Download Midterm I

Midterm 1
Math 133
Lecturer: Quoc P. Ho
April 25, 2014
Name:
Instructions: This midterm has 6 questions for a total of 100 points. Please carefully show your work in
the space provided and unless otherwise specified, please fully justify your answer. The value of each part of
each question is stated. If you need more space, you can use the final page, which is blank, or request more
paper. Please do not hesitate to ask for clarification if something is unclear. No calculator is allowed.
Good Luck!
Question
Points
1
16
2
6
3
18
4
37
5
22
6
1
Total:
100
Score
Name:
Math 133
Midterm 1
1. This question is about inverse functions of trigonometric functions.
(a) [6 points] What are the domains and images (i.e. ranges) of arcsin x, arccos x, arctan x? (2 pts each).
(NOTE. arcsin x, arccos x, and arctan x are inverse functions of sin x, cos x, and tan x respectively.)
•
 ‹˜
2
. The result should not involve any trigonometric function.
(b) [6 points] Compute sin 2 arccos
3
(c) [4 points] Use inverse function theorem to compute the derivative of arcsin x.
Total for Question 1: 16
Page 1 of 8
Name:
Math 133
Midterm 1
2. Compute the following integrals
Z
p
(a) [2 points]
x 1 − x 2 d x.
(b) [2 points]
(c) [2 points]
Z
Z
sin3 x d x.
2048
d x.
(x + 2048)2048
Total for Question 2: 6
Page 2 of 8
Name:
Math 133
Midterm 1
3. Compute the following integrals
Z
e x sin x d x.
(a) [4 points]
(b) [4 points]
(c) [4 points]
(d) [6 points]
Z
Z
Z
sec2 log x
d x.
2x
sin4 x d x.
e x − e−x
d x.
e x + e−x
Total for Question 3: 18
Page 3 of 8
Name:
Math 133
4. Compute the following integrals
Z
(a) [6 points]
(b) [7 points]
(c) [7 points]
log x d x.
Z
1
d x.
cos x
Z
1
dx
p
1 + x2
Page 4 of 8
Midterm 1
Name:
(d) [8 points]
(e) [9 points]
Math 133
Z
Z
Midterm 1
6x 2 − 3x + 1
d x.
(4x + 1)(x 2 + 1)
x cos2 x sin x d x. (HINT. Use integration by parts.)
Total for Question 4: 37
Page 5 of 8
Name:
Math 133
5. Compute the following limits
sin x
.
(a) [2 points] lim
x→0 x
(b) [2 points] lim
x→0
(c) [4 points]
|x|
.
x
lim (x 4 − x 3 ).
x→+∞
Page 6 of 8
Midterm 1
Name:
Math 133
(d) [4 points] lim x x
Midterm 1
2
x→∞
(e) [5 points] lim+
x→1

‹
x
1
.
−
x − 1 log x
(f) [5 points] lim+ log((x + 1)cot x ).
x→0
Total for Question 5: 22
Page 7 of 8
Name:
Math 133
6. [1 point] What do you think about this exam?
Page 8 of 8
Midterm 1