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rties described. 7. (2)-1: f'(2) = 0; concave up for all .x. 3. (-1)-0; f'(x) < 0 for x < -1, "(-1)-0 and f'(x) > 0

rties described. 7. (2)-1: f'(2) = 0; concave up for all .x. 3. (-1)-0; f'(x) < 0 for x < -1, "(-1)-0 and f'(x) > 0

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Q7,Q9&Q11 needed to be solved correctly in 1 hour in the order to get positive feedback These are easy questions so solve these please do.all correctly
In Exercises 7-12, sketch the graph of a function that has the prop- erties described. 146 CHAPTER 2 Applications of the Derivative 7. (2) - 1; /'(2) 0; concave up for all .x. 8. (-1)= 0; f'(x) <0 for x < -1, 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 f'(4) 0; f(x) <0 for x < 2, 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 /(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 A. 19. Refer to the graph in Fig. 19. Fill in each box of the grid with either POS, NEG, or 0. y=f(x) X 0≤x≤2 2 (-1)-0 and f'(x) > 0 B 2<x<3 3 3<x<4 4 4 < x≤6 x the graph; /'(0) = 0 and "(2)-0, f"(x) > 0 for A B C S J' Figure 19 20. The first and second derivatives of the function f(x) have the values given in Table 1. (inw (a) Find the x-coordinates of all relative extreme points. (b) Find the x-coordinates of all inflection points. Table 1 Values of the First Two Derivatives of a Function f"(x) f'(x) Positive Negative 0 Negative Negative Negative 0 Negative Negative 0 Negative 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. 2/ 10 8 6 4 2 Y 3 1 0 -1 y-f(a) 6 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? 10. (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 = (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. y = f'(t) 1 y=f"(t) u-J'(x) A++++++++ 10 12 14 16 18 y = f(t) 3 y-f(x) Confourrure 5 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? x (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
Transcribed Image Text:In Exercises 7-12, sketch the graph of a function that has the prop- erties described. 146 CHAPTER 2 Applications of the Derivative 7. (2) - 1; /'(2) 0; concave up for all .x. 8. (-1)= 0; f'(x) <0 for x < -1, 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 f'(4) 0; f(x) <0 for x < 2, 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 /(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 A. 19. Refer to the graph in Fig. 19. Fill in each box of the grid with either POS, NEG, or 0. y=f(x) X 0≤x≤2 2 (-1)-0 and f'(x) > 0 B 2<x<3 3 3<x<4 4 4 < x≤6 x the graph; /'(0) = 0 and "(2)-0, f"(x) > 0 for A B C S J' Figure 19 20. The first and second derivatives of the function f(x) have the values given in Table 1. (inw (a) Find the x-coordinates of all relative extreme points. (b) Find the x-coordinates of all inflection points. Table 1 Values of the First Two Derivatives of a Function f"(x) f'(x) Positive Negative 0 Negative Negative Negative 0 Negative Negative 0 Negative 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. 2/ 10 8 6 4 2 Y 3 1 0 -1 y-f(a) 6 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? 10. (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 = (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. y = f'(t) 1 y=f"(t) u-J'(x) A++++++++ 10 12 14 16 18 y = f(t) 3 y-f(x) Confourrure 5 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? x (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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