Problems 1d,3,5,6

Taylor Series

1. Find the Taylor series for the following functions. Use the definition and write out the series to a degree 4 polynomial.

(a) f(x) = ln(sec(x)) at c = 0. (Maclaurin Series)

(b) f(x) = x3 at c = −1. (c) f(x) = ln(x) at c = 2.

(d) f(x) = sin(x) at c = π

2 .

2. Use a known Maclaurin series to find the Maclaurin series of the function f(x) = (7x2) sin(3x)

3. Use the known Maclaurin series for f(x) = ex to find a Taylor series for g(x) = cosh(x).

4. Find the degree three Taylor polynomial for the function f(x) = (−2x + 16)4/3 at c = 4.

5. Let f(t) = sin(3t2)

(a) Find the degree 10 Maclaurin polynomial for f(t). Hint: Use a known series and not derivatives. Reduce all fractions.

(b) Find the Maclaurin series for

∫ x 0

f(t) dt =

∫ x 0

sin(3t2) dt

(c) Use (b) to find the degree 11 Maclaurin polynomial for

∫ x 0

f(t) dt =

∫ x 0

sin(3t2) dt

(d) Use (c) to estimate

∫ 0.6 0

sin(3t2) dt. (Use a calculator for d only)

6. Let f(x) = √

9 −x.

(a) Use the Binomial Series to write f(x) as a series.

(b) Use your answer from (a) to write out the 3rd degree Taylor Polynomial of f(x).

(c) Approximate T3(1) for f(x).