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In Problems 15 to 20, graph the families of functions by finding asymptotes and using first and second deriva- tives. In particular, determine how the graph of the functions depends on the parameter a > 0. ах 15. y = x - ax2 16. y = x2 +1 17. y = ae* + e¬* 18. y = ex +ae

In Problems 15 to 20, graph the families of functions by finding asymptotes and using first and second deriva- tives. In particular, determine how the graph of the functions depends on the parameter a > 0. ах 15. y = x - ax2 16. y = x2 +1 17. y = ae* + e¬* 18. y = ex +ae

College Algebra
College Algebra
7th Edition
ISBN:9781305115545
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:James Stewart, Lothar Redlin, Saleem Watson
Chapter2: Functions
Section: Chapter Questions
Problem 30P: In this problem you are asked to find a function that models in real life situation and then use the...
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288 Chapter 4 Applications of Differentiation х — 3 x? 27. In Example 6 of Section 1.5, we develop the 14. У 13. у 3 X +1 Michaelis-Menton model for the rate at which 1+ x4 In Problems 15 to 20, graph the families of functions by finding asymptotes and using first and second deriva- tives. In particular, determine how the graph of the functions depends on the parameter a > 0. an organism consumes its resource. For bacte- rial populations in the ocean, this model was given by 1.2078x f(x) = mg of glucose/hr 1+0.0506x ах 15. у %3D х4 — ах? = xª – ax² 16. У where x is the concentration of glucose (micrograms per liter) in the environment. Use asymptotes and first derivatives to sketch this function by hand. - x2 +1 17. y = ae* +e¯x 18. у — е* + ае * 1 20. у — ах + 28. In Example 5 of Section 2.4 we found that the rate at which wolves kill moose can be modeled by а +x 19. у %3D - 1+ x X 3.36x In Problems 21 and 22, sketch the graph of a function with the given properties. 21. x = 2, x = -2 are vertical asymptotes f(x) = moose/wolf/hundred days 0.42 + x where x is measured in number of moose per square kilometer. Use asymptotes and first derivatives to f is increasing for 0 < x < 2 and x > 2 f is decreasing for x < -2 and -2 < x < 0 sketch this function. graph is concave down on (-∞, -2) and (2, ∞) intercepts are (-1,0), (–3, 0), (3, 0) and (1, 0) 22. y = 1, y = -1 are horizontal asymptotes 29. In Problem 42 in Problem Section 2.4, we examined how wolf densities in North America depend on moose densities. We found that the following func- tion provides a good fit to the data: 3 and for x > 2 3 f is increasing for x < – 58.7(х — 0.03) 2 f(x) = wolves per 1000 km² 0.76 + x f is decreasing for –1 < x < 1 graph is concave down for x < -1 and for 0 < x < 1 where x is number of moose per square kilometer. a. Find the horizontal and vertical asymptotes. graph is concave up for x > 1 and for –1 < x < 0 - b. Determine on which intervals f is increasing and decreasing. c. Determine on which intervals f is concave up Level 2 APPLIED AND THEORY PROBLEMS 23. Consider the graph of y = ax? + bx +c for constants a, b, and c. Use second derivatives to determine what happens to the graph as a changes. and concave down. d. Use the information from parts a-c to sketch the graph of f(x).
Transcribed Image Text:288 Chapter 4 Applications of Differentiation х — 3 x? 27. In Example 6 of Section 1.5, we develop the 14. У 13. у 3 X +1 Michaelis-Menton model for the rate at which 1+ x4 In Problems 15 to 20, graph the families of functions by finding asymptotes and using first and second deriva- tives. In particular, determine how the graph of the functions depends on the parameter a > 0. an organism consumes its resource. For bacte- rial populations in the ocean, this model was given by 1.2078x f(x) = mg of glucose/hr 1+0.0506x ах 15. у %3D х4 — ах? = xª – ax² 16. У where x is the concentration of glucose (micrograms per liter) in the environment. Use asymptotes and first derivatives to sketch this function by hand. - x2 +1 17. y = ae* +e¯x 18. у — е* + ае * 1 20. у — ах + 28. In Example 5 of Section 2.4 we found that the rate at which wolves kill moose can be modeled by а +x 19. у %3D - 1+ x X 3.36x In Problems 21 and 22, sketch the graph of a function with the given properties. 21. x = 2, x = -2 are vertical asymptotes f(x) = moose/wolf/hundred days 0.42 + x where x is measured in number of moose per square kilometer. Use asymptotes and first derivatives to f is increasing for 0 < x < 2 and x > 2 f is decreasing for x < -2 and -2 < x < 0 sketch this function. graph is concave down on (-∞, -2) and (2, ∞) intercepts are (-1,0), (–3, 0), (3, 0) and (1, 0) 22. y = 1, y = -1 are horizontal asymptotes 29. In Problem 42 in Problem Section 2.4, we examined how wolf densities in North America depend on moose densities. We found that the following func- tion provides a good fit to the data: 3 and for x > 2 3 f is increasing for x < – 58.7(х — 0.03) 2 f(x) = wolves per 1000 km² 0.76 + x f is decreasing for –1 < x < 1 graph is concave down for x < -1 and for 0 < x < 1 where x is number of moose per square kilometer. a. Find the horizontal and vertical asymptotes. graph is concave up for x > 1 and for –1 < x < 0 - b. Determine on which intervals f is increasing and decreasing. c. Determine on which intervals f is concave up Level 2 APPLIED AND THEORY PROBLEMS 23. Consider the graph of y = ax? + bx +c for constants a, b, and c. Use second derivatives to determine what happens to the graph as a changes. and concave down. d. Use the information from parts a-c to sketch the graph of f(x).
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