M161, Test 2, Fall 2004
NAME:
SECTION:
Problem
1abc
Points
15
1def
15
2
15
3
19
4
5
5
5
6
8
7
10
8
8
Total
100
Score
INSTRUCTOR:
You may not use calculators.
The Remainder Estimation Theorem: If M is a constant such that |f (n+1) (x)| ≤ M for a ≤ x ≤ b, then
n+1
|Rn (x)| ≤ M|x−a|
(n+1)! .
cos2 θ =
1 + cos 2θ
2
sin2 θ =
1 − cos 2θ
2
1. Evaluate
Z √ the following integrals. You must show your work.
(a) 2 sin x cos x dx
(b)
Z
e
x3 ln(x) dx
1
(c)
Z
16
dx
1 + 16x2
(d)
Z
(e)
Z
0
(f)
Z
x
√
dx
9 − 4x2
∞
(x2
x
dx
+ 1)2
25
dt
(t − 2)(t + 3)
2. Determine whether each sequence converges or diverges. If it converges, find the limit. If it diverges, give
some explanation (short) why.
1 + 3n − 5n3
(a) an =
2n3 + 4n
(b) An =
(c) an =
ln(n + 1)
√
n
n+1
+ (−1)n
n
3. (a) Write out the first 5 terms of the sequence defined by a1 = 1, an+1 =
an
.
n+1
1 1
1
1 1
(b) Find the formula for the n-th term of the sequence 1, − , , − , , − , · · ·
2 3
4 5
6
(c) Consider the sequence {an }, an =
that |an − L| < ǫ for all n > N .
n−1
with limit L = 1. For ǫ = 0.1 find the smallest integer N such
n
(d) Does the inequality hold for aN ? (i.e. is |aN − L| < ǫ?)
4. Set up the partial fraction expansion with constants A, B, C and so on for the rational function
Do not solve for the constants A, B, C, etc.
5. Find the 3rd order Taylor polynomial of f (x) = ln x centered at a = 1.
4 − 2t − 2t3
.
1 − t4
6. Given the 1st order Taylor formula of the function f (x) centered at a = 0
Z x
f (x) = T1 (x) + R1 (x) = f (0) + xf ′ (0) −
(t − x)f ′′ (t) dt,
0
derive the 2nd order Taylor formula.
7. Suppose that f has the following Taylor polynomial expansion centered at a = 0: T4 (x) = 2 + 5x2 + 4x4 .
(a) Calculate f (0) and explain your answer.
(b) What can you say about the first and third derivatives of f at x = 0? Explain your answer (please make
the explanation short).
1
8. The 3rd order Taylor polynomial of f (x) = sin(x) centered at a = 0 is T3 (x) = x − x3 and the error
3!
Z
1 x
term is given by R3 = −
(x − t)3 f (4) dt. For what values of x can we replace the function f (x) = sin(x)
3! 0
by T3 (x) with an error of magnitude no greater than 3 · 10−4 .