Math 142 Week-in-Review #7 (Sections 2.3-2.6 and 3.1-3.3)
Note: This collection of questions is intended to be a brief overview of the exam material (with emphasis on Sections 3.2 and 3.3). When studying, you should also rework your notes, the previous week-in-reviews for this material, as well as your suggested and online homework.
1. Determine where the function f (x) = −1
x2+ 3 is concave upward/downward, and find any inflection points of f .
2. Suppose the domain of f is (−∞, ∞) and f is twice-differentiable on its domain. If we know f (3) = −4, f0(3) = 0, and f 00(3) = −15, what can we conclude, if anything, about the behavior of the graph of f at x = 3? What if f00(3) = 0?
3. Create a chart showing the graphical relationships between f , f0, and f 00.
4. Consider the graph of f below. At what points do f 0(x) and f 00(x) have the same sign?(Courtesy of Joe Kahlig)
5. a) Sketch a function whose slopes are positive and decreasing.
b) Sketch a function whose slopes are getting less negative.
c) Sketch a function that is increasing whose slopes are also increasing.
6. Sketch the graph of a function that satisfies all of the following conditions:
• Domain of f : (−∞, ∞)
• f (0) = 1
• f 0(1) = f 0(−1) = 0
• f 0(x) < 0 on (−1, 1)
• f 0(x) > 0 on (−∞, −1) and (1, 2)
• f 0(x) = −1 on (2, ∞)
• f 00(x) < 0 on (−∞, 0)
• f 00(x) > 0 on (0, 2)
• lim
x→−∞f(x) → −∞
7. Consider a function f that is continuous on its domain of (−∞, 4)∪(4, ∞). Also, f0(x) = −x − 2
(x − 4)3and f00(x) = 2x + 10 (x − 4)4. Find the x-values where any local maximum or minimum values of f occur using the Second Derivative Test, if possi- ble. If it is not possible, explain why.
8. A 10-foot ladder is leaning against a wall. The bottom of the ladder slides away from the wall at a rate of 3 feet per second. How fast is the distance between the top of the ladder and the ground changing when the bottom of the ladder is 5 feet from the wall?
9. A company has a price-demand equation given by p =p3
−0.5x2+ 2500 dollars, where p is the price per item when x items are sold.
a) Find the marginal revenue when 58 items are sold and interpret your answer.
b) Approximate the revenue from selling the 40thitem.
c) Find the exact revenue from selling the 40thitem.
d) Approximate the revenue when 65 items are sold.
e) Find the exact revenue when 65 items are sold.
10. a) Suppose the domain of f is (−∞, ∞), and f is twice-differentiable on its domain. If we know f (−2) = −4, f 0(−2) = 0, and f00(−2) = 7, what can we conclude, if anything, about the behavior of the graph of f at x = −2?
b) Suppose the domain of f is (−∞, ∞), and f is twice-differentiable on its domain. If we know f (5) = 1, f0(5) = −7, and f00(5) = −10, what can we conclude, if anything, about the behavior of the graph of f at x = 5?
11. Find the equation of the line tangent to the graph of the function f (x) = 6 ln x4 + 3x − ex−1at x = 1.
12. Use the graph of f below to find the following:
a) Where is f (x) < 0?
b) Find the partition numbers of f 0.
c) Where is f00(x) > 0?
d) Find the critical values of f .
e) Where is f0(x) < 0?
f) Where is f 0decreasing?
g) Find and classify any local extrema of f .
b
f (x)
c e
a d
x13. Find the derivative of each of the following functions. Do not simplify your final answer.
a) f (x) = ln
e5x4+ 9x log4(x) + e2
b) g(x) = 7
√
3x2+1 6x2 + πx3
c) h(x) = ln 3x4− 5x + 26
9x − 10
!
14. Assuming the graph shown below is f0and f is continuous on its domain of (−∞, d) ∪ (d, ∞), find the following:
a) Find the partition numbers of f0.
b) Find the critical values of f .
c) Where is f increasing?
d) Find and classify any local extrema of f .
e) Where is f concave up?
f) Where is f 00(x) < 0?
c d e g
a b
f ’ (x)
f
xg) Find the partition numbers of f 00.
h) Where does f have inflection points, if any?
15. Find the derivative of f (x) =
9e3x2+4x+ 14x
(log(6x + 5))8. Do not simplify your final answer.
16. Sketch the graph of a function that satisfies all of the given conditions.
• Domain of f : (−∞, 0) ∪ (0, ∞)
• f (−5) = 8 and f (2) = −2
• f 0(4) = 0
• f 0(x) > 0 on (−∞, −5), (2, 4)
• f 0(x) < 0 on (−5, 0), (0, 2), (4, ∞)
• f 00(x) > 0 on (−∞, −2), (0, 3)
• f 00(x) < 0 on (−2, 0), (3, ∞)
• Vertical asymptote: x = 0
• lim
x→−∞f(x) = 1
17. Assuming the graph shown is f 00, f is continuous on its domain of (a, c) ∪ (c, ∞), and f 0is continuous on its domain of (a, c) ∪ (c, ∞), find the following:
a) Find the partition numbers of f00.
b) Where are the slopes of f increasing?
c) Where is f concave down?
d) Where does f have inflection points, if any?
a b c d e f
f ’’(x)
x
18. Determine where the function f (x) =ln(x)
x is increasing/decreasing and concave upward/downward, as well as where any local extrema or inflection points occur.
19. Finddy
dxfor each of the following. You do not need to simplify your final answer.
a) x ln(y) + 2y = 2x3
b) 43xy2− y +√3 x= 5
20. The price p, in dollars, and demand x for a product are related by 2x2+ 5xp + 50p2= 80, 000.
(Source: #29, pg. 262 of Calculus for Business, Economics, Life Sciences, and Social Sciences, 11th ed., by Barnett, Ziegler, and Byleen)
a) If the price is increasing at a rate of $2 per month when the price is $30, find the rate of change of the demand with respect to time.
b) If the demand is decreasing at a rate of 6 units per month when the demand is 150 units, find the rate of change of the price with respect to time.
21. Use calculus to find any asymptotes of the function f (x) = xe−8x2, where the function is increasing/decreasing and concave upward/downward, and any local extrema and inflection points of the function. Use your information to graph
f.