LEBANESE AMERICAN UNIVERSITY
DEPARTMENT OF COMPUTER SCIENCE AND MATHEMATICS
MTH 201 - CALCULUS III
Exam-I, Spring 2015
Duration: 60 minutes
INSTRUCTIONS: This exam consists of 8 pages and 5 problems. Check that none is missing.
Answer the questions in the space provided for each problem; if more space is needed, you may use
the back pages. To receive full credits, you have to justify your answers.
Student’s Name:
Grading scheme
(Keep it empty)
Question 1
/15
Question 2
/20
Question 3
/20
Question 4
/25
Question 5
/20
Total
/100
1. [15 Points] Simplify the following equations
(a) sin(cos−1 x) =
(b) cosh(ln 5) =
2. [20 Points] Calculate the following integrals
Z
1
dx =
2
x(x + 1)
Z
√
1
dx =
−x2 + 4x
3. [20 Points] Calculate the following improper integrals
Z e
1
√
dx =
1 x ln x
Z
1
∞
√
cosh x
√
dx =
x
4. [25 Points] Determine whether the following improper integrals converge or diverge. Justify
your answer and precise the test your are using.
Z ∞√
x+1
dx
x2
1
Z
1
∞
ln x
√ dx
x x
Z
0
∞
ex
1
√ dx
+ x
5. [20 Points] Determine if the following sequences converge or diverge.
√
n
(a) an =
.
(ln n)2
sin(n! + 3n2 + 10)
√
(b) an =
5 n
n 9
2n
(c) an = 1 −
1+
n
n!
(2n + 1)n+1
(d) an =
3n(2n − 4)n