MATH 152, SPRING 2012
COMMON EXAM III - VERSION B
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PART I: Multiple Choice (4 pts each)
1. Which of the following is a unit vector in the direction of b − 3a, where a = h1, −2, 2i and b = h5, 0, −3i?
2
6 9
(a) − , − ,
11 11 11
2 6
9
(b)
, ,−
11 11 11
(c) h2, 6, −9i
2
18
14
, −√
,√
(d) − √
524
524 524
14
2
18
(e) √
,√
, −√
524
524
524
2. Find the sum of the series
∞ (−2)n
P
.
n!
n=0
(a) 0
(b) e−2
(c) e2
(d) sin(−2)
(e) cos(−2)
3. Represent
(a)
1
as a power series about 0 and give the interval of convergence.
1 + 4x2
∞
X
1
=
(−4)n x2n , where |x| < 4.
1 + 4x2
n=0
∞
X
1
1
=
4n x2n , where |x| < .
(b)
1 + 4x2
4
n=0
(c)
∞
X
1
1
=
4n x2n , where |x| < .
2
1 + 4x
2
n=0
(d)
∞
X
1
1
=
(−4)n x2n , where |x| < .
1 + 4x2
4
n=0
(e)
∞
X
1
1
=
(−4)n x2n , where |x| < .
2
1 + 4x
2
n=0
1
4. Find the cosine of the angle, θ, between the vectors a = h2, 0, 1i and b = h1, −2, 1i.
5
(a) cos(θ) = √
30
4
(b) cos(θ) = √
30
5
(c) cos(θ) = √
12
3
(d) cos(θ) = √
30
3
(e) cos(θ) = √
12
5. Find the 15th derivative at x = 0 for f (x) =
∞
P
n=0
(−1)n
xn .
+ 5)
7n (n
(a) f (15) (0) = 0
15
715 (20)
15
(c) f (15) (0) = − 15
7 (20)
15!
(d) f (15) (0) = − 15
7 (20)
15!
(e) f (15) (0) = 15
7 (20)
(b) f (15) (0) =
6. The series
∞
P
n=1
en
1
√ is
+ n
∞ 1
P
1
1
√
and
is a convergent geometric series.
<
n
en + n
en
n=1 e
∞
P
1
1
1
√ is a divergent p series.
divergent because n √ > √ and
e + n
n
n
n=1
∞
P
1
1
1
√ is a divergent p series.
divergent because n √ < √ and
e + n
n
n
n=1
∞
P
1
1
1
√ is a convergent p series.
convergent because n √ < √ and
e + n
n
n
n=1
∞ 1
P
1
1
divergent because n √ > n and
is a divergent geometric series.
n
e
e + n
n=1 e
(a) convergent because
(b)
(c)
(d)
(e)
2
7. Find the Taylor Polynomial of degree 2, T2 (x), for f (x) =
1
(x − 9)2
108
1
T2 (x) = 3 + (x − 9) −
6
1
T2 (x) = 3 + (x − 9) −
6
1
(x − 9)2
T2 (x) = −
216
None of these.
√
x centered at a = 9.
(a) T2 (x) = −
(b)
(c)
(d)
(e)
8. The series
1
(x − 9)2
108
1
(x − 9)2
216
∞ (−1)n n
P
2
n=1 n + 1
(a) is convergent but not absolutely convergent.
(b) is absolutely convergent.
(c) is divergent.
(d) is absolutely divergent but not divergent.
(e) converges absolutely by the ratio test.
9. What does it mean to say
∞
P
an is convergent?
n=1
(a) lim an = 0.
n→∞
(b) lim an exists.
n→∞
(c) There are numbers A and A′ such that A ≤ an ≤ A′ for all n.
(d) There are numbers B and B ′ such that B ≤ a1 + a2 + ... + an ≤ B ′ for all n.
(e) lim (a1 + a2 + ... + an ) exists.
n→∞
10. The series
∞
P
2
ne−n
n=1
(a) converges by the Comparison Test with
∞
P
2
e−n .
n=1
(b) diverges by the Ratio Test.
(c) converges by the Integral Test.
(d) diverges by the Test for Divergence.
(e) diverges by the p-series test.
3
11. If we apply the Ratio Test to the series
∞ (−4)n n2
P
, which of the following statements is true?
5n+1
n=1
(a) lim
an+1
4
= < 1, therefore the series converges.
an
5
(b) lim
4
an+1
= − < 1, therefore the series converges.
an
5
(c) lim
an+1
4
=
< 1, therefore the series converges.
an
25
(d) lim
an+1
= ∞ > 1, therefore the series diverges.
an
(e) lim
an+1
= 0 < 1, therefore the series converges.
an
n→∞
n→∞
n→∞
n→∞
n→∞
12. What is the intersection of the sphere (x − 2)2 + (y − 1)2 + (z − 3)2 = 16 with the xy-plane?
(a) The circle (x − 2)2 + (y − 1)2 = 7.
(b) The circle (x − 2)2 + (y − 1)2 = 1.
(c) The point (2, 1, 0).
(d) The circle (x − 2)2 + (y − 1)2 = 16.
(e) The sphere does not intersect the xy-plane.
13. In three-dimensional space, R3 , the equation x2 + y 2 = 9 describes
(a) a plane.
(b) a circle.
(c) a parabola.
(d) a cylinder.
(e) a sphere.
4
PART II: WORK OUT (48 points total)
Directions: Present your solutions in the space provided. Show all your work neatly and concisely and box your
final answer. You will be graded not merely on the final answer, but also on the quality and correctness of the work
leading up to it.
14. a.) (5 pts) Using the known Maclaurin series for sin x, find the Maclaurin series for sin(x3 ).
sin(x3 ) − x3
.
x→0
x9
b.) (5 pts) Using the Maclaurin series for sin(x3 ), compute lim
5
2
15. a.) (5 pts) Using the known Maclaurin series for ex , find a Maclaurin series for e−x .
2
b.) (5 pts) Expand the Maclaurin series for e−x found above to degree 4. Use this 4th degree Maclaurin polynomial
Z 1
2
to estimate
e−x dx. Do not simplify.
0
6
16. (10 pts) Find a power series about 0 for f (x) = ln(5x + 4). What is the associated radius of convergence?
7
17. (10 pts) Find the interval of convergence of the series
∞ (3x − 1)n
P
. Be sure to test the endpoints for convergence.
n6n
n=1
8
18. Consider the series
∞ (−1)n
P
.
2
n=1 3n + 1
a.) (4 pts) Prove the series is absolutely convergent.
b.) (2 pts) Find s2 , the sum of the first 2 terms, to approximate
∞ (−1)n
P
.
2
n=1 3n + 1
c.) (2 pts) Find an upper bound on |R2 |, the absolute value of the remainder, in using s2 to approximate
9
∞ (−1)n
P
.
2
n=1 3n + 1
Last Name:
First Name:
Section No:
MATH 152, SPRING 2012
COMMON EXAM III - VERSION B
Question
Points Awarded
Points
1-13
52
14
10
15
10
16
10
17
10
18
8
100
10