Midterm 3
Math 104 SAIL, Fall 2018
Name:
Instructions: • Write your name above. • You may use both sides of an 8.5” x 11” sheet of paper with handwritten notes. No other resources are allowed. • Please clearly mark a multiple choice option for each problem. • To obtain credit, you must show your work. You may earn partial credit based on your work, even if your final answer is wrong. Likewise, a correct answer with poor or no work will not receive full credit.
Question Points Score 1
5
2
5
3
5
4
5
5
5
6
5
7
5
8
5
9
5
10
5
Total:
50
Question 1: Find the volume of the solid formed by revolving the region bounded by the graphs of y = sec(x) tan(x), y = 0, x = 0, and x =
π 3
about the line y = −1. (a) 2π
(b)
√
3π
(c)
π2 3
(d)
√ 2+ 3 π
Page 2
(e)
2+
π π 3
(f)
√
3+
π π 3
Question 2: 1 from x = 12x 23 21 (c) (d) 24 12
Find the arc-length of the curve y = x3 + (a) 0
(b)
11 12
Page 3
1 2
to x = 1. (e)
23 12
(f)
47 24
Question 3: Evaluate the integral below. Z 0
(a) 2π
(b) 2π −
1 3
(c) 2π −
2 3
2
x4 dx 4 + x2 (d) 2π −
Page 4
4 3
(e) 2π −
8 3
(f) 2π −
16 3
Question 4: Compute the partial fraction expansion of
(x + 1)3 . x(x2 + 1)
One of the fractions has denominator x2 + 1. What is that fraction’s numerator? (a) 2x + 2
(b) 3x + 2
(c) 4x + 2
(d) x + 3
Page 5
(e) 2x + 3
(f) 3x + 3
Question 5: For a certain positive real number C, the function ( 1 if 1 ≤ X ≤ C f (X) = 2x 0 otherwise. is a probability density function for a continuous random variable X. Find the mean of X. (a)
e 2
(b)
e−1 2
(c)
e−4 4
(d)
Page 6
e2 2
(e)
e2 − 1 2
(f)
e2 − 4 4
Question 6: Determine whether the sequence below converges or diverges and find its limit. s 2n an = n + sin(n) √
(a) converges, lim an = 0
(b) converges, lim an =
(d) diverges, lim an = +∞
(e) diverges, lim an = −∞
n→∞
n→∞
n→∞
n→∞
Page 7
2
(c) converges, lim an = 2 n→∞
(f) diverges, lim an does not exist n→∞
Question 7: Determine which of the following series are convergent. For full credit be sure to explain your reasoning (e.g., say what test was used). (I)
∞ X cos(nπ) n=3
πn
(II)
∞ X n=3
1 n ln(n)2
∞ X n! (III) 3 n 7n n=3
(a) only (I)
(b) only (I) and (II)
(c) only (I) and (III)
(d) only (II)
(e) only (II) and (III)
(f) only (III)
Page 8
Question 8: Find the interval of convergence for the series below. ∞ X 3n−1 (x + 2)n √ n n n 5 n=1
(a) x = −2
(b) − 11 , − 31 3
(c) − 11 , − 13 3
Page 9
(d) − 13 , − 57 5
(e) − 13 , − 75 5
(f) (−∞, ∞)
Question 9: Find the Taylor polynomial P3 (x) of order 3 for ln(cos2 (x)) centered at x = π and evaluate it at x = 0. Then P3 (0) = (a) 0
(b) π
(c) ln(2) − π
(d) − π 2
Page 10
(e) 2π 2
(f) ln(2) − 2π 2
Question 10: Solve the initial value problem x
dy = x3 cos(x) + y, dx
y(π) = π
Find y( π2 ). (a)
π 2
(b)
3π 2
(c)
5π 2
(d)
π2 4
Page 11
(e) π +
π2 4
(f) 2π +
π2 4
