Calculus Exam
April 22, 2017
Name: Department:
Student ID number:
Instructions:
1. There are 7 pages (including the cover page) in this exam.
2. You have 100 minutes to work on the exam.
3. Write your answers above the answer line, if an answer line is provided.
4. The computation processes/proofs of each problem is required. An answer without any explanations will not be graded.
Problem Points Score
1 20
2 15
3 10
4 10
5 10
6 10
7 15
8 10
1. Multiple-Choice Questions (Select ONE answer each question) (a) (5 points) If ex/y = x + y, then what is dy
dx?
(A) xy
x2+ xy + y2 (B) xex/y+ y2
yex/y− y2 (C) xy
xex/y− y2 (D) y
x − 2
xy2ex/y (E) yex/y+ y2
xex/y− y2
(b) (5 points) Let f (x) = 2x3− 9x2+ 12x. Then
(A) f (x) has a local maximum at x = 1, an absolute minimum at x = 2, and an inflection point at x = 3/2;
(B) f (x) has local maxima at x = 1 and x = 3/2, and a local minimum at x = 2;
(C) f (x) has local minima at x = 1 and x = 2, and an inflection point at x = 3/2;
(D) f (x) has a local maximum at x = 1, a local minimum at x = 2, and an inflection point at x = 3/2.
(E) f (x) has inflection points at x = 1, x = 2 and x = 3/2;
(c) (5 points) Let f (x) be continuous and differentiable on [−1, 2]. Suppose that f (−1) = −5 and f(2) = 7. Which of the following statements about f(x) is NOT true?
(A) f (x) has an absolute maximum for some x on [−1, 2].
(B) There is a point c with −1 < c < 2 where f(c) = 0.
(C) There is a point c with−1 < c < 2 where f′(c) = 4.
(D) f′(x) > 0 for some values of x between−1 and 2.
(E) Any of the statements (A), (B), (C), (D) can be false depending on what other properties f (x) has.
(d) (5 points) The series
∑∞ n=1
(−1)n (n2+ 1)p is (A) absolutely convergent for p≥ 12
(B) absolutely convergent for p > 1 and conditionally convergent for 0 < p≤ 1
(C) absolutely convergent for 0 < p ≤ 12 and conditionally convergent for p > 12
(D) divergent for all values of p
(E) absolutely convergent for p > 1 and conditionally convergent for 0 < p≤
2. Calculate
(a) (5 points) lim
x→−∞
( x x + 2
)x
.
(a)
(b) (5 points)
∫ π
4
0
x cos x dx.
(b)
(c) (5 points) Write an equation for the tangent line to the graph of y = xcos(πx) when x = 3.
3. (10 points) Define f (x) =
∫x2
x et2 dt
x− 1 for x ̸= 1. Give a value of f(1) such that f is continuous at 1.
3.
4. (10 points) Suppose that f (0) = 0, f′(0) = 1 and |f′′(x)| ≤ 2 for every x ∈ [0, 3]. Find the largest possible value of f (3).
4.
5. (10 points) Prove that 1− x
1 + x < e−2x for every x∈ (0, 1).
6. (10 points) The volume enclosed by a sphere of radius r is V = 43πr3. If blowing air into a balloon at the rate of 3 cubic inches per minute, how fast is the radius r changing when the volume is 36π?
6.
7. A curve is given by the parametric equations
x = cos3t y = sin3t t∈ [0,π 2].
(a) (5 points) Find all t∈ [0,π2] at which the point tracing the curve has a maximum speed.
(a)
(b) (5 points) Calculate the arc length of the curve on the interval [0,π2].
(b)
(c) (5 points) Calculate the surface area of the solid obtained by rotating the curve about the x-axis.
8. (10 points) Find the value of
∑∞ n=0
2n
n!(n + 2). (Hint: Integrating the Taylor series of f (x) = xex. )
8.
