146 CHAPTER 2 Applications of the Derivative In Exercises 7-12, sketch the graph of a function that has the prop- erties described. 7. (2)-1; f'(2)= 0; concave up for all .x. 8. f(-1)-0; f'(x) < 0 for x < -1, f(-1)-0 and /'(x) > 0 for x>-1. 9. f(3) = 5: f'(x) > 0 for x <3, f'(3)-0 and f'(x) > 0 for x>3. 10. (-2,-1) and (2, 5) are on the graph; /'(-2)-0 and /'(2) - 0; f"(x) > 0 for x < 0, "(0)-0, f(x) < 0 for x > 0. 11. (0, 6), (2, 3), and (4,0) are on the graph; f(0) = 0 and f'(4)=0; f(x) <0 for x<2, f"(2)-0, f"(x) > 0 for x > 2. 12. f(x) defined only for x20; (0,0) and (5, 6) are on the graph; f'(x) > 0 for x ≥ 0; f(x) < 0 for x < 5, "(5)-0, f(x) > 0 for x>5. In Exercises 13-18, use the given information to make a good sketch of the function f(x) near x = 3. 13. f(3) = 4, f'(3)=-"(3) = 5 14. f(3) = -2, f'(3)=0, f(3) = 1 15. f(3) = 1, f'(3) = 0, inflection point at x = 3, f'(x) > 0 for x > 3 16. f(3) = 4, f'(3)=-."(3) = -2 17. f(3) = -2, f'(3) = 2, f(3) = 3 18. f(3) = 3, f'(3) = 1, inflection point at x = 3, f"(x) < 0 for x > 3 19. Refer to the graph in Fig. 19. Fill in each box of the grid with either POS, NEG, or 0. 3
146 CHAPTER 2 Applications of the Derivative In Exercises 7-12, sketch the graph of a function that has the prop- erties described. 7. (2)-1; f'(2)= 0; concave up for all .x. 8. f(-1)-0; f'(x) < 0 for x < -1, f(-1)-0 and /'(x) > 0 for x>-1. 9. f(3) = 5: f'(x) > 0 for x <3, f'(3)-0 and f'(x) > 0 for x>3. 10. (-2,-1) and (2, 5) are on the graph; /'(-2)-0 and /'(2) - 0; f"(x) > 0 for x < 0, "(0)-0, f(x) < 0 for x > 0. 11. (0, 6), (2, 3), and (4,0) are on the graph; f(0) = 0 and f'(4)=0; f(x) <0 for x<2, f"(2)-0, f"(x) > 0 for x > 2. 12. f(x) defined only for x20; (0,0) and (5, 6) are on the graph; f'(x) > 0 for x ≥ 0; f(x) < 0 for x < 5, "(5)-0, f(x) > 0 for x>5. In Exercises 13-18, use the given information to make a good sketch of the function f(x) near x = 3. 13. f(3) = 4, f'(3)=-"(3) = 5 14. f(3) = -2, f'(3)=0, f(3) = 1 15. f(3) = 1, f'(3) = 0, inflection point at x = 3, f'(x) > 0 for x > 3 16. f(3) = 4, f'(3)=-."(3) = -2 17. f(3) = -2, f'(3) = 2, f(3) = 3 18. f(3) = 3, f'(3) = 1, inflection point at x = 3, f"(x) < 0 for x > 3 19. Refer to the graph in Fig. 19. Fill in each box of the grid with either POS, NEG, or 0. 3
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter5: Graphs And The Derivative
Section5.3: Higher Derivatives, Concavity, And The Second Derivative Test
Problem 74E
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Transcribed Image Text:146 CHAPTER 2 Applications of the Derivative
In Exercises 7-12, sketch the graph of a function that has the prop-
erties described.
7. (2)-1; f'(2) 0; concave up for all .x.
8. f(-1)=0; f'(x) < 0 for x < -1, (-1)-0 and f'(x) > 0
for x>-1
9. f(3)-5; f'(x) > 0 for x <3, f'(3) = 0 and f'(x) > 0 for
x > 3.
10. (-2,-1) and (2, 5) are on the graph; /'(-2) = 0 and /'(2) = 0;
f"(x) > 0 for x < 0, "(0) = 0, "(x) < 0 for x > 0.
11. (0, 6), (2, 3), and (4,0) are on the graph; /'(0) = 0 and
f'(4)=0; f(x) <0 for x<2, f"(2)-0, "(x) > 0 for
x> 2.
12. f(x) defined only for x = 0; (0, 0) and (5, 6) are on the
graph; f'(x) > 0 for x ≥ 0; f(x) < 0 for x < 5, "(5)=0,
f(x) > 0 for x > 5.
In Exercises 13-18, use the given information to make a good
sketch of the function f(x) near x = 3.
13. f(3) = 4, f'(3) = -f(3) = 5
14. f(3) = -2, f'(3) = 0, f"(3) = 1
15. f(3) = 1, f'(3) = 0, inflection point at x = 3, f'(x) > 0 for
x > 3
16. f(3) = 4, f'(3)=-₁/"(3) = -2
17. f(3) = -2, f'(3) = 2, f"(3) = 3
18. f(3) = 3, f'(3)= 1, inflection point at x = 3, f"(x) < 0 for
x > 3
19. Refer to the graph in Fig. 19. Fill in each box of the grid with
either POS, NEG, or 0.
Y
4
4< x≤6
A
y = f(x)
B
x
X
0≤x≤2
2
2<x<3
3
3 < x < 4
A
B
Figure 19
20. The first and second derivatives of the function f(x) have the
values given in Table 1.
C
(a) Find the x-coordinates of all relative extreme points.
(b) Find the x-coordinates of all inflection points.
f'(x)
Positive
0
Negative
Negative
Negative
0
Negative
S J' S"
Table 1 Values of the First Two Derivatives of a Function
f"(x)
Negative
Negative
Negative
0
Positive
0
Negative
21. Suppose that Fig. 20 contains the graph of y= s(t), the dis-
tance traveled by a car after t hours. Is the car going faster at
t = 1 ort=2?
22. Suppose that Fig. 20 contains the graph of y = v(t), the velocity
of a car after t hours. Is the car going faster at t = 1 or t = 2?
Figure 20
23. Refer to Fig. 21.
AME
10
2/1
8
6
4
2
Y
3
1
0
Figure 21
(a) Looking at the graph of f'(x), determine whether f(x) is
increasing or decreasing at x = 9. Look at the graph of
f(x) to confirm your answer.
(b) Looking at the values
of f'(x) for 1 ≤ x < 2 and
2<x≤ 3, explain why the graph of f(x) must have a
relative maximum at x = 2. What are the coordinates of
the relative maximum point?
(c) Looking at the values of f'(x) for x close to 10, explain
why the graph of f(x) has a relative minimum at x = 10.
(d) Looking at the graph of f"(x), determine whether f(x) is
concave up or concave down at x = 2. Look at the graph
of f(x) to confirm your answer.
(e) Looking at the graph of f"(x), determine where f(x) has an
inflection point. Look at the graph of f(x) to confirm your
answer. What are the coordinates of the inflection point?
(f) Find the x-coordinate of the point on the graph of f(x)
at which f(x) is increasing at the rate of 6 units per unit
change in x.
24. In Fig. 22, the t-axis represents time in minutes.
-1
v-f(x)
6 8 10 12 14 16 18
y = f'(t)
y=f(x)
4++++++++x
1
y=J"(t)
-J'(x)
y = f(t)
malam
3 mp 5
7
t
Figure 22
(a) What is f(2)?
(b) Solve f(t) = 1.
(c) When does f(t) attain its greatest value?
(d) When does f(t) attain its least value?
(e) What is the rate of change of f(t) at t = 7.5?
(f) When is f(t) decreasing at the rate of 1 unit per minute?
That is, when is the rate of change equal to -1?
(g) When is f(t) decreasing at the greatest rate?
(h) When is f(t) increasing at the greatest rate?
Exerci
the de
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