Taylor polynomials in several variables Worksheet by Mike May, S.J. - maymk@slu.edu Revised by Russell Blyth - blythrd@slu.edu restart; with(plots): with(Student[MultivariateCalculus]):

Recall from calculus of one variable that we can approximate a nice function at a point x=a by using Taylor polynomials. We get the nth degree part of the approximation near x=a by evaluating the nth derivative at that point and multiplying by NiMqJiksJiUieEciIiIlImFHISIiJSJuR0YnLSUqZmFjdG9yaWFsRzYjRipGKQ==. We can compute the Taylor polynomial using the TaylorApproximation command from the Student[MultivariableCalculus] package . It is instructive to note that the difference between the nth degree Taylor polynomial and the (n-1)st degree Taylor polynomial is the term we just described. TaylorDegree := 10; NthTermOfTaylor := TaylorApproximation(f(x),[x]=[a],TaylorDegree)- TaylorApproximation(f(x),[x]=[a],TaylorDegree-1);

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Consider an example: f := x -> sin(x) + cos(2*x); a:= Pi/3;

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Now we compute the Taylor polynomials up to degree four for our function f at the given point a. Deg0 := TaylorApproximation(f(x),[x]=[a],0); Deg1 := TaylorApproximation(f(x),[x]=[a],1); Deg2 := TaylorApproximation(f(x),[x]=[a],2); Deg3 := TaylorApproximation(f(x),[x]=[a],3); Deg4 := TaylorApproximation(f(x),[x]=[a],4);

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Visually we notice that we get better and better approximations to the function f by adding in more and more terms to the Taylor polynomial. del := Pi/2; plot([f(x), Deg0, Deg1, Deg2, Deg3, Deg4], x=a-del..a+del, y=-5..5, axes=normal, legend = ["func", "Deg0", "Deg1", "Deg2", "Deg3", "Deg4"]);

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1. For each Taylor polynomial approximation above give an interval on which you think it is a good approximation to the original function.

To check the accuracy of an approximation we graph the difference between the function and the approximation. del := .6; plot([Deg1 - f(x), Deg2 - f(x), Deg3 - f(x), Deg4 - f(x)], x=a-del..a+del, y=-0.1..0.1, axes=normal, legend=["LinErr","QuadErr","CubeErr","QuarticErr"]);

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2. For each of the Taylor approximations above, give an interval on which the error is less than .02.

If we want values rather than graphs, we can evaluate the Taylor polynomials with the subs command. We also use the evalf command to convert the answer to a decimal value. "f(1.2)" = f(1.2); "Deg 3 Approx" = eval(Deg3, x=1.2); "Deg 3 Approx" = evalf(eval(Deg3, x=1.2)); "Error in Approx" = f(1.2) - evalf(eval(Deg3, x=1.2));

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3. Find the values of f(x) and the 4th degree Taylor polynomial above at x=0.7. Find the value of the error in the approximation.

We are interested in using Taylor polynomials when we either have information about the function and its derivatives at a point but no formula (hence the Taylor polynomial is the best description of the function we have) or when we want a numerical value where a symbolic manipulation is hard or impossible for the given function but easy using a polynomial approximation. (For example, finding NiMtJSRpbnRHNiQtJSRzaW5HNiMqJCklInhHIiIlIiIiL0YrOyIiIUYt.)

To generalize Taylor polynomials to functions of two variables we need to take into account all the partial derivatives. We have two first partials, four second partials, eight third partials, and so on. We are helped out by the fact that for nice functions the mixed partials are equal, which simplifies the computations. That leads to the following procedures for computing the linear through cubic terms of the Taylor polynomials. LinTerm2 := TaylorApproximation(g(x,y),[x,y]=[a,b],1)- TaylorApproximation(g(x,y),[x,y]=[a,b],0); QuadTerm2:= TaylorApproximation(g(x,y),[x,y]=[a,b],2)- TaylorApproximation(g(x,y),[x,y]=[a,b],1); CubeTerm2 := TaylorApproximation(g(x,y),[x,y]=[a,b],3)- TaylorApproximation(g(x,y),[x,y]=[a,b],2);

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(The rule for the factor added to each derivative is, if we take the partial n times with respect to x and the partial m times with respect to y then we multiply by NiMqKCksJiUieEciIiIlImFHISIiJSJuR0YnKSwmJSJ5R0YnJSJiR0YpJSJtR0YnKiYtJSpmYWN0b3JpYWxHNiNGKkYnLUYyNiNGL0YnRik=. Note that this becomes the usual familiar factor if we only have one variable.) We start with a function and a point. g := (x,y) ->sin(x+3*y)+cos(2*x-y); a:= Pi/2; b:=Pi/2;

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Next we compute the Taylor polynomials for g at the point (a,b), and then graph these approximations. Deg0TP := g(a,b); Deg1TP := TaylorApproximation(g(x,y),[x,y]=[a,b],1); Deg2TP := TaylorApproximation(g(x,y),[x,y]=[a,b],2); Deg3TP := TaylorApproximation(g(x,y),[x,y]=[a,b],3);

IiIh

LChJInhHNiIhIiJJInlHRiQiIiVJI1BpRyUqcHJvdGVjdGVkRyMhIiQiIiM=

LChJInhHNiIhIiJJInlHRiQiIiVJI1BpRyUqcHJvdGVjdGVkRyMhIiQiIiM=

LDBJInhHNiIhIiJJInlHRiQiIiVJI1BpRyUqcHJvdGVjdGVkRyMhIiQiIiMqJCwmRiMiIiJGKCNGJUYsIiIkIyIiKCIiJyomLCZGJkYvRihGMEYvRi5GLCMhIihGLComRi5GL0Y2RixGNyokRjZGMSMhIzlGMQ==

del := Pi/6: plot3d({g(x,y),Deg1TP, Deg2TP, Deg3TP}, x=a-del..a+del, y=b-del..b+del, view=-5..5, style=patch);

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We can make the graph clearer by specifying colors for the graphs del := Pi/6: plot3d([g(x,y),Deg1TP,Deg2TP,Deg3TP], x=a-del..a+del, y=b-del..b+del,view=-5..5, color=[orange,red,green,blue], axes=boxed, style=patch);

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