330`g$f(4!=\"F*$\"3[es=peM3QF97$$\"3&oOhy?<3C\"F*$\"3![m6&RFB
SKF97$$\"3XnP+Q(3/J\"F*$\"3S'*='\\%4buDF97$$\"3/%yp@B_EP\"F*$\"35*fH')
y+&o>F97$$\"3cNK@zn,R9F*$\"3WGV'*zY)RJ\"F97$$\"3*p@f'=h`-:F*$\"3@%*fWC
6s?oF37$$\"3+++lBjzq:F*F+-%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!Fa[lF`[l-F$
6$7S7$F($!3yx<'>X0qL#F97$F/$!3'\\UfS/0[G\"F97$F5$!3yrt7W_TuSF37$F;$\"3
Y$>^.RV$o`F37$F@$\"3s<hcVN-U9F97$FE$\"3+z*o3lQxH#F97$FJ$\"3M1+()R!Q30$
F97$FO$\"3'eT_([%4)*y$F97$FT$\"3N-!*f2(p.^%F97$FY$\"3S3FpeIT%=&F97$Fhn
$\"3Z[$\\=W,<$eF97$F]o$\"3CUL4-M:jjF97$Fbo$\"3'36!GCI4=pF97$Fgo$\"3V76
N!39\"HuF97$F\\p$\"3]%)e.%zwx(yF97$Fap$\"3m5Q2e(fzC)F97$Ffp$\"37_;0O<+
U')F97$F[q$\"3Y_Iqk%)eO*)F97$F`q$\"3\"4vb&HE+M#*F97$Feq$\"3OJ9#)QAlc%*
F97$Fjq$\"3exFW(Rdql*F97$F_r$\"3&G/30$y@0)*F97$Fdr$\"3Q[ZVZ,V:**F97$Fi
r$\"3E(p?5S@n(**F97$F^s$\"2!\\!3t(z)*****F*7$Fds$\"3u'*Gs([uq(**F97$Fi
s$\"3Wn5eS)p!=**F97$F^t$\"3b;3$yW3N\")*F97$Fct$\"3]J=F:>&4m*F97$Fht$\"
3')GcuQz!zY*F97$F]u$\"3UJ<7Zm*)R#*F97$Fbu$\"3D91`C)>#R*)F97$Fgu$\"3_b4
9chSE')F97$F\\v$\"33v'4s5byC)F97$Fav$\"3gI'*yr>4lyF97$Ffv$\"3^VjnZxR.u
F97$F[w$\"3V>'*>!HVx#pF97$F`w$\"3+)Qr]I4yQ'F97$Few$\"3xCMaxAp<eF97$Fjw
$\"3%\\[zaM#4w^F97$F_x$\"3>()y_S0*[^%F97$Fdx$\"3Y*)=mh8\"[z$F97$Fix$\"
3qS[$yM[o.$F97$F^y$\"333cP$Gj=I#F97$Fcy$\"30oo\"opWTT\"F97$Fhy$\"3Qfa'
fZ#H\"z&F37$F]z$!3'yqhGak%QNF37$Fbz$!37BvPRR2)G\"F97$FgzFf[l-Fjz6&F\\[
lF`[lF][lF`[l-%+AXESLABELSG6$Q\"x6\"Q!6\"-%%VIEWG6$;$!+Fjzq:!\"*$\"+Fj
zq:Feel%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 
45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
33 "Now find how fast the difference " }{XPPEDIT 18 0 "cos(x)-p2(x);" 
"6#,&-%$cosG6#%\"xG\"\"\"-%#p2G6#F'!\"\"" }{TEXT -1 11 " goes to 0:" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "limit( (cos(x) - (1 - x^2/
2))/x^3, x=0 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "limit( (cos(x) - (1 - x^2/2))/x^4, \+
x=0 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\"\"#C" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "p4 := 1 - x^2/2 + x^4/24;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#p4G,(\"\"\"F&*&#F&\"\"#F&*$)%\"xGF)F&F&!
\"\"*&#F&\"#CF&)F,\"\"%F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
38 "plot( \{cos(x),p4(x)\}, x=-Pi/2..Pi/2 );" }}{PARA 13 "" 1 "" 
{GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7S7$$!+Cjzq:!\"*$\"*,
'*o*>!#57$$!+X&=B]\"F*$\"*D?kP)F-7$$!+&ROFW\"F*$\"+*z0yR\"F-7$$!+O+tv8
F*$\"+7%f$H?F-7$$!+;&z#38F*$\"+#*GniEF-7$$!+w&\\6C\"F*$\"+:e['G$F-7$$!
+H:\"*y6F*$\"+pbobQF-7$$!+OwY96F*$\"+')QeKWF-7$$!+p'>y/\"F*$\"+^qj7]F-
7$$!+NW&Q\")*F-$\"+;5\"4d&F-7$$!+nn\\I\"*F-$\"+))*z77'F-7$$!+g\\fG&)F-
$\"+?wf$e'F-7$$!+'3'*4&yF-$\"+LbRwqF-7$$!+E\\hqrF-$\"+`;FRvF-7$$!+2;%
\\^'F-$\"+\\1%G&zF-7$$!+>u_>fF-$\"+207*H)F-7$$!*OA:@&F*$\"+Axts')F-7$$
!*?\\<h%F*$\"+oeVb*)F-7$$!*WqS\"RF*$\"+I=yV#*F-7$$!*_1lH$F*$\"+&os:Y*F
-7$$!*-T*=EF*$\"+dv,f'*F-7$$!*VQP(>F*$\"+l,&e!)*F-7$$!*AP0I\"F*$\"+[$
\\b\"**F-7$$!)NGBoF*$\"+L/tw**F-7$$!(d1b\"F*$\"+xz)*****F-7$$\")&Q8x'F
*$\"+ZK3x**F-7$$\"*Pz+G\"F*$\"+;<==**F-7$$\"*0x7$>F*$\"++\")39)*F-7$$
\"*(z-/EF*$\"+5y'Gm*F-7$$\"*^$=iKF*$\"+Pmis%*F-7$$\"*#G)*)*QF*$\"+Lf_
\\#*F-7$$\"**)Rgg%F*$\"+pS(z&*)F-7$$\"*7i8C&F*$\"+d@&yl)F-7$$\"*?/(>fF
*$\"+r>-*H)F-7$$\"*&HQMlF*$\"+We0TzF-7$$\"*=*Q1sF*$\"+Q,x:vF-7$$\"*n%p
QyF*$\"+#)f0&3(F-7$$\"*5O'*\\)F*$\"+oYF0mF-7$$\"*;Fe9*F*$\"+w0A4hF-7$$
\"*JHB#)*F*$\"+(4FRc&F-7$$\"+H\")QZ5F*$\"+l4L;]F-7$$\"+H(=S6\"F*$\"+!z
]lV%F-7$$\"+%f(4!=\"F*$\"+&pR\\%QF-7$$\"+3s\"3C\"F*$\"+i<b*G$F-7$$\"+Q
(3/J\"F*$\"+$))fFk#F-7$$\"+KAls8F*$\"+yEMe?F-7$$\"+zn,R9F*$\"+Ae&GV\"F
-7$$\"+>h`-:F*$\"*]HgN)F-7$$\"+Cjzq:F*F+-%'COLOURG6&%$RGBG$\"#5!\"\"$
\"\"!F^[lF][l-F$6$7S7$$!3+++lBjzq:!#<$\"3&pI?mi'*[9$!#E7$$!3WNzQW&=B]
\"Fe[l$\"3/+Gv?xUUo!#>7$$!3.42![ROFW\"Fe[l$\"3gb6eB?5x7!#=7$$!3?xt3O+t
v8Fe[l$\"3m&\\2))o:$Q>Fd\\l7$$!3xxIt:&z#38Fe[l$\"3?'*G%4c>^f#Fd\\l7$$!
3\")zz#fd\\6C\"Fe[l$\"314EY&>)3PKFd\\l7$$!3W:#)\\G:\"*y6Fe[l$\"3%zp)3e
+J>QFd\\l7$$!3,%4``jnW6\"Fe[l$\"3-PRxsVb1WFd\\l7$$!3)yYu)o'>y/\"Fe[l$
\"3mj(RcL6Y*\\Fd\\l7$$!3EWfpKW&Q\")*Fd\\l$\"3#39Fb09(ebFd\\l7$$!3(fe1V
w'\\I\"*Fd\\l$\"3`Z6(>o^L6'Fd\\l7$$!333*H!e\\fG&)Fd\\l$\"3ugk,u;KylFd
\\l7$$!35`u@%3'*4&yFd\\l$\"3!p!G<*fyJ2(Fd\\l7$$!3%ewEV#\\hqrFd\\l$\"3x
M,Zt3SPvFd\\l7$$!3SI*3_gT\\^'Fd\\l$\"3#)=n:Ymy^zFd\\l7$$!3B!RBoTF&>fFd
\\l$\"3#>F(fhm_)H)Fd\\l7$$!3!=qRrNA:@&Fd\\l$\"3gh=&o!3Ys')Fd\\l7$$!3Z[
5T-#\\<h%Fd\\l$\"33=YcbFIb*)Fd\\l7$$!3#4J\\*R/29RFd\\l$\"34?j_E?tV#*Fd
\\l7$$!3wFQ0=l]'H$Fd\\l$\"3))3cV'*[bh%*Fd\\l7$$!3)eE=r,T*=EFd\\l$\"3u.
9#>38!f'*Fd\\l7$$!3eSV%*H%QP(>Fd\\l$\"39eW&[M\\e!)*Fd\\l7$$!3)*RWU;s`+
8Fd\\l$\"3p=!\\7G\\b\"**Fd\\l7$$!3.gSQ<NGBoF^\\l$\"3sT?<J/tw**Fd\\l7$$
!3y]Py0ul]:!#?$\"2#e/Jxz)*****Fe[l7$$\"3O=#e5YQ8x'F^\\l$\"3'e4OgC$3x**
Fd\\l7$$\"3I^(**zOz+G\"Fd\\l$\"3V4EMb;==**Fd\\l7$$\"3dNv()\\qFJ>Fd\\l$
\"3!oa$Hzt39)*Fd\\l7$$\"3y0j*\\(z-/EFd\\l$\"3Q&QuL[jGm*Fd\\l7$$\"3_\"o
Md]$=iKFd\\l$\"3CKyKJ*4EZ*Fd\\l7$$\"3SH4XBG)*)*QFd\\l$\"3A%yk%psZ\\#*F
d\\l7$$\"3=DLY%*)Rgg%Fd\\l$\"3=Y2HT>%y&*)Fd\\l7$$\"3U6;S?@OT_Fd\\l$\"3
5X&f]glvl)Fd\\l7$$\"3KMYS.Uq>fFd\\l$\"3q,*ok,G%)H)Fd\\l7$$\"3g!oi?&HQM
lFd\\l$\"3WFkMvG)*RzFd\\l7$$\"3Jr$zA=*Q1sFd\\l$\"3temLGG%Q^(Fd\\l7$$\"
3gHJCuYpQyFd\\l$\"3i\"H<$e!p=3(Fd\\l7$$\"3)fG\"*Q5O'*\\)Fd\\l$\"31%)f
\\U[5+mFd\\l7$$\"3#=wm/;Fe9*Fd\\l$\"3#3&>#)fC@,hFd\\l7$$\"37E5$3JHB#)*
Fd\\l$\"3i[%GE;n;b&Fd\\l7$$\"31h>eG\")QZ5Fe[l$\"3MF&R1Y\\$)*\\Fd\\l7$$
\"3PK#)fG(=S6\"Fe[l$\"3YPFIeOe5WFd\\l7$$\"330`g$f(4!=\"Fe[l$\"3[es=peM
3QFd\\l7$$\"3&oOhy?<3C\"Fe[l$\"3![m6&RFBSKFd\\l7$$\"3XnP+Q(3/J\"Fe[l$
\"3S'*='\\%4buDFd\\l7$$\"3/%yp@B_EP\"Fe[l$\"35*fH')y+&o>Fd\\l7$$\"3cNK
@zn,R9Fe[l$\"3WGV'*zY)RJ\"Fd\\l7$$\"3*p@f'=h`-:Fe[l$\"3@%*fWC6s?oF^\\l
7$$\"3+++lBjzq:Fe[lFf[l-Fgz6&FizF][lFjzF][l-%+AXESLABELSG6$Q\"x6\"Q!6
\"-%%VIEWG6$;$!+Fjzq:F*$\"+Fjzq:F*%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 
1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 34 "plot( \{cos(x),p4(x)\}, x=-Pi..Pi );" }}
{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7go7
$$!+ZEfTJ!\"*$\"*8*4R7F*7$$!+o[6tIF*$!(dfx&F*7$$!+)3PY+$F*$!**o1!=\"F*
7$$!+Q\\0XHF*$!*sp@-#F*7$$!+*ysa)GF*$!*,4eu#F*7$$!+IkY=GF*$!*3gdU$F*7$
$!+s+Y^FF*$!*\\hA(RF*7$$!+_&4So#F*$!*P'3'R%F*7$$!+J!flh#F*$!*A;:q%F*7$
$!+iS*He#F*$!*L>>\"[F*7$$!+#4H%\\DF*$!*k/g*[F*7$$!+AT'e^#F*$!*bUZ&\\F*
7$$!+^\"*H#[#F*$!*w!4*)\\F*7$$!+G,=^CF*$!*Tr***\\F*7$$!+/61?CF*$!*HV9*
\\F*7$$!+!3U*)Q#F*$!*RSU'\\F*7$$!+cI#yN#F*$!*K(3>\\F*7$$!+j\"zLH#F*$!*
=z:x%F*7$$!+q_$*GAF*$!*4rjb%F*7$$!+P$Rc4#F*$!+ROCAR!#57$$!+')3xi>F*$!+
5/QyIFfq7$$!+_$*4E=F*$!+iu$*R?Ffq7$$!+\"**=dq\"F*$!+r)y--\"Ffq7$$!+;#*
>q:F*$\"*mA@0#Ffq7$$!+%)H7M9F*$\"+7v(*y9Ffq7$$!+?$))HI\"F*$\"+w)G@r#Ff
q7$$!+#[0R=\"F*$\"+\\5T5QFfq7$$!+qWIU5F*$\"+nFyf]Ffq7$$!*R)\\B#*F*$\"+
&H7z/'Ffq7$$!*(39GyF*$\"+=xZ#4(Ffq7$$!*-8If'F*$\"+4hL0zFfq7$$!*-#)yB&F
*$\"+k@ff')Ffq7$$!*%oZZRF*$\"+)o))4B*Ffq7$$!*Uu5g#F*$\"+4yij'*Ffq7$$!+
bb'G)>Ffq$\"+Bj0/)*Ffq7$$!*pcYO\"F*$\"+6,.2**Ffq7$$!++\"\\$yp!#6$\"+.7
mv**Ffq7$$!(885$F*$\"+4>&*****Ffq7$$\"++HF;mFjv$\"+]/7y**Ffq7$$\"*rnUN
\"F*$\"+-\"Q%3**Ffq7$$\"+NK@d>Ffq$\"+Ss24)*Ffq7$$\"*we,c#F*$\"+q$pSn*F
fq7$$\"*7aD'QF*$\"+6#3LE*Ffq7$$\"*(f03_F*$\"+X>Yu')Ffq7$$\"*.nV_'F*$\"
+M48ZzFfq7$$\"*nlzz(F*$\"+M^l8rFfq7$$\"*\")z?@*F*$\"+pr%p0'Ffq7$$\"+EC
F[5F*$\"+@1w3]Ffq7$$\"+V3%R=\"F*$\"++-45QFfq7$$\"+$fwoI\"F*$\"+3)*yvEF
fq7$$\"+RyFT9F*$\"+v\"\\:T\"Ffq7$$\"+P*Qxc\"F*$\"*Xh(zAFfq7$$\"+Bs#**p
\"F*$!*PqJp*Ffq7$$\"+Na;H=F*$!+S\"yZ1#Ffq7$$\"+leYk>F*$!+IYF!4$Ffq7$$
\"+gix%4#F*$!+$)4R<RFfq7$$\"+gu.GAF*$!*iBHb%F*7$$\"+Dj6%H#F*$!*)4mtZF*
7$$\"+!>&>gBF*$!*/XJ#\\F*7$$\"+(*\\b!R#F*$!*h(4m\\F*7$$\"+0[\"4U#F*$!*
qH>*\\F*7$$\"+7YF^CF*$!*7o***\\F*7$$\"+>Wj\"[#F*$!*;J&*)\\F*7$$\"+&=Ik
^#F*$!*#o&R&\\F*7$$\"+]fA^DF*$!*(R:#*[F*7$$\"+:<-'e#F*$!*8bI![F*7$$\"+
zu\"3i#F*$!*>ybo%F*7$$\"+u41$o#F*$!*A$=,WF*7$$\"+nWIXFF*$!*8og,%F*7$$
\"+:!p;\"GF*$!*&*Rr[$F*7$$\"+iN.yGF*$!*S\\\"GGF*7$$\"+,HbTHF*$!*!)=z1#
F*7$$\"+TA20IF*$!*]lM<\"F*7$$\"+XCLtIF*$!(l?R&F*7$$\"+]EfTJF*$\"*=*4R7
F*-%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!FcclFbcl-F$6$7en7$$!3*****4tk#fTJ!
#<$FaclFccl7$$!35'\\8!o[6tIFjcl$!3EdR#\\7jl(**!#=7$$!3w\"*pr)3PY+$Fjcl
$!3ng@plBO1**Fadl7$$!3)**z*4R\\0XHFjcl$!3y?$G=_&[2)*Fadl7$$!3?3E[*ysa)
GFjcl$!3-/?(R3-Qn*Fadl7$$!3gX&))>2g9v#Fjcl$!3wsjRdke[#*Fadl7$$!3)4577.
flh#Fjcl$!3m\"))3;*32`')Fadl7$$!3u\"QM::*H#[#Fjcl$!3Xt%*)4+_U!zFadl7$$
!3y\\AhcI#yN#Fjcl$!3[dVKLTd#3(Fadl7$$!3&e=d-FN*GAFjcl$!3gz^S3dX;hFadl7
$$!3u%*GBP$Rc4#Fjcl$!3HBRr;:x5]Fadl7$$!3mCj&f)3xi>Fjcl$!3WB/R/'R,#QFad
l7$$!3(or4AN*4E=Fjcl$!3E5L?OiQDDFadl7$$!3QQQ*3**=dq\"Fjcl$!3IXjd$yO^M
\"Fadl7$$!3e!>jg@*>q:Fjcl$\"3+_hBip5rf!#@7$$!3`;m,%)H7M9Fjcl$\"3g*)e)Q
R#[i8Fadl7$$!3v(4F,K))HI\"Fjcl$\"3[\"4??=#=YEFadl7$$!3S*4!R#[0R=\"Fjcl
$\"3%Rft'f*3Jx$Fadl7$$!3pY@QqWIU5Fjcl$\"3#[Z2$)H:B/&Fadl7$$!3Wp3w$R)\\
B#*Fadl$\"3I-2e(\\*[RgFadl7$$!3EJc8o39GyFadl$\"3GV=%RO;$*3(Fadl7$$!3'=
[BP-8If'Fadl$\"3:P!R#>U?/zFadl7$$!3CB3<@?)yB&Fadl$\"3ce2))\\nIf')Fadl7
$$!3_*)R<YoZZRFadl$\"3a]32!GO4B*Fadl7$$!38TqX=W2,EFadl$\"3;UpZ8Nij'*Fa
dl7$$!3S\"4gMblG)>Fadl$\"3w5p#*ya0/)*Fadl7$$!3qTJY)ocYO\"Fadl$\"3x_tH@
+.2**Fadl7$$!3c.@Z/\"\\$yp!#>$\"3eOID,7mv**Fadl7$$!38,z7VKJ,J!#?$\"3[%
Q>$4>&*****Fadl7$$\"3p0t!3)GF;mFc\\m$\"3Si.#)\\/7y**Fadl7$$\"39SFf3xEa
8Fadl$\"3$y;bu,Q%3**Fadl7$$\"312**yIK@d>Fadl$\"3Pq7lgk24)*Fadl7$$\"3'R
2()Hve,c#Fadl$\"3c+KPla1u'*Fadl7$$\"3IcwR<TbiQFadl$\"3q:a.6AEj#*Fadl7$
$\"3*4*=Jof03_Fadl$\"31X2bIh=u')Fadl7$$\"3q]1XIqOClFadl$\"3%H8%pqx1YzF
adl7$$\"3%HoBlmlzz(Fadl$\"39bB(y$fc5rFadl7$$\"3^u'f#4)z?@*Fadl$\"31dw$
GV'e[gFadl7$$\"39Hv<ECF[5Fjcl$\"3k.%3IU)o!*\\Fadl7$$\"3Olj%G%3%R=\"Fjc
l$\"31u!\\))[\"ysPFadl7$$\"3\\-)REfwoI\"Fjcl$\"3)HVd1fl'3EFadl7$$\"3cM
2vQyFT9Fjcl$\"3_fG.(ym:H\"Fadl7$$\"3!y32s$*Qxc\"Fjcl$\"3#)fgeC[QdIFi\\
m7$$\"35+K?Bs#**p\"Fjcl$!3+X(fwBBxG\"Fadl7$$\"3<#H$eMa;H=Fjcl$!379c4G8
/bDFadl7$$\"3%>>CZ'eYk>Fjcl$!3_-=qm$)zNQFadl7$$\"3kYuyfix%4#Fjcl$!3#oH
!ec0I.]Fadl7$$\"3T4q))fu.GAFjcl$!3$>4B*z.N4hFadl7$$\"3y?w'**=&>gBFjcl$
!3!*Gdan.I*4(Fadl7$$\"3!*>3a=Wj\"[#Fjcl$!3Yyl.w$y,!zFadl7$$\"3A?c*)yu
\"3i#Fjcl$!333*G2]PVn)Fadl7$$\"3-i-HnWIXFFjcl$!3G:DUS4+D#*Fadl7$$\"3u=
RWhN.yGFjcl$!3'[\"[-g))oa'*Fadl7$$\"3tX=#4!HbTHFjcl$!3n!=4Vz'e+)*Fadl7
$$\"3ss(*RSA20IFjcl$!3(Qny3`bp!**Fadl7$$\"3/')[UXCLtIFjcl$!3Qtebk<rw**
Fadl7$$\"3!)***\\/l#fTJFjclF[dl-F\\cl6&F^clFbclF_clFbcl-%+AXESLABELSG6
$Q\"x6\"Q!6\"-%%VIEWG6$;$!+aEfTJF*$\"+aEfTJF*%(DEFAULTG" 1 2 0 1 10 0 
2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "You can see that the polynomial " 
}{XPPEDIT 18 0 "p4;" "6#%#p4G" }{TEXT -1 49 " is an excellent approxim
ation over the interval " }{XPPEDIT 18 0 "[-Pi/2, Pi/2];" "6#7$,$*&%#P
iG\"\"\"\"\"#!\"\"F)*&F&F'F(F)" }{TEXT -1 81 ", but we will need more \+
terms if we want to approximate over the larger interval " }{XPPEDIT 
18 0 "[-Pi, Pi];" "6#7$,$%#PiG!\"\"F%" }{TEXT -1 1 "." }}}{SECT 0 
{PARA 4 "" 0 "" {TEXT -1 10 "Question 2" }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 34 "Find the next 2 approximations to " }{XPPEDIT 18 0 "cos(x
);" "6#-%$cosG6#%\"xG" }{TEXT -1 2 " ." }}}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 10 "Question 3" 
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "What is the pattern of the coeff
icients in the approximating polynomials for sine and cosine?" }}}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 4 "" 0 "" 
{TEXT -1 10 "Question 4" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 162 "Use yo
ur approximating polynomials to guess what the derivatives of the sine
 and cosine function are.  Confirm your guess by differentiating these
 functions with " }{TEXT 258 5 "Maple" }{TEXT -1 1 "." }}}}}}{MARK "0 \+
0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }