M161, Test 1, Fall 03
NAME:
SECTION:
INSTRUCTOR:
You may not use alulators on this exam.
1
Problem
1
Points
22
2
18
3ab
12
3d
12
3ef
12
4
6
5
6
6
12
Total
100
Sore
1. (a) Simplify tan(sin
1
x).
(b) Give the denition of the natural logarithmi funtion (ln x).
() Answer True or False
(i) ln e = 0
(ii) ln x > 0 for all x, 0 < x < 1
(iii) 2 = e ln 2
(iv) lim ln x = e
x!1
(v) ex = y if and only if ln y = x
2
2. Calulate the following derivatives (you do not have to simplify).
(a)
d
log3 (x2
dx
(b)
d 3 sin x
x e
dx
()
d
sin 1 (e 3x )
dx
4)
3
3. Evaluate
the following integrals. You must show your work.
Z
1
dx
(a)
9 + 4x2
(b)
Z
os d
4
()
(d)
Z
2
xe5x dx
Z e
3
e2 x
dx
5
(e)
(f)
Z
x2
Z
x2
p1
1 + x2
dx
1
dx
5x + 6
6
4. Derive the formula for integration by parts.
7
5. For any positive number a (a =
6 1), derive the formula
loga x =
8
ln x
:
ln a
6. For the funtion f (x) =
p
x
2, answer the following questions
(a) Show that f is one-to-one.
(b) Use Theorem 7 to nd g 0 (2) where g = f
1.
1 ).
()State the domain and range of f and g (g = f
(d) Sketh the graphs of f and g on the same axes.
0
10
8
6
4
2
0
−2
−4
−6
−8
−10
−10
−8
−6
−4
−2
0
x
2
4
6
8
10
9