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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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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Question: we have plotted four surfaces and yet only three are visible. Why is one surface apparently missing? (Hint: try removing one of the visible surfaces at a time from the plot.) As in the one variable case, we are interested in plotting an error term to see where the approximation is good, and in evaluating the approximation at a specified point. plot3d(g(x,y)-Deg3TP, x=a-del..a+del, y=b-del..b+del, view=-0.1..0.1, axes=normal); "Deg 3 Approx" = evalf(eval(Deg3TP, {x=1.7, y=1.5})); "computed value" = g(1.7,1.5); "Error" = g(1.7,1.5)-evalf(eval(Deg3TP, {x=1.7, y=1.5}));

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4. For what region is Deg2TP a good approximation of g(x,y)? (It may be helpful to rotate the graph above to view the (visible part of the) surface from above.) 5. Find the error the approximation Deg3TP gives for g(x,y) at (1.5, 1.6).

Since you may be asked to evaluate linear and quadratic Taylor polynomials, it seems worthwhile to put some useful code for such questions in one section so you can easily use Maple for those problems (or to check your work). g := (x,y) ->sin(x+3*y)+cos(5*x-y); a:= Pi/2; b:= Pi/2; Deg0TP := g(a,b); Deg1TP := TaylorApproximation(g(x,y),[x,y]=[a,b],1); Deg2TP := TaylorApproximation(g(x,y),[x,y]=[a,b],2); "Approx" = evalf(eval(Deg2TP, {x=a+0.1, y=b-0.1})); "Function Value" =g(a+0.1, b-0.1);

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As noted above, we can get Maple to compute Taylor polynomials of arbitrary degree using the TaylorApproximation command. Taypoly1 := proc(func,xval,maxdeg) TaylorApproximation(func,[x]=[xval],maxdeg); end: This code lets us compare a function with its 15th degree Taylor polynomial, for example. f := x -> sin(x) + cos(2*x); a:= Pi/2; del := 2*Pi; f(x); "TaylorDeg15" = TaylorApproximation(f(x),[x]=[a],15); plot([f(x), TaylorApproximation(f(x),[x]=[a],15)], x=a-del..a+del, y=-5..5, axes=normal, legend = ["f(x)", "Deg15TP"]); plot(f(x)-TaylorApproximation(f(x),[x]=[a],15), x=a-del..a+del, y=-0.1..0.1, axes=normal, legend = "Deg15Error");

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The second graph is a plot of the error in using the Taylor polynomial to approximate the function. Once again, we use the subs and evalf commands if we want to evaluate at a point. "Taylor Approximation"=evalf(eval(TaylorApproximation(f(x),[x]=[a],15),x=0.3)); "Function value"=f(0.3);

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L1EvRnVuY3Rpb25+dmFsdWU2IiQiK0FlJjM3IiEiKg==

6. What degree Taylor polynomial do we need to use for the graph of the function above and the graph of its Taylor polynomial polynomial to appear the same between -5 and 5? What error do you allow in two graphs that you say look the same? 7. What degree Taylor polynomial do we need to use for the graph of the function NiMsJi0lJHNpbkc2IyomIiIkIiIiJSJ4R0YpRiktJSRjb3NHNiMqJkYqRikiIiMhIiJGMA== and the graph of its Taylor polynomial to appear the same between 0 and 6? (The Taylor approxiamtion should be centered at x=3.) 8. Evaluate the function NiMsJi0lJHNpbkc2IyomIiIkIiIiJSJ4R0YpRiktJSRjb3NHNiMqJkYqRikiIiMhIiJGMA== and its 20 degree Taylor polynomial approximation at x=4.

We can use the same approach for functions of 2 variables. Once again we use the TaylorApproximation command. Taypoly2 := proc(func,xval,yval,maxdeg) TaylorApproximation(func,[x,y]=[xval,yval],maxdeg); end: Now we would like to look at the 5th degree Taylor polynomial approximation to the function g(x,y) that we used earlier. Once we have the polynomial we compare the graphs of the original function with its Taylor polynomial. g := (x,y) ->sin(x+3*y)+cos(2*x-y): a:= Pi/2: b:=Pi/2: del := 1: ApproxPoly := TaylorApproximation(g(x,y),[x,y]=[a,b],5); plot3d([g(x,y),ApproxPoly], x=a-del..a+del, y=b-del..b+del, view=-3..3, color=[red, blue], axes=boxed, style=patch);

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Next we look at the error, or difference, between the function and its Taylor polynomial approximation. We are interested in the region where the error is no more than 0.1. g := (x,y) ->sin(x+3*y)+cos(2*x-y): a:= Pi/2: b:=Pi/2: TaylorDeg := 5: del := 1: maxerr := 0.1: ApproxPoly := TaylorApproximation(g(x,y),[x,y]=[a,b],TaylorDeg): plot3d(g(x,y) - ApproxPoly, x=a-del..a+del, y=b-del..b+del, view=-maxerr..maxerr, axes=boxed, style=patchcontour);

6(-%+AXESLABELSG6%Q"x6"Q"y6"Q!6"-%*AXESSTYLEG6#%$BOXG-%%GRIDG6%;$"/!\zEjzq&!#9$"0!\zEjzqD!#9;$"/!\zEjzq&!#9$"0!\zEjzqD!#9X,I)anythingG%*protectedG6"6"[gl'!%"!!#\bm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The shape of the error surface is fairly typical. The error is close to 0 in a region around the point of contact, then the error blows up further away. Changing either the degree of the approximation or the definition of acceptable error simply changes the size of the good patch.

9. What degree Taylor approximation do we need to use if we want to approximate the function NiMsJi0lJHNpbkc2IywmJSJ4RyIiIiomIiIkRiklInlHRilGKUYpLSUkY29zRzYjLCYqJiIiI0YpRihGKUYpRiwhIiJGKQ== for all points within 1 of (NiMqJiUjUGlHIiIiIiIjISIi, NiMqJiUjUGlHIiIiIiIjISIi) with an error of at most 0.1?