Math 161 Final
Problem
Value
1
20
December 12, 2011
2
9
3
9
4
32
5
8
6
20
7
40
Name:
8
20
9
10
10
10
11
10
12
12
Total
200
Points
SHOW ALL WORK.
NO CALCULATORS/CELL PHONES.
DO NOT SIMPLIFY ANSWERS.
1. Find each of the following limits:
x2 + 6x + 8
x→−2
x2 − 4
(a) lim
7x − 2x3
x→∞ x3 + 3x + 1
(b) lim
(c) lim eln(x−4)
x→5
(d) lim− f (x) and lim+ f (x) where f (x) =
x→0
x→0
x2 if x ≥ 0
1
if x < 0
x
2. USE THE LIMIT DEFINITION of the derivative to find f 0 (2) when f (x) =
1
.
2x + 1
Page 2 of 8
3. Find the equation of the tangent line to the graph of f (x) = e(3x
4. Differentiate each of the following.
√
(a) f (x) = (3 x + π 2 )4
(b) g(x) =
ln(x3 )
3x5 + x
(c) h(x) = ex ln x
2 +1)
at the point (0, e).
Page 3 of 8
(x2 − 5)(2x + 1)3
(d) f (x) = ln
4x2 + 5
5. Use implicit differentiation to find y 0 =
dy
when x2 ey + 4x3 = ln(y 2 ).
dx
6. Given f (x) = −2x3 − 9x2 + 24x − 2 (Show all work!)
(a) Find all critical numbers of f (x).
(b) The x-value of the relative maximum of f (x) is
.
The x-value of the relative minimum of f (x) is
.
Page 4 of 8
(c) The x-value of the inflection point of f (x) is
(Show all required work)
.
(d) The absolute maximum value of f (x) on the interval [−1, 2] is
.
The absolute minimum value of f (x) on the interval [−1, 2] is
.
7. Find each of the following indefinite integrals.
Z √
1
(a)
x+
dx
x
Z
(b)
√
2x2
dx
x3 + 4
Page 5 of 8
Z
(c)
Z
(d)
x2 e4x
3 +1
dx
1
dx
x ln x
8. Evaluate the following definite integrals.
Z
2
√
( 3 x − ex )dx
5
6x2 − 14
dx
x3 − 7x
(a)
0
Z
(b)
3
Page 6 of 8
9. Find the average value of f (x) =
√
x on the interval [0, 1].
10. Find the area of the region enclosed by the graphs of f (x) = x3 − 3x and g(x) = x.
Page 7 of 8
11. A blue car approaches an intersection at 60 mph. A red car has already passed the intersection
traveling at 20 mph. How fast is the distance between the cars increasing (or decreasing) when the
blue car is 4 miles from the intersection and the red car is 3 miles past the intersection?
∧ red
>
blue
Page 8 of 8
12. A wolverine pen with two dividers is made out of two types of fencing. Wood fencing that costs $5
per foot goes along one side. Chain link fencing that costs $3 per foot is used for the rest of the
pen. If our wolverine rancher has $3600 to spend on fencing, what is the maximum total area she
can enclose?
wood