Mathematics 36, section 5 -- AP Calculus

Lab Day 3: Taylor Polynomials, Approximations

September 28, 1999

Goals

In today's lab will use Maple to compute Taylor polynomial approximations, and use graphical and numerical methods to analyze the error in Taylor approximations.

Background

Recall from class on Monday that if

|f⁽ⁿ⁺¹⁾(x)| <= M

|f(x) - p_n(x)| <= (M/(n+1)!) |x-a|ⁿ⁺¹

Taylor Polynomials in Maple

Maple has a ``built-in'' function called taylor that can be used to compute Taylor polynomials of functions. The general format is

taylor(f(x),x=a,d);

where f(x) is the function to be approximated, a is the x-value where the Taylor polynomials will be expanded, and d >= 1 is an integer. For example, try entering:

taylor(exp(x),x=0,6);

Note two things:

1) The output

1 + x + (1/2)x² + (1/6)x³ +(1/24)x⁴+ (1/120)x⁵ + O(x⁶)

is a polynomial, plus another term -- O(x⁶) -- that describes the size of the error. The way to interpret this is that the error will go to zero like x⁶ (at least) as x -> 0. To get rid of the error term, you can ``nest'' the taylor command inside a convert command like this:

convert(taylor(exp(x),x=0,6),polynom);

Try this and note the output.

2) The 6 in the Taylor command is one more than the degree of the polynomial. To get the nth degree polynomial, you will always want to take d = n + 1.

Lab Questions

In these questions, you will generate plots of sin(x), together with its Taylor polynomials of degrees n = 3,5,7,9,11 at a = 0 and compare the accuracy of the Taylor approximations.

1. First plot sin(x) and it Taylor polynomial of degree 3 together on the same axes with -3 <= x <= 3. You can use these commands, for instance, first to compute the Taylor polynomial, assign it the name p3, then plot it with the sine function:

   p3 := convert(taylor(sin(x),x=0,4),polynom);

   plot({sin(x), p3}, x=-3..3);

   Note: There is no (x) after the p3 in the plotting command; we computed the Taylor polynomial as an expression in the command before, assigned it to the symbolic variable p3 and then used that expression in the plot command.

2. Now plot the absolute error function: abs(sin(x)-p3), together with the function abs(x)^4/24 using the same range of x values: -3 <= x <= 3. Which graph is which? Also, explain how your graphs relate to the Theoretical Error Bound for Taylor approximations to the sine function (see the Background section above).

3. Repeat part 2, but replacing the function abs(x)^4/24 with abs(x)^5/120. Use the same range of x values: -3 <= x <= 3. Which graph is y = abs(sin(x)-p3) and which is y = abs(x)^5/120? You should see that abs(sin(x)-p3) <= abs(x)^5/120 on the whole interval. Why is this true? (Hint: What is the 4th degree Taylor polynomial for sin(x) at a = 0?)

4. Repeat part 1 for the Taylor polynomial of degree 5 of sin(x) at a = 0.

5. Does using the fifth degree polynomial to approximate sin(x) seem to yield better results than using the polynomial of degree 3? For instance, is the absolute error for this polynomial smaller than the absolute error for p₃ (x) on the whole interval?

6. Repeat part 1 for the Taylor polynomials of degree 7, 9, 11 of sin(x) at a = 0. When you graph the polynomials of degree 9 and 11 together with the sine function, you may actually only see one graph on the interval -3 <= x <= 3. What does that mean? Try plotting a bigger range of x-values until you see the graphs start to "diverge". When does that happen for the degree 9 polynomial? When for the degree 11 polynomial?

7. What is the smallest value of n for which the Taylor polynomial of degree n (always with a = 0) approximates sin(x) with 3 decimal digit accuracy for all x between -5 and 5? Use the Theoretical Error Bound above first to get a rough idea how big n should be. Does a smaller n than that "work", though?

Lab Writeups Due:

Friday, October 1.