Corporate Finance—Renewable Energy: Renewable energy companies have recently attempted to...

Corporate Finance—Renewable Energy: Renewable energy companies have recently attempted to benefit from the same tax advantages oil and coal companies receive. “It would be a real boon to the renewable energy industry,” said John McKenna, of Hamilton Clark Securities. (Source: The San Francisco Chronicle.) The net profits for renewable energy companies from 1990 to 2011 can be modeled by

 f(x) = (1.16x + 4.2)1.37 0 ≤ x ≤ 21

where x represents the number of years since 1990 and f(x) represents the net profits for renewable energy companies, measured in billions of dollars. (Source: U.S. Census Bureau.)

(a) Use the Chain Rule to determine f′(x).

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(b) Evaluate and interpret f′(8).

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(c) Graph f′ in the viewing window [0, 21] by [0, 6],

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(d) Compare the solution to part (b) of Exercise 1. Which model was growing at a faster rate?

Exercise 1

Corporate Finance—Oil Profits: An economic sector that has done well in the past few decades is petroleum- and coal-based energy companies. “The earnings reflect continued leadership in operational performance during a period of strong commodity prices,” said Exxon’s chairman, Rex W. Tillerson. (Source: The New York Times). The net profits for petroleum and coal corporations from 1990 to 2011 can be modeled by

 f(x) = (1.8x + 6.3)1.26 0 ≤ x ≤ 21

where x represents the number of years since 1990 and f(x) represents the net profits for petroleum and coal corporations, measured in billions of dollars. (Source: U.S. Census Bureau.)

(a) Use the Chain Rule to determine f′(x).

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(b) Evaluate and interpret f′(8).

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(c) Graph f′ in the viewing window [0, 21] by [0, 7],

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(d) Use the value command to verify your answer to part (b).