Calculus Exam
April 25, 2015
Name: Department:
Student ID number:
Instructions:
1. There are 9 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 10
2 15
3 10
4 20
5 15
6 10
7 10
8 10
1. Evaluate the following integrals.
(a) (5 points) ∫ 2
1
ln x dx =
(a)
(b) (5 points)
∫ 1
4
0
√ dx
1− 4x2 =
(b)
2. Let
F (θ) = 1
1
2sin θ + cos θ, 0≤ θ ≤ π 2.
(a) (5 points) Find the critical point ˆθ of F (θ), i.e. solve F′(θ) = 0 for θ.
(a)
(b) (5 points) Find the relative/local extreme values of F (θ) in (0,π2) and justify your answer.
(b)
(c) (5 points) Find the absolute extreme values of F (θ) in [0,π2].
3. Let S be the region bounded by y = sin x, x = π2, and y = 0.
(a) (5 points) Find the volume of the solid obtained by rotating S about the x-axis.
(a)
(b) (5 points) Find the volume of the solid obtained by rotating S about the y-axis.
(b)
4. Let
F (x) =
∫ x 0
3tdt.
(a) (5 points) Prove that F (x) is strictly increasing.
(b) (5 points) Find the indefinite integral
∫
3tdt =
(b)
(c) (5 points) Find F−1(ln 32 ), i.e. solve F (x) = ln 32 for x.
(c)
(d) (5 points) Find (F−1)′(ln 32 ).
Hint: Use the chain rule F′(F−1(x))· (F−1)′(x) = 1, and plug x = ln 32 into this equation.
5. Let
f (x) = 1 2ln
(1 + x 1− x
)
,−1 < x < 1.
(a) (5 points) Find f′(x).
(a)
(b) (5 points) Find the Taylor series of f′(x) at x = 0.
(b)
(c) (5 points) Find the Taylor series of f (x) at x = 0.
(c)
6. (10 points) Determine whether the following series is convergent and justify your answer.
∑∞ n=2
1 n ln n Hint: Compute the following improper integral
∫ ∞
2
dx x ln x
7. (10 points) Prove that for all x > y > 0,
√1 + x−√
1 + y < 1
2(x− y).
8. Let
f (x) = {
e−x21 if x̸= 0, 0 if x = 0.
(a) (3 points) Determine whether f (x) is continuous at x = 0 and justify your answer.
(b) (7 points) Determine whether f (x) is differentiable at x = 0 and justify your an- swer.
Hint: Use the L’Hospital’s rule。
