MATH 142
Test 2 Review
1. Find the derivatives of the following functions:
(a) f (x) = 4x3 + 3x4 − x2
(b) y =
8
x5
− 6 ln(x)
(c) H(x) = x2 ln(4x)
(d) g(x) = 5x8 3−2x
MATH 142
Test 2 Review
2. Find the second derivatives of the following functions:
(a) f (x) = 4x3 + 3x4 − x2
(b) g(x) =
√
√
x+56x
(c) r(t) = t3 log3 (2t)
(d) G(t) = 3t6 e4t
MATH 142
Test 2 Review
3. Below is a table of values of r, t, r0 , and t0 .
x r(x) t(x) r0 (x) t0 (x)
8
11
8
8
11
9
10
8
9
12
10 12
10
10
10
11 11
11
9
9
12
9
12
9
10
d
Compute dx
(r(t(x)) at x = 12.
4. The price demand function for 100 packs of whiteboard erasers is x =
the price of one 100 pack of erasers.
√
360 − 4p. p is
(a) What is the elasticity of demand at p = $18? Is the demand elastic, inelastic, or
unit elasticity at this price?
(b) What price should the manufacturer charge in order to maximize revenue?
MATH 142
Test 2 Review
5. The cost function for a particular brand of computers is C(x) = x2 + 38x + 2300
(a) Use the marginal cost function to estimate the cost of making the 25th computer.
(b) What is the exact cost of making the 25th computer?
(c) What is the average cost per cup if the company made 467 computers?
(d) What is the marginal average cost function?
MATH 142
6. f (x) = 170 +
Test 2 Review
15x4
2
− 6x5 . 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?
MATH 142
Test 2 Review
7. g(x) = 22x log4 (3x)
(a) What is the domain of g?
(b) Where is g increasing?
(c) Find the x-coordinates of the relative extrema of g.
i. Relative Maximums:
ii. Relative Minimums:
MATH 142
Test 2 Review
8. Below is the graph of H 0 (x). The domain of H(x) is all real numbers.
8
6
4
2
0
-2
-4
-4
-3
-2
(a) What are the critical values of H(x)?
(b) Where is H increasing?
(c) Where is H concave down?
-1
0
1
2
MATH 142
Test 2 Review
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.
(a) We are also given that f (20) = 12, f 0 (20) = −8, and f 00 (20) = 0. Given this
information, what can we conclude about the point (20, 12)?
(b) We are also given that f (−100) = 1, f 0 (−100) = 0, and f 00 (−100) = 5. Given this
information, what can we conclude about the point (−100, 1)?
(c) We are also given that f (3) = 16, f 0 (3) = 0, f 00 (3) = 0. Given this information,
what can we conclude about the point (3, 16)?
MATH 142
Test 2 Review
10. Calculate the following limits:
(a) lim −4x3 + 3x2 + 2x + 12
x→∞
(b) lim −4x3 + 3x2 + 2x + 12
x→−∞
(c) lim 2x3 + 3x + 1200000000
x→∞
−2x4 +3x2 +2
(d) lim e 50x3 −200x2 −300
x→∞
MATH 142
Test 2 Review
11. 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
f 0 (x) > 0 if − 1 < x < 3 or x > 3
f 0 (x) < 0 if x < −1
00
f (x) > 0 if x < 0 or x > 3
f 00 (x) < 0 if 0 < x < 3
7
6
5
4
3
2
1
0
-1
-2
-3
-2
-1
0
1
2
3
4
12. Use calculus to find the x-coordinates of the absolute minimum of the function k(x) =
x2 −1
on the interval [−1, ∞).
x2 +1
5
6
MATH 142
Test 2 Review
13. Find two positive numbers, x and y, whose product is 324 and the sum x + 4y is a
minimum.
14. 1200 cm2 of material is available to make a box with a square base and an open top.
What is the largest possible volume of the box?