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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 14 "Taylor Series:" }}}
{EXCHG {PARA 257 "" 0 "" {TEXT -1 34 "a visual approach to approximati
on" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 ""
{TEXT -1 81 "One of the ongoing themes of calculus is polynomial appro
ximation of functions. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 93 "The tangent line can be described as the \"best li
near approximation\" to a curve at a point. " }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 209 "When we first introduced
second derivatives, we looked at the \"best quadratic approximation\"
of a curve at a point. This was a quadratic polynomial that p(x) tha
t approximated f(x) at a given point x=c, with " }}{PARA 258 "" 0 ""
{TEXT -1 49 "p(c) = f(x), p'(c) = f'(c), and p''(c) = f''(c)." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 518 "We have \+
continued this theme by studying Taylor series, which comes at the en
d of the chapter on sequences and series. In the flurry of computatio
ns on sequences and series, it is all to easy to loose sight of the id
ea that a Taylor polynomial is the \"best nth degree polynomial approx
imation\" to a function at a point. A visual approach makes this clea
r. (If a picture is worth 1000 words, how much is a movie worth?) Th
e visual approach also makes concepts like the interval of convergence
easier to understand." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0
{PARA 4 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 141 "W
e start with the technical details. Put your cursor in the red sectio
n below, and hit the return key. This should load the plots routines.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 193 "We next view an example. The exa
mple we look at is the sin function expanded about the point x=0. We \+
look at the plot of sin(x) plotted on the same axis with the nth degre
e Taylor polynomial " }{XPPEDIT 18 0 "T[n]" "&%\"TG6#%\"nG" }{TEXT -1
33 ", where n goes from 2 to 20 by 2." }}}{SECT 0 {PARA 4 "" 0 ""
{TEXT -1 72 "The first example, an animation on a nice sequence of Tay
lor polynomials" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 608 "with(plo
ts):\nfunc := sin(x);\nminx := -3: maxx := 5: aboutx := 0:\nminy := -
1.25: maxy := 2:\nmindeg := 2: degsteps := 9: bydeg :=2:\nA := display
(seq(\nplot(convert(taylor(func,x=aboutx, mindeg + bydeg*i), polynom),
\nx=minx..maxx,y=miny..maxy, color=blue,\ntitle=`T`.(mindeg + bydeg*i)
.` = `.(convert(convert(taylor(func,x=aboutx, mindeg + bydeg*i),polyno
m),string)), titlefont=[HELVETICA, 10]), i=0..degsteps), insequence=tr
ue):\nB := animate(func,x=minx..maxx,y=miny..maxy,frames=degsteps+1, c
olor=green):\nprint(` `.(convert(func, string)).` vs its Taylor polyno
mial`);\ndisplay(A,B,view=[minx..maxx,miny..maxy]);" }}}{PARA 4 "" 0 "
" {TEXT -1 20 "To run the animation" }}{EXCHG {PARA 0 "" 0 "" {TEXT
-1 365 "To run the animation, click once on the picture above. A box \+
should appear aroun the picture. If you see the box, you should see a
series of buttons, like you might see on a tape recorder. From left \+
to right, the buttons are: stop, start, advance one frame, forward, re
verse, stop after one cycle, and run in a loop. Click the on button a
nd watch the animation." }}}{PARA 4 "" 0 "" {TEXT -1 9 "Exercises" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "1) Run the animation above with t
he frames in a loop going about 3 frames per second." }}}{EXCHG {PARA
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 ""
}}{PARA 0 "" 0 "" {TEXT -1 356 "The code for animating sequences of Ta
ylor polynomials is more complicated than I can expect you to work out
on your own. Nevertheless it is useful for you to be able to experim
ent with such sequences. The code was written as a template that can \+
easily be modified by copying and pasting, then modifying the variable
s in the first 4 lines of the section." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 76 "func is the function. minx, maxx, mi
ny, and maxy give the viewing window. " }}{PARA 0 "" 0 "" {TEXT -1
72 "aboutx tells the x-coordinate of thepoint that series is expanded \+
about." }}{PARA 259 "" 0 "" {TEXT -1 169 "mindeg, bydeg, and degsteps,
are the degree of the first approximation, the amount the degree is c
hanged for each frame, and the number of extra frames in the animation
." }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 193 "2) Modify the section of
code so that the Taylor polynomials are centered about x = Pi. (Chan
ge the value of aboutx and execute the section again.) Execute the se
ction and run the animation." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1
0 0 "" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 313 "3) Look at the Taylo
r polynomials for sin(x) centered around x=0 of degrees 10, 20, 30, an
d 40. For what range of x values doues each polynomial give a good ap
proximation of sin(x)? (Set aboutx := 0; mindeg := 11; bydeg := 10;
degsteps := 3;. You will also need to adjust minx and maxx to be la
rge enough.)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG
{PARA 259 "" 0 "" {TEXT -1 200 "4) Modify the section of code so that \+
the Taylor polynomials approximate the function y = cos(2x). (Change \+
the value of func and execute the section again.) Execute the section
and run the animation" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 ""
}}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 28 "Back to dry boring formulas.
" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 175 "In our animation, the printin
g of the high degree Taylor polynomials get truncated when printed on
the graph. The loop below prints the polynomials so we can look at
them." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "func := sin(x);
\naboutx := 0:\nmindeg := 2: maxdeg := 20: bydeg := 2:\nfor i from min
deg by bydeg to maxdeg do\nT[i] :=(taylor(func, x=aboutx, i)); od;" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "You should notice several things \+
about the Taylor polynomials. " }}{PARA 0 "" 0 "" {TEXT -1 37 "1) Th
ey all end with an error term, " }}{PARA 259 "" 0 "" {TEXT -1 251 "2) \+
The front of the polynomials is unchanged as the order of the polynomi
al goes up. This means that instead of running a loop to look at a se
quence of Taylor polynomials, we get the same information by looking a
t the last polynomial in the sequence." }}{PARA 259 "" 0 "" {TEXT -1
240 "3) Maple understands the order of the Taylor to be the order of \+
the error term, rather than the order of the polynomial itself. This \+
means that the Maple command for finding the ith Taylor polynomial of \+
func center at the point aboutx is:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT
-1 28 "taylor(func, x=aboutx, i+1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT
-1 78 "It cleans things up if we formally convert to a polynomial and \+
use the command" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "convert(
taylor(sin(x), x=0, 20), polynom);" }}}{EXCHG {PARA 4 "" 0 "" {TEXT
-1 10 "Exercises:" }}{PARA 0 "" 0 "" {TEXT -1 38 "Find the indicated t
aylor polynomials." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0
"" 0 "" {TEXT -1 40 "5) The 6th degree Taylor polynomial of " }
{XPPEDIT 18 0 "sqrt(x)" "-%%sqrtG6#%\"xG" }{TEXT -1 13 " about x = 9.
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 ""
{TEXT -1 40 "6) The 8th degree Taylor polynomial of " }{XPPEDIT 18 0
"1/(1 - 2 x)" "*&\"\"\"F#,&F#F#*&\"\"#F#%\"xGF#!\"\"F(" }{TEXT -1 12 "
about x= 5." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0
"" 0 "" {TEXT -1 41 "7) The 10th degree Taylor polynomial of " }
{XPPEDIT 18 0 "exp(3*x)" "-%$expG6#*&\"\"$\"\"\"%\"xGF'" }{TEXT -1 13
" about x = 0." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA
0 "" 0 "" {TEXT -1 58 "8) Do three more Taylor polynomials of your ow
n choosing." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 65 "A second ex
ample: Taylor polynomials and interval of convergence." }}{EXCHG
{PARA 0 "" 0 "" {TEXT -1 313 "The first animation we looked at has a T
aylor series that works everywhere. To make the Taylor polynomial wor
k as an approximation on a bigger interval we simply increased the deg
ree of the polynomial. We now consider a case where the approximation
is only good in a small region, no matter how high the degree." }}
{PARA 0 "" 0 "" {TEXT -1 32 "Execute the following animation:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 641 "func := 1/(1 + x^2);\nminx \+
:= -2: maxx := 5: aboutx := 1.5:\nminy := -0.5: maxy := 1.5:\nmindeg \+
:= 2: degsteps := 10: bydeg :=4:\nfor i from mindeg by bydeg to mindeg
+ bydeg * degsteps do\nT[i] :=convert(taylor(func, x=aboutx, i), poly
nom): od:\nA := display(seq(plot(T[mindeg + bydeg*i],\nx=minx..maxx,y=
miny..maxy, color=blue,\ntitle=`T`.(mindeg + bydeg*i).` = `.(convert(T
[mindeg + bydeg*i],string)), titlefont=[HELVETICA, 10]), i=0..degsteps
), insequence=true):\nB := animate(func,x=minx..maxx,y=miny..maxy,fram
es=degsteps+1, color=red):\nprint(` `.(convert(func, string)).` vs its
Taylor polynomial`);\ndisplay(A,B,view=[minx..maxx,miny..maxy]);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 168 "Notice that the Taylor polynomial
approximation of 1/(1 + x^2) centered at x = 1.5 only gives a good ap
proximation for x between 0 and 3, no matter how high the degree." }}}
{EXCHG {PARA 4 "" 0 "" {TEXT -1 9 "Exercises" }}}{EXCHG {PARA 259 ""
0 "" {TEXT -1 234 "9) Do animations for the Taylor polynomials of 1/(
1 + x^2) centered at x = 2, 4, and -3. (You will need to change the v
iewing range appropriately.) For each animation, find the range where
the polynomials make a good approximation." }}}{EXCHG {PARA 0 "> " 0
"" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 123 "10) Gu
ess at a generalization of your results in the previous exercise. The
taylor polynomials of high enough degree for " }{XPPEDIT 18 0 "1/(1 +
x^2)" "*&\"\"\"F#,&F#F#*$%\"xG\"\"#F#!\"\"" }{TEXT -1 64 " centered a
t x =a, make a good approximation on the interval ..." }}}{EXCHG
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1
0 0 "" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 172 "11) Do another anima
tion of the Taylor polynomials approximating a non-polynomial rational
function of your choice. What is the range where it makes a good app
roximation?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 4 "" 0 "" {TEXT
-1 19 "Final Instructions:" }}{PARA 0 "" 0 "" {TEXT -1 61 "When you co
mplete the worksheet, print it out and hand it in." }}}{EXCHG {PARA 0
"> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "2 2 2 0 0" 12 }{VIEWOPTS 1 1 0
1 1 1803 }