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OPMT 5701 Midterm Study Questions Applied Maximum and Minimum MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 500 1) The demand equation for a monopolist's product is p = , where p is the price per unit (in dollars) for q units. q If the total cost c (in dollars) of producing q units is given by c = 5q+ 2000, then the level of production at which profit is maximized is A) 50 units. B) 100 units. C) 750 units. D) 1235 units. E) 2500 units. 2) A manufacturer has determined that the total cost c of producing q units of a product is given by c = 0.04q2 + 4q + 6400. Average cost will be a minimum at a production level of A) 100 units. B) 200 units. C) 400 units. D) 800 units. E) none of the above SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 3) Determine the intervals on which the function is increasing and on which it is decreasing. Also determine the points of relative maxima and relative minima. x2 f(x) = ex 4) The demand equation for a monopolist's product is p = 2700 - q2 , where p is the price per unit (in dollars) when q units are demanded. (a) Find the value of q for which revenue is maximum. (b) What is the maximum revenue? 5) The demand function for a monopolist's product is p = 100 - 3q, where p is the price per unit (in dollars) for q 100 units. If the average cost c (in dollars) per unit for q units is c = 4 + , find the output q at which profit is q maximized. 6) Find all the critical values of f(x) = x 4 - 8x 2 + 3. 7) The profit equation for a taco stand is given by P(x) = -0.4x 2 + 100x - 100, where x is the number of tacos sold, and P(x) is the profit in dollars. Use the first-derivative test to find when relative extrema occur. 8) The cost equation for a bakery is given by C(x) = 10(x - 2)2 - 120(x - 2) + 450, where x is the number of doughnuts made (in dozens), and C(x) is the cost in dollars. Use the first-derivative test to find when relative extrema occur. 9) Suppose that c = (0.2q - 20)3 + 20,000 is a total cost function, where c is the total cost (in dollars) of producing q units of a product. How fast is marginal cost changing when q = 150? 1 10) The cost equation for a company is C(x) = 2x 3 - 15x 2 - 84x + 3100. Use the second-derivative test, if applicable, to find the relative maxima and the relative minima. 11) The cost equation for a company is C(x) = 2x 3 - 39x 2 + 180x + 21,200. Use the second-derivative test, if applicable, to find the relative maxima and the relative minima. 12) A company owns an apartment building containing 80 units. If the company charges $300 per month rent, then all units can be rented out, and for every increase of $10 per month in rent, the company will lose one customer. What rent should be charged to maximize revenue? 350 13) If a manufacturer's average cost equation is c = - 422q + 17q2 , find the total cost function c, and the q dc marginal cost function . What is the marginal cost when 20 units are produced? dq 14) The revenue equation for a company is given by R(x) = 68.04x - 0.07x 3 . Determine when relative extrema occur on the interval (0, «). 15) The cost equation for a company is C(x) = 3x 3 - 27x 2 + 45x + 100. Use the second-derivative test, if applicable, to find the relative maxima and the relative minima. 16) If y = (7x + 5)11, then find y'''. 17) If y = e2x 2 -3, then find y'' at x = 0. exy ∂z 18) If z = , find . 2x + 3y ∂x 19) A sporting goods store determines that the optimal quantity of athletic shoes (in pairs) to order each month is 2CM given by the Wilson lot size formula: Q(C, M, s) = , where C is the cost (in dollars) of placing an order, M s is the number of pairs sold each month, and s is the monthly storage cost (in dollars) per pair of shoes. Find ∂Q ∂Q . Then find and interpret . ∂M ∂M (100, 500, 3) 20) A sporting goods store determines that the optimal quantity of athletic shoes (in pairs) to order each month is 2CM given by the Wilson lot size formula: Q(C, M, s) = , where C is the cost (in dollars) of placing an order, M s ∂Q is the number of pairs sold each month, and s is the monthly storage cost (in dollars) per pair of shoes. Find . ∂s ∂Q Then find and interpret . ∂s (100, 500, 3) x2 + 1 ∂z ∂z 21) If z = , find (a) and (b) . y ∂x ∂y 2 22) A company manufactures two products, X and Y, and the joint-cost function for these products is given by c = x x + 4y , where c is the total cost of producing x units of X and y units of Y. Determine the marginal cost with respect to x when x = 36 and y = 16. 23) If w = f (x, y, z) = x 2 yz - yz2 + xz2 , find: ∂w (a) ∂x ∂w (b) ∂y ∂w (c) ∂z ∂2 w (d) ∂y 2 ∂2 w (e) ∂x∂z 24) If f(x, y) = exy, find: (a) fx (x, y) (b) fxx(x, y) (c) fxy(x, y) 25) If f(x, y) = 2x4y3 - 3x3y3 + 4xy - x + 2y + 4, find: (a) fx(x, y) (b) fy(x, y) (c) fxy(x, y) (d) fxy(-1, 1) (e) fyyx(x, y) 3 Answer Key Testname: 5701-2010_APPLIED_MAX_MIN-STUDYQUESTIONS 1) E 2) C 3) Decreasing on the intervals (-«, 0) and (2, «); increasing on (0, 2); relative maximum at x = 2; relative minimum at x = 0. 4) (a) 30 (b) $54,000 5) 16 6) 0, ± 2 7) There is a relative maximum when x = 125. 8) There is a relative minimum when x = 8. 9) 2.4 10) relative maximum when x = -2, relative minimum when x = 7 11) relative maximum when x = 3, relative minimum when x = 10 12) $550 dc dc 13) c = 350 - 422q 2 + 17q3 , = -844 + 51q2 , = 19,566 dq dq q=20 14) relative maximum when x = 18 15) relative maximum when x = 1, relative minimum when x = 5 16) 339,570(7x + 5)8 1 17) 2 exy(3y 2 + 2xy - 2) 18) (2x + 3y) 2 ∂Q C ∂Q 19) = ; ‘ 0.18; When C = 100, M = 500, and s = 3, the optimal quantity increases by about 0.18 ∂M 2Ms ∂M (100, 500, 3) pair per order for each 1-pair increase in the number of pairs sold each month. ∂Q CM ∂Q 20) =- ; ‘ -30.43; When C = 100, M = 500, and s = 3, the optimal quantity decreases by about 30 ∂s 2s3 ∂s (100, 500, 3) pairs per order for each dollar increase in the monthly storage cost per pair. 2x (x 2 + 1) 21) (a) ; (b) - y y2 22) 11.8 23) (a) 2xyz + z2 (b) x 2 z - z2 (c) x 2 y - 2yz + 2xz (d) 0 (e) 2xy + 2z 24) (a) ye xy (b) y 2 exy (c) exy(xy + 1) 25) (a) 8x 3 y 3 - 9x 2 y 3 + 4y - 1 (b) 6x 4 y 2 - 9x 3 y 2 + 4x + 2 (c) 24x 3 y 2 - 27x 2 y 2 + 4 (d) -47 (e) 48x 3 y - 54x 2 y 4