Week 12: Week In Review
MATH 131
Final EXAM Review
DROST
Section 1.1
1. A rectangular storage container with an
open top has a volume of 12𝑚3. The
length of the base is twice the width.
The costs of the materials is $10/𝑚2 for
the base and $8/𝑚2 for the sides.
Express the cost of materials as a
function of the width of the base.
2. Find the domain of (𝑥) =
√𝑥+3
log⁡(𝑥+1)
.
Section 1.2
3. Which of the following are
polynomials?
5
2
a) 𝑓(𝑥) = 2𝑥 6 − 𝑥 −2 + 3 𝑥 3 + √2
4
b) 𝑔(𝑥) = 5𝑥 − 𝑥 + 1 + 2
c) ℎ(𝑥) = √𝑥 − 1 + 9
𝑥+3
d) 𝐹(𝑥) = 𝑥−2
𝑥
e) 𝐺(𝑥) = 3𝑥 2 − 3𝑥 + ln⁡|𝑥 − 2|
Section 1.3
4. Describe the transformations which
result in: 𝑓(𝑥) = −√𝑥 − 2 + 3
5. Sketch the graph of
𝑔(𝑥) = |𝑎𝑥 2⁡ + 𝑏𝑥 + 𝑐| given the graph
below of 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐⁡.
Section 1.5
8. Simplify the expression:
𝑥 2𝑛 ⋅ 𝑥 3𝑛−1
𝑥 𝑛+2
9. A bacterial culture starts with 500
bacteria and doubles in size every half
hour.
a) How many bacteria are there
after 2hrs?
b) How many after t hours?
c) Estimate the time for the
population to reach 80,000.
Section 1.6
10. Find the inverse of (𝑥) =
.
11. Write as a single function:
ln|𝑎 + 𝑏| + ln|𝑎 − 𝑏| − 2ln⁡|𝑐| .
12. Evaluate: log 3 12, round answer to 4
decimal places.
Section 2.1
13. Write the equation of the tangent to
the curve 𝑓(𝑥) = 2𝑥 + 5𝑥 2 at = −2 .
14. A ball is thrown upward with a velocity
of 10 m/s , its height in meters t
seconds later is given by
𝑦 = 10𝑡 − 1.86𝑡 2 .
a) Find the average velocity over
the time interval [1, 1.1].
b) Find the instantaneous velocity
at 𝑡 = 1⁡.
Section 2.2
15. Find the
lim+ 𝑔(𝑡), lim− 𝑔(𝑡),⁡⁡⁡lim 𝑔(𝑡),⁡⁡⁡lim 𝑔(𝑡)⁡⁡⁡
𝑡→0
6. Given 𝑓(𝑥) = 2𝑥 − 1, 𝑔(𝑥) = ln|𝑥|,
and ℎ(𝑥) = 𝑒 𝑥 , find (𝑔⁡ ∘ ⁡ℎ⁡ ∘ ⁡𝑓)(𝑥) .
7. Find 𝑔(𝑥) given 𝑓(𝑥) = 𝑥 − 3,⁡ and
(𝑓 ∘ 𝑔)(𝑥) = ℎ(𝑥)⁡ where
ℎ(𝑥) = (𝑥 − 5)2 − 2𝑥 + 3 .
4𝑥−1
2𝑥+3
𝑡→0
𝑡→3
𝑡→4
𝑥 2 −2𝑥
Section 2.8
16. lim ⁡ 𝑥 2 −𝑥−2
𝑥→2
Section 2.3
√𝑡 2 +9⁡−3
𝑡2
𝑡→0
1
1
lim ( − 2 )
𝑥 +𝑥
𝑥→0 𝑥
17. Find lim
18. Find
Section 2.4
19.
sin 𝑥
Evaluate lim ⁡⁡ 3+cos 𝑥
𝑥→𝜋
27. Given the graph of 𝑓 ′ (𝑥) , shown
below,
a) State the intervals where 𝑓(𝑥) is
increasing.
b) At what x-values does 𝑓(𝑥) have a
rel. maximum?
20. State the values at which 𝑓(𝑥)is not
continuous.
Section 3.1
28. Find the derivative of
𝑓(𝑥) = 𝑏𝑥 3 + 𝑏 2 + 𝑏 2𝑥
29. Find the derivative of
𝑏 𝑐
𝑦 = 𝑎𝑒 2𝑣 + + 2
𝑣 𝑣
Section 3.2
Section 2.5
21. lim
𝑥→∞
6𝑥 4 +5
(𝑥 2 −2)(2𝑥 2 −1)
22. lim −⁡⁡
𝑥→−5
𝑥−4
𝑥+5
Section 2.6
′ (𝑎)⁡𝑔𝑖𝑣𝑒𝑛⁡𝑓(𝑥)
23. Find 𝑓
= √1 − 2𝑥
24. The displacement of a particle (in
meters) moving in a straight line is
given by 𝑠 = 𝑡 2 − 8𝑡 + 18, where t is
measured in seconds. Find the average
velocity over the interval [4,5].
Section 2.7
25. Find the derivative of 𝐹(𝑡) =
4𝑡
2𝑡−1
26. State the values where the function is
not differentiable.
𝑥2
30. Given 𝑓(𝑥) = 𝑔(𝑥) , 𝑔(2) = −3,
𝑔′ (2) = 1 Find 𝑓′(2)
31. Find 𝑦′ given 𝑦 = 𝑥 3 ∙ tan 𝑥
Section 3.3
32. Find the derivative of 𝑦 = 𝑥𝑒 𝑥 ∙ csc 𝑥
𝑑
33. Show 𝑑𝑥 (csc 𝑥) = ⁡ − csc 𝑥⁡ ∙ cot 𝑥
Section 3.4
34. Find the derivative of
𝑦 = (2𝑥 + 3𝑥 4 )5
35. Differentiate 𝑦 = 𝑒 cos 𝑥
Section 3.7
36. Differentiate 𝑓(𝑥) = log 5(𝑥𝑒 −𝑥 )
37. Let 𝑓(𝑥) = log 𝑎 (3𝑥 2 − 2) For what
value of a is 𝑓 ′ (1) = 3⁡?
Section 3.8
6
38. Given 𝑠(𝑡) = 2𝑡 3 − 𝑡 + 5𝜋,⁡ s is
measured in feet, and t in seconds.
23
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
Find the acceleration at any time t , and
at 3 seconds.
If a ball is thrown vertically of 80𝑓𝑡/𝑠𝑒𝑐
then its height after t seconds is
𝑠 = 80𝑡 − 16𝑡 2 .
a) What is the maximum height of
the ball?
b) What is the velocity of the ball
when it is 96 ft above the
ground on the way up?
Section 4.2
Find the critical numbers of
𝑓(𝑥) = 3𝑥 4 + ⁡4𝑥 3 − 6𝑥 2 ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡
Find the absolute extrema of
𝑦 = 𝑥 3 − 6𝑥 2 + 9𝑥 + 2 on the interval
[−1, 4].
Section 4.3
𝑓(𝑥) = 2𝑥 3 + 3𝑥 2 − 36𝑥,
a) Find where the function is ↘.
b) Find where the function is ↗.
c) Find where the function is ∪ .
Find the critical numbers of
𝑓(𝑥) = 𝑥 4 (𝑥 − 1)3 .
What does the 2nd Derivative Test tell
you about each critical number?
Section 4.6
Two numbers, x and y: what is the
smallest possible value of the sum of
their squares if 𝑥 + 2𝑦 = 30 ?
If 1200⁡𝑐𝑚2 of material is available to
make a box with a square base and an
open top, find the largest possible
volume of the box.
Section 4.8
Find the antiderivative of
𝑓(𝑥) = 𝑥(6 − 𝑥)2
Find the antiderivative of
𝑥 3 − 4𝑥 2 + 10
𝑔(𝑥) =
√𝑥
Find the antiderivative of
ℎ(𝑥) = 5𝑒 𝑥 + 4 cos 𝑥
49. Given 𝑓 ′′ (𝑥) = √𝑥 ⁡, 𝑓′(0) = 3, 𝑓(1) =
3
5, find 𝑓(𝑥).
Section 5.1
50. Estimate the area under the graph of
𝑦 = |𝑥 − 2| from [−3, 6] by find 𝑅3 ⁡.
51. ∆𝑥 =
Section 5.2
21
52. Approximate ∫1 ⁡𝑑𝑥 by finding 𝑀5 ⁡.
𝑥
53. Write as a single integral, a<b<c<d:
𝑐
𝑐
𝑎
⁡∫𝑎 𝑓(𝑥)𝑑𝑥 − ∫𝑑 𝑓(𝑥)𝑑𝑥 + ∫𝑏 𝑓(𝑥)𝑑𝑥
Section 5.3
9 1
⁡𝑑𝑥
2𝑥
𝜋/4
∫0 sec 𝜃 tan 𝜃⁡𝑑𝜃
54. Evaluate: ∫1
55. Evaluate:
Section 5.4
56. Find the derivative of
𝑡
𝑓(𝑡) = ∫ √3𝑥 − 8 ⁡𝑑𝑥
0
57. Find the derivative of
3𝑥 2
𝑔(𝑥) = ∫
2𝑥
𝑡 −4
⁡𝑑𝑡
𝑡2 + 1
Section 5.5
𝑥
58. ∫ 𝑒 √4 + 𝑒 𝑥 ⁡ ⁡𝑑𝑥
59. ∫
(ln 𝑥)4
𝑥
⁡𝑑𝑥
Section 6.1
60. Find the area bounded by
𝑓(𝑥) = 4𝑥 − 𝑥 2 and 𝑔(𝑥) = −3𝑥.
61. Find the area bounded by:
𝑥 = −𝑦 2 + 7 and 𝑥 = 𝑦 2 − 1.
62. Find the number c ⁡such that c bisects
1
the area under the curve 𝑦 = 𝑥 on the
interval [2,10].
Section 6.5
63. Find the average value of
𝑓(𝑥) = (3 − 2𝑥)−1 ,⁡⁡⁡⁡[−1, 1]
64. Find the number(s) b, such that the
average value of 𝑓(𝑥) = 2 + 6𝑥 − 3𝑥 2
on the interval [0, 𝑏] is 3.
65. Find the average value of
𝑦 = 6𝑥 2 − 4𝑥 − 1 on [−𝑎, 2𝑎].
Section 6.7
66. The demand function is
𝑝 = 20 − 0.05𝑥.⁡⁡Find the consumer
surplus when the sales level is 300.
67. Use Poiseuille’s Law to calculate the
rate of flow in a human artery where
𝜂 = 0.027, 𝑅 = .006⁡𝑐𝑚, 𝑙 = 2⁡𝑐𝑚
𝑃 = 4000⁡𝑑𝑦𝑛𝑒𝑠/𝑐𝑚2
68. Find the area bounded by
1
𝑓(𝑥) = sin 𝑥⁡ and 𝑔(𝑥) = 2 𝑥,
⁡𝑦 = ⁡ 𝑥 2 − 1⁡.⁡⁡
69. Find A, B, and C such that 𝑦 = 𝐴𝑥 2 +
𝐵𝑥 + 𝐶⁡ satisfies the differential
equation 𝑦 ′′ − 𝑦 ′ + 2𝑦 = 𝑥 2 + 3𝑥 + 2.
70. Write in terms of 𝑢 = 𝑥 − 1,
𝑒
3
∫ (𝑥 + 1)√𝑥 − 1 ⁡𝑑𝑥
2