12.11 Applications of Taylor Polynomials Infinite Sequences & Series LAST ONE! ☺

Goal: Approximate a Function with a Taylor Polynomial Why? ◦ Polynomials are the easiest functions to work with

How? ◦ Physics & Engineering Relativity Optics, Blackbody radiation Electric dipoles Velocity of water waves Constructing highways across a dessert

Approximating Functions by Polynomials Suppose that f(x) is equal to the sum of its Taylor series at a: ∞

f ( x) = ∑ n=0

f ( n ) (a) n x − a ( ) n!

nth partial sum for a Taylor Series is the nth-degree Taylor Polynomial n

Tn ( x) = ∑ i =0

f (i ) (a ) i ( x − a) i!

f '(a ) f ''(a) f ( n ) (a) 2 n = f (a ) + ( x − a) + ( x − a ) + ... + ( x − a) 1! 2! n!

1st Degree Taylor Polynomials Same as the linearization of f at a f '(a) T1 ( x) = f (a ) + ( x − a) 1!

Notice: and its derivative have the same values at T a1that f and f’ have Goal: Show that the derivatives of Tn at a agree with those of f up to and including derivatives of order n.

Look at the graph:

3 Methods for Estimating the Size of the Error: 1.

If a graphing device is available, we can use it to graph Rn ( x) and thereby estimate the error.

2.

If the series happened to be an alternating series, we can use the Alternating Series Estimation Theorem

3.

In all cases we can use Taylor’s Inequality

Example 1A: Approximate the function by a Taylor polynomial of degree 2 at a=8 f ( x) = x 3

Example 1B: How accurate is the approximation when 7 ≤ x ≤ 9

Example 2A: What is the maximum error possible in using the−0.3 ≤ x ≤ 0.3 approximation below when ? x x5 sin x ≈ x − + 3! 5!

Example 2B: Use the approximation to find sin12⁰ sin12⁰ correct to six decimal places.

Example 2C: For what values of x is the approximation accurate to within 0.00005?

Example 2: With Taylor’s Inequality

Graphically

Homework: Page 791 Problems 3-9 odd, 13-21 odd