Problem
1
Points
15
2
15
3ab
10
3d
10
3ef
10
SECTION:
4
5
INSTRUCTOR:
5
20
6
10
7
5
Total
100
M161, Test 1, Spring 04
NAME:
You may not use alulators on this exam.
os2 =
1 + os 2
2
sin2 =
1
1
os 2
2
Sore
1. (a) Simplify os(sin
(b) Evaluate os
1
1
2y
).
3
p
3
2
() Use the denitions of sinh, osh tanh, et to prove
osh2 x =
osh(2x) + 1
2
2
2. Calulate the following derivatives (you do not have to simplify).
d
sin 1 (2x)
(a)
dx
(b)
d osh(3x) + sinh
dx
()
d
dx
p
2x + 1
tan 1 (ex )
3
3. Evaluate
the following integrals. You must show your work.
Z
x2
(a)
(b)
x2
Z
+1
dx
x ln(x) dx
4
()
(d)
1
Z
1
Z
1
x1:001
1
x2 + 2x
dx
dx
5
(e)
(f)
Z
Z
ex
dx
1 + e2x
p 162x
8x + 1
dx
6
4. Derive the formula for integration by parts.
7
5. Calulate the following limits.
x sin(3x)
(a) lim
x!0 os(x)
1
(b) lim xe
!1
x
x
() lim
x
!0 os(x)
x
(d) lim
!
x
x2 25
5 x+5
8
6. (a) Whih of the funtions x2 and x ln(x) grows faster (or do they grow at the same speed) as x
approahes innity? Explain.
x
(b) Whih of the funtions ln(x) and ln(2x) grows faster (or do they grow at the same speed) as
approahes innity? Explain.
9
dy
7. Find the solution to the dierential equation x = 4(x2 + x2 y 2 ). Write your solution, y , as a
dx
funtion of x.
10