Animating Taylor polynomials in several variables Demonstration Worksheet by Mike May, S.J.- maymk@slu.edu \302\2512006 When working with Taylor polynomials there are two features that are worth demonstrating with animations that compare higher and higher degree Taylor polynomials with the original function. The first feature is that, as we would hope with something that is called an approximation, the Taylor polynomials morph into the function. The second feature is the radius of convergence. If the Taylor series has a finite radius of convergence, it will only morph to the function in that region of convergence, and will never become a good approximation outside that region. As with all concepts in multi-variable calculus, we will start by reviewing the one variable calculus concept. The animations are memory intensive. Since this often causes memory problems the animation is done here as a separate worksheet. Before doing this worksheet, you should have worked through the worksheet on Taylor polynomials in several variables. First a technical detail. Rather than working out the Taylor polynomials from the definition, we are going to call the Maple library for multivariate Taylor polynomials. restart: with(plots): with(Student[MultivariateCalculus]):

We start with the single variable case. We first want to demonstrate the idea of convergence. Note that with this function, the Taylor series converges everywhere. with(plots): func := sin(x); aboutx := 0; width := 12: height:= 1.75: yval := subs(x=aboutx, func); minx := aboutx-width: maxx := aboutx+width: miny := yval-height: maxy := yval+height: mindeg := 2: degsteps := 10: bydeg :=4: fudge:= (maxy-miny)/20: framer2d := proc(i) local A, B, C, D, TaylorDeg: TaylorDeg := mindeg + bydeg*i: A := plot([func,TaylorApproximation(func,[x]=[aboutx], TaylorDeg)], x=minx..maxx,y=miny..maxy, color=[green,blue]): C := textplot([minx,maxy, `The Taylor polynomial of degree `||(mindeg + bydeg*i)|| ` (blue)`], align={ABOVE,RIGHT}, font = [HELVETICA, BOLD, 12], view=[minx..maxx,miny..maxy]): D := textplot([maxx,maxy-fudge, `graphed against the function (green)`], align={ABOVE,LEFT}, font = [HELVETICA, BOLD, 12], view=[minx..maxx,miny..maxy]): display({A,C, D},view=[minx..maxx,miny..maxy]); end: display([seq(framer2d(i), i=0..degsteps)], insequence = true);

Next we want to demonstrate the idea of radius of convergence. We want to look at what happens if the series has a finite radius of convergence. with(plots): func := 1/(1 + x^2); aboutx := 1.5; width := 3.5: height:= 1.2: yval := eval(func, x=aboutx); minx := aboutx-width: maxx := aboutx+width: miny := yval-height: maxy := yval+height: mindeg := 2: degsteps := 10: bydeg :=4: fudge:= (maxy-miny)/20: framer2d := proc(i) local A, C, D, TaylorDeg: TaylorDeg := mindeg + bydeg*i: A := plot([func,TaylorApproximation(func,[x]=[aboutx], TaylorDeg)], x=minx..maxx,y=miny..maxy, color=[green,blue]): C := textplot([minx,maxy, `The Taylor polynomial of degree `||(mindeg + bydeg*i)|| ` (blue)`], align={ABOVE,RIGHT}, font = [HELVETICA, BOLD, 12], view=[minx..maxx,miny..maxy]): D := textplot([maxx,maxy-fudge, `graphed against the function (green)`], align={ABOVE,LEFT}, font = [HELVETICA, BOLD, 12], view=[minx..maxx,miny..maxy]): display({A,C, D},view=[minx..maxx,miny..maxy]); end: display([seq(framer2d(i), i=0..degsteps)], insequence = true);

Now we want to show the same idea with a function of two variables. We start with the same function we used in the last worksheet, but expanded about a different point. Now we do a sample animation. g := (x,y) -> sin(x+3*y)+cos(2*x-y): a:= Pi/4: b:=Pi/3: del := 2: height := 3; mindeg := 2: degsteps := 7: bydeg :=2: lowx := a-del: highx := a+del: lowy := b-del: highy := b+del: zval := evalf(g(a, b)); lowz := zval-height: highz := zval+height: xrange:= lowx..highx: yrange := lowy..highy: zrange := lowz..highz: framer3d := proc(i) local A, B, T1, deg: deg := mindeg + bydeg*i: A := plot3d(mtaylor(g(x,y),[x=a, y=b], mindeg + bydeg*i), x=xrange, y=yrange, view= zrange, color=blue): B := plot3d(g(x,y),x=xrange, y=yrange, view= zrange, color=green): T1 := textplot3d([(highx+lowx)/2,lowy,highz, `Degree `||deg||` Taylor (blue)`], align={BELOW,RIGHT}, font=[HELVETICA, BOLD, 12]): display3d({A,B,T1},axes=boxed); end: display([seq(framer3d(i), i=0..degsteps)], insequence = true, style=patch, view=zrange); (The block of code above is designed so that all the variables to be changed for different examples are in the first 5 lines.)

In single variable Taylor polynomials we learned that the Taylor series of some functions have a finite radius of convergence. It is instructive to look at what happens in the 2 variable case. Consider the function h(x,y) = 1/(x^2+y^2) approximated by Taylor polynomials centered at (2,1). It is clear that we will never have a good approximation that includes the origin. What we get is an approximation that is good in a circular region with a radius small enough to exclude the trouble spot at the origin. The following animation plots the error between the function and its Taylor approximations with the degree growing. h := (x,y) -> 1/(x^2+y^2): a := 2: b := 3: del := 4: errortolerance := .1; mindeg := 2: degsteps := 10: bydeg :=4: lowx := a-del: highx := a+del: lowy := b-del: highy := b+del: lowz := -errortolerance: highz := errortolerance: xrange:= lowx..highx: yrange := lowy..highy: zrange := lowz..highz: framer3de := proc(i) local A, T1, deg: deg := mindeg + bydeg*i: A := plot3d( mtaylor(h(x,y),[x=a, y=b], deg)-h(x,y), x=lowx..highx, y=lowy..highy, view= lowz..highz, orientation=[135,45]): T1 := textplot3d([(highx+lowx)/2,lowy,highz, `Degree `||deg||` Taylor error`], align={BELOW,RIGHT}, font=[HELVETICA, BOLD, 12]): display3d({A,T1},axes=boxed); end: display([seq(framer3de(i), i=0..degsteps)], insequence = true, style=patch, view=lowz..highz); Note that increasing the degree does not increase the region where the series converges to the function.