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Unformatted text preview: Math 1325 Exam Two Review Problems
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the relative extrema of the function, if they exist. 1) y = x3 - 3x2 + 4x - 4 A) Relative minimum at (1, 0) C) Relative maximum at (- 1, 0) B) Relative maximum at (2, 0) D) No relative extrema exist 1) Find the absolute maximum and absolute minimum values of the function, if they exist, over the indicated interval, and indicate the x -values at which they occur. 2 1 2) 2) f(x) = x3 - x2 - x + 3; [0, 2] 3 2
10 8 6 4 2 y 1 2x A) Absolute maximum = 3 at x = 2; absolute minimum = 2.17 at x = 1 B) Absolute maximum = 4.33 at x = 2; absolute minimum = 3.29 at x = 0 C) Absolute maximum = 3.29 at x = 0; absolute minimum = 2.17 at x = 1 D) Absolute maximum = 4.33 at x = 2; absolute minimum = 2.17 at x = 1 Find the absolute maximum and absolute minimum values of the function, if they exist, over the indicated interval. When no interval is specified, use the real line ( -∞, ∞). 3) 3) f(x) = 20x - x2 A) No absolute maximum; absolute minimum: 10; B) Absolute maximum: 100; absolute minimum: 0 C) Absolute maximum: 200; no absolute minimum D) Absolute maximum: 100; no absolute minimum 2 3 4) f(x) = x3 + x2 - 27x + 2; (-∞, 0) 3 2 A) No absolute maximum; absolute minimum = - B) Absolute maximum = C) Absolute maximum = D) No absolute extrema 95 2 4) 745 95 ; absolute minimum = - 8 2 745 ; no absolute minimum 8 1 Solve the problem. 5) For a dosage of x cubic centimeters (cc) of a certain drug, assume that the resulting blood pressure B is approximated by B(x) = 0.06x2 - 0.2x3 . Find the dosage at which the resulting blood pressure is maximized. Round your answer to the nearest hundredth. A) 0.20 cc B) 0.18 cc C) 0.30 cc D) 0.45 cc 5) Find the absolute maximum and absolute minimum values of the function, if they exist, over the indicated interval. When no interval is specified, use the real line ( -∞, ∞). 16 6) f(x) = x2 + ; (0, ∞) 6) x
200 y 14 x A) No absolute maximum; absolute minimum: 2 B) No absolute maximum; absolute minimum: 12 C) Absolute maximum: 196; absolute minimum: 2 D) No absolute extrema Use a graphing calculator or computer graphing software to solve the problem (correct to one decimal place). 7) Find the absolute maximum value for the function f(x) = x(x - 9)2/3 over the interval [- 1, 8]. 7) A) 9.3 B) 8.5 C) 12.7 D) 8.7 Solve the problem. 8) The total- revenue and total- cost functions for producing x clocks are R(x) = 520x - 0.01x2 and C(x) = 120x + 100,000 , where 0 ≤ x ≤ 25,000. What is the maximum annual profit? A) $3,900,000 B) $4,200,000 C) $4,100,000 D) $4,000,000 9) A hotel has 230 units. All rooms are occupied when the hotel charges $110 per day for a room. For every increase of x dollars in the daily room rate, there are x rooms vacant. Each occupied room costs $34 per day to service and maintain. What should the hotel charge per day in order to maximize daily profit? A) $77 B) $170 C) $177 D) $187 10) Suppose that the weekly profit, in dollars, of producing and selling x cars is P(x) = - 0.006x3 - 0.3x2 + 960x - 900, and currently 60 cars are produced and sold weekly. Use P(60) and the marginal profit when x = 60(P′ (60)) to estimate the weekly profit of producing and selling 61 cars. Round to the nearest dollar. A) $55,182 B) $55,183 C) $56,083 D) $55,269 8) 9) 10) 2 11) The average cost for a company to produce x thousand units of a product is given by the function 1024 + 1200x A(x) = x Use A′ (x) to estimate the change in average cost if production is increased by one thousand units from the current level of 16 thousand. A) Average cost will decrease by $64 B) Average cost will increase by $4 C) Average cost will increase by $64 D) Average cost will decrease by $4 Find dy. 12) y = x2 + x + 4 x + 5 x2 + 10x + 1 dx (x + 5) 2 B) (2x + 1) dx C) x2 + 10x + 1 dx x + 5 D) x2 + 10x + 9 dx (x + 5) 2 11) 12) A) Find dy/dx by implicit differentiation. 13) 9xy + 2y - 2 = 0 - 9y - 9y(x + 1) B) A) 9x + 2 2 13) - 9y C) 9xy + 2 - 9(x + y) D) 2 For the given demand equation, differentiate implicitly to find dp/dx. 14) 7p2 + x2 = 1000 dp - 7p = A) x dx dp - p B) = dx 7x dp - x C) = dx 7p dp - 7x D) = dx p 14) Provide an appropriate response. 15) A man 6 ft tall walks at a rate of 5 ft/sec away from a lamppost that is 13 ft high. At what rate is the length of his shadow changing when he is 65 ft away from the lamppost? 30 325 30 15 ft/sec B) ft/sec C) ft/sec D) ft/sec A) 19 6 7 19 16) A 26- foot ladder is placed against a wall. If the top of the ladder is sliding down the wall at 2 feet per second, at what rate is the bottom of the ladder moving away from the wall when the bottom of the ladder is 10 feet away from the wall? A) 9.6 ft/sec B) 2.4 ft/sec C) 5.2 ft/sec D) 4.8 ft/sec 17) A company is manufacturing a new digital watch and can sell all it manufactures. The revenue (in x2 dollars) is given by R(x) = 50x - , where the production output in one day is x watches. If 50 production is increasing at 5 watches per day when production is 375 watches per day, find the rate of increase in revenue. A) $250 per day B) $75 per day C) $175 per day D) $150 per day 18) Given the revenue and cost functions R = 26x - 0.3x2 and C = 3x + 10, where x is the daily production, find the rate of change of profit with respect to time when 20 units are produced and the rate of change of production is 7units per day per day. A) $77.00 per day B) $149.00 per day C) $156.80 per day D) $98.00 per day 15) 16) 17) 18) 3 19) Evaluate dy/dt for the function at the point. x + y = x2 + y 2 ; dx/dt = 12, x = 1, y = 0 x - y A) 12 B) - 1 12 C) - 12 D) 1 12 19) Solve the problem. 20) A beverage company works out a demand function for its sale of soda and finds it to be x = D(p) = 3100 - 24p where x = the quantity of sodas sold when the price per can, in cents, is p. At what prices, p, is the elasticity of demand inelastic? A) For p < 129 cents B) For p > 37,200 cents C) For p < 65 cents D) For p > 258 cents 20) 4 Answer Key Testname: MATH 1325 EXAM2REV 1) D 2) D 3) D 4) C 5) A 6) B 7) C 8) A 9) D 10) B 11) D 12) A 13) B 14) C 15) D 16) D 17) C 18) A 19) A 20) C 5 ...
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