Math 161- Final Exam
December 17, 2010
Name
Prob.
1
2
3
4
5
6
7
8
9
10
11
12
Total
Value
20
9
9
32
8
20
50
10
10
10
10
12
200
Points
SHOW ALL WORK.
NO CALCULATORS.
NO CELL PHONES.
DO NOT SIMPLIFY ANSWERS.
1. Find each of the following limits, if they exist. If the limit equals ±∞ , indicate so.
(a)
2
lim ln  x 1
x 4
2
(b)
lim
x −4
x −16
2
x x−12
3
(c)
3x −4x
lim
3
x∞
1−x
{
2
x
if x3
lim f  x where f x= 9
if x=3
(d) x 3
2x2 if x3
2. Use the LIMIT DEFINITION of the derivative to find
f ' 3 when
f  x =
1
.
2x1
3. Find an equation of the tangent line to the graph of
f  x=ln  x 2 at the point (1, 0).
4. Differentiate each of the following. (DO NOT simplify answers.)
(a)
f  x= xe
(b)
g  x =
3x
4
3x −1
 1− x 2
(c) h x = ln [x 3−42 4x−1]
(d) k  x = e  ln 2
2
5. Use LOGARITHMIC DIFFERENTIATION to find y ' when
6. For
f  x=2x 3−9x 2−24x 1
y =
4
63x −1
 x 75
(Show all necessary work.)
(a) Find all critical numbers of f  x.
(b) Find all relative extrema of
f  x.
The x-value of the relative maximum of f  x  is ________.
The x-value of the relative minimum of f  x is ________.
(c) Find the x-values of the inflection point(s) of f  x.
(d) Find the absolute maximum value and absolute minimum value of f  x on [-2, 0].
7. Find each of the following indefinite integrals:
(a)
∫ 3x 2− 3 x dx
(b)
∫ xe x 1
2
dx
(c)
∫
6x 22
dx
 x 3x
(d)
∫
ln4x 
dx
x
(e)
∫ 2x1 x−5
dx
8.
2
∫0 e x −7x3 dx
9. Find the average value of
3
f  x= x −x on the interval [-1, 2].
10. Find the area of the region enclosed by the graphs of
f  x= x
4
and g  x=x.
11. Two highways intersect, one going north-south, the other east-west. Two cars are approaching the
intersection; one car is traveling north at 60 mph while the second car is traveling west at 50 mph.
How fast is the distance between the cars changing when the northbound car is 3 miles from the
intersection, and the westbound car is 4 miles from the intersection?
12. Carol Calculus wants to send a big box of presents home for the holidays. US Postal service
regulations specify that the maximum combined length plus girth cannot exceed 108 inches. Find
the dimensions of a package with a square base that has the largest volume that can be shipped.
(Girth is distance around the box.)
L
x