Calculus Exam
April 28, 2018
Name: Department:
Student ID number:
Instructions:
1. There are 10 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 15
6 10
7 10
8 10
Total 100
1. Multiple-Choice Questions (Select ONE answer each question)
Given function f (x), problems (a) and (b) concern the graph of its derivative f′(x) sketched below:
The graph is circular on 0 ≤ x ≤ 2, straight line on 2 ≤ x ≤ 3. Further assume that f (0) = 0.
(a) (5 points) f (3) =?
(A) 2π (B) 12 −π2 (C) 34 (D) 0 (E) π2 − 12
(b) (5 points) Which of the following statements must be false?
(A) f has an inflection point at x = 1.
(B) f is a continuous function on [0, 3].
(C) f has a local minimum at x = 1.
(D) f has a local maximum at x = 2.
(E) f (4) = 0.
(c) (5 points) Consider the improper integral
∫ ∞
1
f (x) dx,
which of the following statements is true?
(A) If limx→∞f (x) = 0, then integral converges.
(B) If f (x) is a nonzero polynomial, then integral diverges.
(C) If ∫∞
1 f (x) dx diverges, then ∫∞
1 cf (x) dx diverges for all real number c.
(D) If∫∞
1 f (x) dx converges, then there exists M > 0 such that for all x > M , f (x) > 0.
(E) All of the above are true.
(d) (5 points) Let P (x) = ∑∞
j=0ajxj be the Taylor series of a function f (x) at 0. Which of the following statements is true?
(A) If f(j)(0) exist for all j, then f (x) = P (x) near x = 0.
(B) aj = f(j)(0) for all j.
(C) f′(x) =∑∞
j=1jajxj−1 for all j.
(D) There is an even function f (x) with a3 = 1.
(E) None of the above is true.
Here, f(j) denotes the jth derivative of f .
2. Calculate (a) (5 points)
tlim→∞
( 1 + 1
4t )t
=?
(a)
(b) (5 points) ∫ 1
0
sin√
x dx =?
(b)
(c) (5 points) ∫
x− 2
x2− 4x + 13 dx =?
(c)
3. (10 points) Describe the range of a so that the curve
y + y3 − x3 − ax2− x = 0 has horizontal tangent line(s).
3.
4. (10 points) Given a metal wire of length 1 meter, fold it perpendicularly into an ”L”
shape at certain point. Place two ends on x and y-axis with edges parallel to axes (see the figure below).
What is the maximum possible volume of revolution formed when rotating the wire around y-axis?
4.
5. (15 points) Compute
xlim→2
∫x2
−4cos(t4+ t2) sin7t dt
x− 2 .
You must provide sufficient reasons to receive full credit.
5.
6. (10 points) Prove that there exists no differentiable function f (x) such that f (0) = 0, f (1) = 4, and f′(x) < ex for all x.
6.
7. (10 points) Find all positive values of p so that the integral
∫ 1 0
cos4px + 1 x2p dx converges.
7.
8. (10 points) Prove that the function
f (x) = ecos x
√e has an inverse on (0, π) and find (f−1)′(1).
8.
