{VERSION 2 3 "APPLE_PPC_MAC" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Title" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }{PSTYLE "" 18 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 18 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Hanging " 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -10 -1 -1 0 0 0 0 0 0 0 0 }} {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 }