Homework #4 Math 126
These problems use the techniques of section 5 except for differentiation and inte-
gration of series. Each problem can be derived from the basic series given in Examples
4.2.
(a) In problems 1-6, find the Taylor series for f(x) based at b. Your answer should have
one Sigma (Σ) sign. On some problems you might want to describe the coefficients
using a multi-part notation as in Example 5.5.
(b) Then write the solution in expanded form: a0+a1(x−b) + a2(x−b)2+... where you
write at least the first three non-zero terms explicitly.
(c) Then give an interval Iwhere the Taylor series converges.
Note that there are some hints below.
1. f(x) = cos(3x2) based at b= 0.
2. f(x) = sin2(x) based at b= 0.
3. f(x) = e4x−5based at b= 2.
4. f(x) = sin(x) based at b=π
6.
5. f(x) = 1
4x−5−1
3x−2based at b= 0.
6. f(x) = x
(2x+ 1)(3x−1) based at b= 1.
7. The “sinh” and “cosh” functions are used, for example, in electrical engineering,
and are defined by sinh(x) = (ex−e−x)/2,and cosh(x) = (ex+e−x)/2.Do questions (a)
and (b) above for the function h(x) = 2 sinh(3x)−4 cosh(3x) based at b= 0.
8. Find the 6th degree Taylor polynomial for f(x) = sin(3x−5) based at b= 0,
without differentiating.
Hints:
Change the base from bto 0 by substituting u=x−b.
Be sure that the terms in your answers are numbers (coefficients) times powers of
x−b.
Use the double angle formula in problem 2.
Use partial fractions in problem 6.
Use the addition formulae for sin(A±B) in problems 4 and 8.
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