9.3 Taylor's Theorem

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AP Calculus BC 9.3 Taylor’s Theorem Objective: able to approximate a function with a Taylor polynomial; to analyze the truncation error of a series both graphically & using the Remainder Estimation Theorem

1. Given the function f (x) = sin 2 x, construct the Taylor polynomial of order 4 at x = 0 , and use it to approximate the function at x = 0.1 . (hint: Double Angle Trig. Identity = ).

2. For the function f (x) = e x, give the 3 rd order Taylor polynomial, then use your grapher to calculate the largest error on the interval [-1, 2].

3. From problem 2, what order Taylor polynomial gives an error of less than 0.01 on the interval [-1, 2]?

4. Find a formula for the truncation error if P (x) is used to approximate = . 4

For most functions, approximating Taylor polynomials are just that, approximations. They are not usually exact. The terms we leave off an infinite polynomial create a truncation error. The truncation error for a geometric series is simply the sum of the rest of the terms of the series, which itself is a geometric series. It is slightly more complex for Taylor series.

If you truncate a Taylor series you create 2 parts, the approximating polynomial, and the remainder.

Taylor’s Theorem with Remainder

Taylor’s Theorem is an existence theorem. It claims the existence of a value for the remainder which makes the polynomial plus remainder exactly equal to the function value.

-1 5. Using your grapher, find the maximum error when P5(x) is used to approximate f (x) = tan x on | x|≤0.2.

6. Use the 4 th order Taylor polynomial for f (x) = cos x and Taylor’s Theorem to determine the error in estimating f (0.1 ) = cos x.

Remainder Estimation Theorem

Note that Mr n+1 is the maximum value of the absolute value of the first unused derivative on the working domain.

7. Use the 3 rd order Taylor polynomial for f (x) = e x and Taylor’s Theorem to determine the maximum error in estimating f (x) = e x on |x| ≤ 0.5.

8. Prove that the Taylor Series for cos x converges for all real x.

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