Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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51. The cost (in dollars) of producing x units of a certain com- modity is C(x) = 5000 + 10x + 0.05x². (a) Find the average rate of change of C with respect to x when the production level is changed (i) from x = 100 to x = 105 100 to x = 101 (ii) from x = (b) Find the instantaneous rate of change of C with respect to x when x = 100. (This is called the marginal cost. Its significance will be explained in Section 2.7.) 52. If a cylindrical tank holds 100,000 gallons of water, which can be drained from the bottom of the tank in an hour, then Torricelli's Law gives the volume V of water remaining in the tank after t minutes as 10 V(t) = 100,000(1-² 0≤t≤ 60 Find the rate at which the water is flowing out of the tank (the instantaneous rate of change of V with respect to t) as a function of t. What are its units? For times t == 0, 10, 20, 30, 40, 50, and 60 min, find the flow rate and the amount of water remaining in the tank. Summarize your findings in a sentence or two. At what time is the flow rate the greatest? The least? 53. The cost of producing x ounces of gold from a new gold mine is C = f(x) dollars. (a) What is the meaning of the derivative f'(x)? What are its units? (b) What does the statement f'(800) = 17 mean? (c) Do you think the values of f'(x) will increase or decrease in the short term? What about the long term? Explain. 54. The number of bacteria after t hours in a controlled laboratory experiment is n = f(t). (a) What is the meaning of the derivative f'(5)? What are its th Se (
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Transcribed Image Text:51. The cost (in dollars) of producing x units of a certain com- modity is C(x) = 5000 + 10x + 0.05x². (a) Find the average rate of change of C with respect to x when the production level is changed (i) from x = 100 to x = 105 100 to x = 101 (ii) from x = (b) Find the instantaneous rate of change of C with respect to x when x = 100. (This is called the marginal cost. Its significance will be explained in Section 2.7.) 52. If a cylindrical tank holds 100,000 gallons of water, which can be drained from the bottom of the tank in an hour, then Torricelli's Law gives the volume V of water remaining in the tank after t minutes as 10 V(t) = 100,000(1-² 0≤t≤ 60 Find the rate at which the water is flowing out of the tank (the instantaneous rate of change of V with respect to t) as a function of t. What are its units? For times t == 0, 10, 20, 30, 40, 50, and 60 min, find the flow rate and the amount of water remaining in the tank. Summarize your findings in a sentence or two. At what time is the flow rate the greatest? The least? 53. The cost of producing x ounces of gold from a new gold mine is C = f(x) dollars. (a) What is the meaning of the derivative f'(x)? What are its units? (b) What does the statement f'(800) = 17 mean? (c) Do you think the values of f'(x) will increase or decrease in the short term? What about the long term? Explain. 54. The number of bacteria after t hours in a controlled laboratory experiment is n = f(t). (a) What is the meaning of the derivative f'(5)? What are its th Se (
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