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Week 13

Exercise 1. Find the Taylor polynomial of order 3 for the functiong(x) =9 + x. (Hint: The coefficients are rational, and some have very large denomi-

nators.)

Exercise 2. Consider the function f(x) = 2x3 + 7x2 9x 11.

(A) Find the order 3 Taylor polynomial for f(x) centered at x = 0.

(B) Find the order 3 Taylor polynomial for f(x) centered at x = 1. (Hint:If you do this correctly, this polynomial equals f(x), although it has adifferent format.

Exercise 3. Suppose that g(x) is a function such that g(0) = 8, g(0) = 5,g(0) = 5, g(0) = 16, g4(0) = 44 and g5(0) = 44.

(A) Find the order 5 Taylor polynomial for g(x) centered at x = 0.

(B) Find the order 4 Taylor polynomial for h(x) = g(x) centered at x = 0.

Exercise 4. Suppose that the order 5 Taylor polynomial, centered at 0, for afunction F (x) is

3

2+

3x

2+

9x2

4 x3 x

4

12 5x

5

28.

What is F (0)? What is F (0)? What is F (5)(0)?