Problem
1
Points
15
2
20
3
15
M161, Test 1, Fall 04
NAME:
4
15
5
5
SECTION:
6
15
INSTRUCTOR:
7
5
8
5
9
5
Total
100
You may not use calculators on this exam.
Score
1. (a) Define the natural logarithm ln(x).
−1
(b) Evaluate cos
−3π
. You must give an exact answer—no decimal approximations.
4
(c) Simplify the expression 2 sinh(ln 3x).
2. Calculate the following derivatives (you do not have to simplify).
d
(ln x3 )2
(a)
dx
(b)
√ i
d h 2√
x cosh x + x cosh x
dx
(c)
√
d
arcsin 3x + 2
dx
(d)
i
d h x3
2 + log5 x1/7
dx
3. Evaluate the following integrals. You must show your work. For definite integrals give the exact
answer—do not give a decimal approximation.
Z π/2
4 cos θ
(a)
dθ
−π/2 3 + 2 sin θ
(b)
Z
(c)
Z
16
dx
16x2 + 9
5
−1
−2e−x + 10e5x dx
4. Suppose that f (x) = ex + 1.
(a) Give the domain and range of f .
(b) On the axes given below plot f . Use that plot to explain why or why not the function has an
inverse.
0
5
4
3
2
1
0
−1
−2
−3
−4
−5
−5
−4
−3
−2
−1
0
x
1
2
(c) If possible, find f −1 and give the domain and range of f −1 .
3
4
5
5. Use logarithmic differentiation to calculate
6. Calculate the following limits.
3x − sin(3x)
(a) lim
x→0
x
x2 + 1
x→1 x + 1
(b) lim
x3 − 1
x→1 x − 1
(c) lim
dy
if y = (sin x)2x
dx
7. Show mathematically (a graph is nice but not enough) that 17x3 + 5x2 − 3 grows more slowly
than ex as x approaches infinity.
8. Suppose that f (a) = g(a) = 0, that f ′ (a) and g ′ (a) exist and that g ′ (a) 6= 0. Show that (prove
that)
f ′ (a)
f (x)
= ′
.
lim
x→a g(x)
g (a)
9. Find the solution to the differential equation y
p
dy
= t2 1 + y 2 with initial condition y(0) = 1.
dt