Mr. Orchard’s Math 142 WIR
Test 2 Review
1. Find the derivatives of the following functions:
(a) f (x) = 3x3 + 2x2 + x
(b) g(x) =
(x2 +2x)3
(x2 −2x)4
(c) h(x) =
x ln(3x2 +22)
54x
√
(d) k(x) = x5 log8 ( 3x2 + 1 + 15x)
Week 8
Mr. Orchard’s Math 142 WIR
Test 2 Review
2. Find the second derivatives of the following functions:
(a) f (x) = 3x3 + 2x2 + x
(b) g(x) = x2 52x
(c) h(x) =
x+2 2
x−2
(d) k(x) = log2 (8x2 + 3x + 4)
Week 8
Mr. Orchard’s Math 142 WIR
Test 2 Review
Week 8
3. Use calculus to find the x-coordinates of the absolute maximum of the function k(x) =
−x4 + 4x3 − 4x2 + 22 on the interval [0, ∞).
4. The cost function for a particular brand of coffee cups is C(x) = x2 + 22x + 1500.
(a) Use the marginal cost function to estimate the cost of making the 48th coffee cup.
(b) What is the exact cost of making the 48th coffee cup?
(c) What is the average cost per cup if the company made 388 cups?
(d) What is the marginal average cost function?
Mr. Orchard’s Math 142 WIR
Test 2 Review
Week 8
5. A rancher wants to create two rectangular pens side by side, using an existing fence as
one side. There is 696 feet of fencing available to her.
(a) What dimensions should be used to maximize the total area of the enclosure?
(b) What is the total area of the enclosure?
6. The price demand function for a transforming toys is x = (96 − 4p)3 .
(a) What is the elasticity of demand at p = $17? Is the demand elastic, inelastic, or
unit elasticity at this price?
(b) What price should the manufacturer charge in order to maximize revenue?
Mr. Orchard’s Math 142 WIR
Test 2 Review
Week 8
x
7. f (x) = x+2
is defined everywhere except x = −2. Use calculus to answer the following
questions.
(a) Find the relative extrema of f .
i. Relative Maximums:
ii. Relative Minimums:
(b) Where is f increasing?
(c) Where is f decreasing?
(d) Where is f concave up?
(e) Where is f concave down?
(f) What are the x-coordinates of the inflection points?
Mr. Orchard’s Math 142 WIR
Test 2 Review
Week 8
8. Below is the graph of H 0 (x). The domain of H(x) is (−3, ∞).
4
3.5
3
2.5
2
1.5
1
0.5
0
-0.5
-1
-3
-2.5
-2
(a) What are the critical values of H(x)?
(b) Where is H increasing?
(c) Where is H concave down?
-1.5
-1
-0.5
0
0.5
1
Mr. Orchard’s Math 142 WIR
Test 2 Review
Week 8
9. We are given the information that f has domain of all real numbers and is differentiable everywhere. Furthermore, we know f 00 (x) is continuous everywhere. Conclude
whether the following information gives us a relative min or max, neither, or not enough
information.
(a) We are also given that f (1) = 20, f 0 (1) = 0, f 00 (1) = 0. Given this information,
what can we conclude about the point (1, 20)?
(b) We are also given that f (10) = 12, f 0 (10) = −2, and f 00 (10) = 0. Given this
information, what can we conclude about the point (10, 12)?
(c) We are also given that f (−6) = 1, f 0 (−6) = 0, and f 00 (−6) = 2. Given this
information, what can we conclude about the point (−6, 1)?
Mr. Orchard’s Math 142 WIR
10. Calculate the following limits:
(a) lim 2 − 9x9
x→∞
(b) lim (7x5 − 3)x3
x→−∞
11x5 −x+3
5
2
x→∞ 5x +5x −2
(c) lim
6x2 −9
(d) lim e 5x6 +7
x→∞
1+12e4x
4x
x→∞ 5+3e
(e) lim
Test 2 Review
Week 8
Mr. Orchard’s Math 142 WIR
Test 2 Review
Week 8
11. Below is a table of values of f , g, f 0 , and g 0 .
x f (x) g(x) f 0 (x) g 0 (x)
3
6
3
3
6
4
5
3
4
7
5
7
5
5
5
6
6
6
4
4
7
4
7
4
5
d
Compute dx (f (g(x)) at x = 7.
12. We are given the following information about f , f 0 (x), and f 00 (x). Sketch a graph of
f (x) below.
f (0) = 4
0
f (x) > 0 if x < −1
f 0 (x) < 0 if − 1 < x < 3 or x > 3
f 00 (x) > 0 if 0 < x < 3
00
f (x) < 0 if x < 0 or x > 3
7
6
5
4
3
2
1
0
-1
-2
-3
-2
-1
0
1
2
3
4
5
6
Mr. Orchard’s Math 142 WIR
Test 2 Review
13. g(x) = x5 ln(2x)
(a) What is the domain of g?
(b) Where is g increasing?
(c) Find the relative extrema of g.
i. Relative Maximums:
ii. Relative Minimums:
Week 8