1
E3-151-09b-copyright Joe Kahlig
1. (18 points) Find
dy
for the following.
dx
(a) y = (x5 + 7)x
(b) y = log3 (x2 + 2) + arctan(5x)
(c) y = sin−1 (x2 ) + 5x
3
2. (6 points) You have been given a sample of a chemical that has a half life of 4 months. How long will it
take until 20% is gone?
E3-151-09b-copyright Joe Kahlig
3. (9 points) Simplify. When possible, give exact value.
7π
=
(a) arcsin sin
5
7π
=
(b) arccos cos
5
x
=
(c) sin cos−1
3
(x − 4)2 (9 − x)
.
(x − 1)3
Find the interval(s) where f (x) is increasing and the intervals where f (x) is decreasing.
4. (6 points) The domain of f (x) is all real numbers except x = 1. Use f ′ (x) =
2
3
E3-151-09b-copyright Joe Kahlig
5. (10 points) Compute these limits. Give exact values of these limits.
x
1
(a) lim
−
x − 1 ln(x)
x→1+
−1/x2
(b) lim (cos (3x))
x→0
4
E3-151-09b-copyright Joe Kahlig
6. (6 points) Find the absolute maximum and the absolute minimum of f (x) on the interval [−2, 0] when
2
f (x) = 50xe−x
7. (12 points) Below is the graph of the derivative f ′ (x) of a function f (x). Assume that the function f (x)
is continuous for all real numbers. Use the graph to answer the following questions.
5
4
3
2
1
−6
−5
−4
−3
−2
−1
−1
1
2
3
4
5
6
−2
−3
−4
(a) Classify the critical values of f (x) as local max, local min, or neither.
(b) Find the interval(s) where f (x) is both increasing and concave down.
(c) Give the x-values of all inflection points.
5
E3-151-09b-copyright Joe Kahlig
8. (6 points) Find the constants a and b such that the point (1, −4) is an inflection point for
f (x) = ax3 + bx2 + 10.
9. (10 points) Find f (x).
√
3
x2 − 7
′
(a) f (x) =
x
(b) f ′ (x) =
30
+ 2sec2 (x)
e3x
E3-151-09b-copyright Joe Kahlig
6
10. (7 points) Find f (x) if f (1) = 10 and
f ′ (x) =
1
+9
5x2
11. (10 points) A poster is to have an area of 180 in2 with 1-inch margins at the bottom and sides and a
2-inch margin at the top. What dimensions for the poster will give the largest printed area? Be sure to
show that your answer satisfies the above conditions.