M161, Test 2, Spring 2005
NAME:
SECTION:
Problem
1abc
Points
18
1def
18
2
18
3
12
4
6
5
10
6
6
7
6
8
6
Total
100
Score
INSTRUCTOR:
You may not use calculators.
Taylor Inequality: If M is a constant such that
|f (n+1)(x)| ≤ M for a ≤ x ≤ b, then |Rn(x)| ≤
M |x−a|n+1
.
(n+1)!
1 + cos 2θ
2
cos θ =
2
1 − cos 2θ
2
sin θ =
2
1. Evaluate the following integrals. You must
show
work.
Z your
8
1
√
√
(a)
dx
x( x + 1)
2
(b)
Z
x2 − x + 1
dx (Hint: Divide)
2
x −x
(c)
Z
x3 + 1
dx (Hint: Divide)
2
x +1
(d)
Z 1
(e)
Z 1
1
(f)
Z
1
0 (4 − x2)3/2
0 x1/3
dx
t sin 3t dt
dx
2. Determine whether each sequence converges
or diverges. If it converges, find the limit. If it
diverges, give
explanation
(short) why.
some
1
n+1
1−
(a) an =
2n
n
ln(n + 1)
(b) an =
n2
n cos nπ
(c) an =
n+1
3. Consider the function f (x) = 1/(x + 1). We
compute the 3rd order Taylor polynomial about
a = 2 by writing the following table.
n
0
1
2
3
4
5
f (n)(x)
(x + 1)−1
−(x + 1)−2
2(x + 1)−3
−3!(x + 1)−4
4!(x + 1)−5
−5!(x + 1)−6
f (n)(2)
3−1
−3−2
2!3−3
−3!3−4
4!3−5
−5!3−6
1 x2 − 1 x3. Use
We get T3(x) = 31 − 19 x + 27
81
the Taylor Inequality to bound the error due to
the approximation of f(x)=1/(x+1) by T3 on the
interval [1.5, 2.5]. (Since you do not have calculators, you do not have to do the arithmetic.)
4. Let g(x) = −6x4 + 3x3 − 17x2 + 2x + 5 and
let Tn(x) be the nth degree Taylor polynomial
of g(x) at a = 0.
(a) What is T7(x) − T5(x)? Justify your answer.
(b) Find the remainder term R3(x). Justify your
answer
5. Find the 3rd order Taylor polynomial of f (x) =
e3x centered at a = 0.
6. (a) Assume that the 2nd order Taylor polynomial of the function f about a = 0 is given by
T2(x) = a + bx + cx2. What can you say about
the signs of a, b and c if the graph of T2(x) is
given below in Figure 1? Explain.
Figure 1
2
1.5
(a) Plot of
T2(x)
1
(b) Plot of f(x)
0.5
0
−0.5
−1
−1.5
−2
−1.5
−1
−0.5
0
x
0.5
1
1.5
2
(b) What can you say about the signs of a, b and
c if the graph of f (x) is given above in Figure
1? Explain.
7. Let Rn(x) be the nth order remainder term
for the Taylor polynomial of the function f expanded about a = 0. Assume that the remainder term R2(x) is given by
Z x
1
f (3)(t)(x − t)2 dt.
R2(x) =
2! a
UseZ integration by parts to show that R3(x) =
1 x (4)
f (t)(x − t)3 dt.
3! a
8. Use the Nondecreasing Sequence Theorem
[The Nondecreasing Sequence Theorem: A nondecreasing sequence of real numbers converges if
and only if it is bounded from above. If a nondecreasing sequence converges, it converges to its
least
bound.] to prove that the sequence
upper
2n + 1
converges. Also use the Theorem to
3n + 4
prove that the sequence converges to 2/3.