MATH 152, SPRING 2010
FINAL EXAM - VERSION C
1. Find the area of the region bounded by y = 0, y =
√
16 − 4 2
(a)
3
√
8 2
(b)
3
√
6 3
(c)
2
√
4 2
(d)
3
10
(e)
3
ˆ e
2. Compute
x ln x dx.
√
x + 1, and y =
√
3 − x.
1
(a) e
(b) e2
1
(c)
2
e2 + 1
(d)
4
3e2 − 1
(e)
4
3. Find the average value of the function g(x) =
√
2 3
(a) 6 −
3
√
3
(b) 3 −
3
√
3
3
(c) −
2
6
√
(d) 18 − 2 3
√
(e) 9 − 3
√
∞
X
(−1)n ln(n)
4. The series
:
n3/2
n=1
(a) diverges by oscillation
(b) converges, but is not absolutely convergent
(c) converges to 0
(d) is absolutely convergent
(e) converges to 4
1
1 + 2x on the interval [1, 4].
ˆ
5. With an appropriate substitution,
x2
p
x2 − 4 dx is equivalent to which of the following?
ˆ
(a) 8 sin2 θ cos θ dθ
ˆ
(b) 16 sec2 θ tan3 θ dθ
´
(c) 16 sec3 θ tan2 θ dθ
ˆ
(d) 8 sec2 θ tan θ dθ
´
(e) 16 sin2 θ cos2 θ dθ
6. Which statement is true about the sequence an = (−1)n
n+1
?
n
(a) an diverges by the Test for Divergence
(b) an converges to 1
(c) an converges to −1
(d) an diverges by the Alternating Series Test
(e) an diverges by oscillation
7. The series
∞
X
2
:
n(n
+ 2)
n=3
(a) converges to 0
(b) diverges by comparison with
∞
X
1
n
n=3
7
12 3
(d) converges to ln
5
3
(e) converges to
2
(c) converges to
ˆ
8. Which of the following is the Maclaurin Series for
∞
X
(a)
(−1)n x4n+3
(4n + 3)(2n + 1)!
n=0
(b)
∞
X
(−1)n (x2 )2n+2
(2n + 2)!
n=0
∞
X
(−1)n+1 x2n+4
(c)
(2n + 4)(2n + 1)!
n=0
∞
X
(d)
(−1)n x2n+4
(2n + 4)(2n + 1)!
n=0
(e)
∞
1 X (−1)n+1 x4n+2
2 n=0 (4n + 3)(2n + 1)!
2
sin(x2 ) dx ?
9. Which integral gives the length of the curve y = x2 −
ˆ
2
2x −
(a)
1
ˆ
2
1
8x
r
4x2 −
(b)
1
ˆ
2
r
x4 −
(c)
1
ˆ
2
1
ln x from x = 1 to x = 2?
8
dx
1
+ 1 dx
64x2
x2 ln x (ln x)2
+
+ 1 dx
4
64
r
x6
x2
1
+ 1 dx
−
+
9
12 64x2
1
ˆ 2
1
(e)
2x +
dx
8x
1
(d)
10. Which is the graph of the polar equation r = 8 + 8 sin θ ?
3
h πi
11. The graphs of y = 2 sin x and y = sec x are shown below on the interval 0, :
2
(a) Find the volume obtained by rotating Region I about the y-axis.
(b) Find the volume obtained by rotating Region II about the x-axis.
12. The curve parametrized by x = 3t − t3 , y = 3t2 , t ∈ [0, 1] is rotated about the x-axis. Find the
area of the surface formed.
13. Given the power series
∞
X
(x − 1)2n
√
:
n + 1 (9n )
n=0
(a) Find the radius of convergence.
(b) Find the interval of convergence.
14. .
(a) Find the third-degree Taylor Polynomial of f (x) = ln x centered at x = 2.
(b) Estimate the error in your approximation on the interval [1.5, 2.5]. (Taylor’s Inequality may
M
|x − a|N +1 where M ≥ |f (N +1) (x)| )
be helpful here: |Rn (x)| ≤
(N + 1)!
15. The points (−1, 3, 2) (0, −1, 0), and (−2, 4, 3) lie in a plane.
(a) Find a UNIT vector n which is orthogonal to this plane.
(b) Show that the vector n is orthogonal to one of the vectors which lie in the plane.
4