Spring 2014
Math 151
Week-in-Review 10
1
1. Find the absolute maxima and the absolute minima for
a. 𝑓(𝑥) = 𝑥 3 − 3𝑥 2 − 1 on the interval [−1, 1]
b. 𝑓(𝑥) = √6𝑥 − 𝑥 2
c. 𝑓(𝑥) = 𝑥 2 𝑒 −𝑥 on the interval [1, 4]
3𝜋
d. 𝑓(𝑥) = sin2 𝑥 + cos 𝑥 on [0, ]
4
2. Find the intervals where the following functions are concave up or concave down.
a. 𝑓(𝑥) = ln(𝑥 2 + 6𝑥 + 13)
b. 𝑓(𝑥) = 𝑥 3 𝑒 −2𝑥
3. Find the intervals where the following functions are increasing, decreasing, concave up
and concave down. Locate all the local extrema and points of inflection. Sketch the graphs.
2𝑥−4
a. 𝑓(𝑥) =
𝑥+2
4. A box with an open top has a volume of 400 cubic feet. If the height of the box is twice its
width, find the dimensions of the box with minimum surface area.
5. A circular cylinder with an open top has a volume of 192𝜋 in2. If the cost of the material
for the bottom of the cylinder is 15 cents per square inch and the cost of the material for
the sides of the cylinder is 5 cents per square inch, what is the ideal height and radius of
the cylinder that will minimize the material cost?
3
1
3
6. Find the most general antiderivative for 𝑓(𝑥) = √𝑥 + 2 +
+ sin 𝑥 + 𝑥 7
2
′ (𝑥)
𝑥
6
√1−𝑥
𝑥
7. Given 𝑓
= + 𝑒 − 4 and 𝑓(1) = 6, find 𝑓(𝑥)
𝑥
8. The population of bacteria quadruples every 3 days. How long will it take for the population
to be 10 times its initial size?
9. Calculate the following limits:
a. lim arcsin(
𝑥→∞
1−3𝑥 2
6𝑥 2 −𝑥
3
)
b. lim ln(ln(1 + ))
𝑥→∞
c. lim+(cos 3𝑥)
𝑥
1
𝑥2
𝑥→0
1
10. Solve for 𝑥: 2 log 9 (𝑥 − 2) − log 9(8 − 𝑥) =
2
11. Find the derivative of
a. 𝑓(𝑥) = arctan(ln 𝑥) + 𝑥 2 arcsin(𝑒 2𝑥 )
b. (𝑥 3 + 1)tan 𝑥
12. Find the value of
13𝜋
16𝜋
19𝜋
22𝜋
a. arccos (cos ) , arccos (cos ) , arccos (cos ) , arccos (cos )
6
−1 2
tan(sin
)
9
3
2
6
6
6
b.
13. If 𝑓(𝑥) = 𝑥 − 2𝑥 + 5𝑥 and 𝑔 = 𝑓 −1 , find 𝑔′(10)
14. If the acceleration of a particle is given by 𝑎⃑(𝑡) =< cos 𝑡, 𝑡 >, 𝑣(0) =< 2,3 >
and 𝑟(0) =< 1,1 >, find the position function of the particle for any time t.
15. An open top cardboard box is made from a 10 inch by 12 inch cardboard piece by cutting
identical squares from the corners and then by folding up the flaps. Find the dimensions of
the box that will maximize its volume.