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Math 117 Exam 2 _______________________________________________ Name Instructor: Nancy Goodisman 1. Let f ( x) 2 x e x . a. Find f (x) . b. Find the critical value where f (x) = 0. c. What is the interval on which f is increasing. d. What is the interval on which f is decreasing. e. Find a relative extreme point and identify it as a relative maximum or a relative minimum. f. Is the relative extreme point an absolute maximum or minimum? How can you be sure? 2. Use implicit differentiation to find the tangent line at the point (5, 0). dy if xy 2 y x 2 25 . Then find the slope of dx 3. The demand function, p 1000 4q , gives the demand q at price p. The cost of producing q units is C q 2 40q 1000 . a. What is the revenue function? ( R p q ) b. What is the profit function? ( P R C ) c. Find the marginal profit. d. Find the maximum profit. e. How do you know it is and absolute maximum? f. What are the quantity and price for a maximum profit? dR dC dR dC when the profit is maximized. Hint. Find and and dq dq dq dq evaluate them at the value for q results in a maximum profit. g. Show that h. Find the maximum revenue and corresponding quantity and price. Maximum revenue = __________ Quantity = _________ Price = ________ 4. If the demand function, p 1000 4q , gives the demand q at price p. p q a. Find the point elasticity of demand, , when q = 50. dp dq b. When the demand is 50 units, a 2% increase in price would result in what decrease in demand? 5. Let W be the weekly sales of a product and t be weeks from the start of an advertizing campaign. a. What conclusion can you draw if W (8) 0 and W (8) 5 ? b. What does this tell you about the effectiveness of the ad campaign at week 8?