Study Questions for OPMT 5701 Most Questions come from Chapter 17. The Answer Key has a section code for each. Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 1) If z = (x2 + y2) 10 where x = 4r2s3 and y = e2r+3s-3, then by means of the chain rule, (a) find z z ; (b) evaluate when r = 0 and s = 1. r r 1) 2) The Cobb-Douglas production function for a company is given by P(l, k) = 70l1/4k3/4, where P is the monthly production value when k is the amount of the company's capital investment (in dollars per month) and l is the size of the labor force (in work hours per month). Find 2P l k (2401, 10,000) and 2P k2 (2401, 10,000). 2) 3) If z = x 2 e y + y 2 e x where x = 2rs 2 and y = ln r 2 + ln s 2, find z s. 3) 4) Determine the critical points of f(x, y) = 2xy - 3x - y - x2-3y2 and also determine by the second-derivative test whether each point corresponds to a relative maximum, to a relative minimum, to neither, or whether the test gives no information. 4) 5) An empirical formula relating the surface area A (in square inches) of an average human body to the weight w (in pounds) and the height h (in inches) of the person is A(w, h) = 15.64w0.425h0.725. Find 2A and 2A. h w (105, 64) w2 (105, 64) 5) 6) Determine the critical points of f(x, y) = x3 + 1 2 y 2-3xy - 4y + 2 and also determine by the 6) second-derivative test whether each point corresponds to a relative maximum, to a relative minimum, to neither, or whether the test gives no information. 7) An empirical formula relating the surface area A (in square inches) of an average human body to the weight w (in pounds) and the height h (in inches) of the person is A(w, h) = 15.64w0.425h0.725. Find Ah(w, h). Then find and interpret Ah(105, 64). 7) 8) A television manufacturing company makes two types of TV's. The cost of producing x units of type A and y units of type B is given by the function C(x, y) = 100 + x3 + 64y3-96xy. How many units of type A and type B televisions should the company produce to minimize its cost? 8) 1
9) 10 2p B The demand function for product A is q A =, and the demand function for product p A 9) B is q B = 20 + 3p A -2p B, where q A and q B are the quantities demanded for A and B, respectively, and p A and p B are their respective prices. Determine: (a) the marginal demand for A with respect to p B (b) the marginal demand for B with respect to p A (c) whether A and B are competitive, complementary, or neither 10) A sporting goods store determines that the optimal quantity of athletic shoes (in pairs) to 2CM order each month is given by the Wilson lot size formula: Q(C, M, s) =, where C is s the cost (in dollars) of placing an order, M is the number of pairs sold each month, and s is the monthly storage cost (in dollars) per pair of shoes. Find Q. Then find and interpret s Q s (100, 500, 3). 10) 11) If w = f (x, y, z) = x2yz - yz2 + xz2, find: w (a) x (b) (c) (d) (e) w y w z 2w y2 2w x z 11) 12) A firm has an order of 10,000 units of its product and has two plants at which to manufacture these units. Let q 1 be the number of units to be produced at the first plant and q 2 denote the number to be manufactured at the second plant. It is known that the cost function is given by 12) C = 48q 3 1 + 3 q 3 2 + 25,000. Use the method of Lagrange multipliers to determine how many units should be produced at each plant to minimize this cost function. 13) For ln(xyz) + e = e y + 1, the partial derivative z x evaluated at x = e -2, y = 1, z = e3 13) 14) For 2x2 + 3y2 + 2z2 = 16, evaluate z when x = 1, y = 2, z = -1. 14) y 2
15) Let f(x, y, z) = ln(x 4 + 6y 2 ) - 2z 4 x 2 e 3y + x 20 y 3 3f. Find x y z. 15) 16) If 2x2 + 3y2 + 2z2 = 16, find z y. 16) 17) If f(p, q) = 3p2-2q+ p, find f(-1, 2). 17) 18) Use the method of Lagrange multipliers to find the critical points of f(x, y, z) = 2x + 4y - 4z subject to the constraint x2 + y2 + z2 = 9. 19) A company's production function is given by P = 2.1L0.6k0.4, where P is the total output generated by L units of labor and k units of capital. Determine: (a) the marginal production function with respect to L (b) the marginal production function with respect to k 18) 19) 20) 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. 20) 21) Let f(x, y, z) = xz3ez2+1 + x2y3z4 + y ln(e2-1). Find 3f x y z. 21) 22) If f(x, y) = e2xy ln(y + 2), find f(0, -1). 22) 23) Use implicit partial differentiation to find z y from exy + 7x 3 + 8z - 19 = 0. 23) 24) Given h(r, s, t, u) = rs, find h(2, 3, 2, 15). 24) 3 - t2 25) Let f(x, y) = x 2 e 3y + y 3 ln 2x. Find: 2f x2, 2f y2, 2f y x 25) 26) Find f f 5xy2 and where f(x, y) = x y (x3 + y3). 26) 27) 27) 28) If y = 3x4-6x2, use the second-derivative test to find all values of x for which (a) relative maxima occur (b) relative minima occur. 28) 29) The cost equation for a company is C(x) = 3x3-27x2 + 45x + 100. Use the second-derivative test, if applicable, to find the relative maxima and the relative minima. 29) 30) The revenue equation for a company is given by R(x) = 1296x - 0.12x3. Determine when relative extrema occur on the interval (0, «). 30) 3
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 31) If 7x2 + 4y2 = 1, then dy dx = A) 7x + 4y. B) - 7x 1-14x. C) 14x + 8y. D). E) 4y 4y 8y 7x. 31) SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 32) Use implicit differentiation to find dy dx explicitly in terms of x and y from 3x 2 + 7xy + y2 = 19. 33) Suppose that a company can produce 12,000 units when the number of hours of skilled labor y and unskilled labor x satisfy 384 = (x + 2)3/4(y + 3)1/3. Find dy, the rate of change dx of skilled labor hours with respect to unskilled labor hours. 34) If z = 2x2y + 3xy + y2 where x = r2 + 2rs and y = 2r - 4s, then by means of the chain rule, (a) find z ; (b) evaluate when r = 1 and s = 0. s 32) 33) 34) 35) Find an equation of the plane that is parallel to the y, z-plane and that passes through the point (1, 2, 3). 36) The Cobb-Douglas production function for a company is given by P(k, l) = 70k3/4l1/4 where P is the monthly production value when k is the number of units of capital and l is the number of units of labor. Suppose that capital costs $450 per unit, labor costs $75 per unit, and the total cost of capital and labor is limited to $60,000. Use Lagrange multipliers to write the system of equations you would use to find the number of units of capital and labor that maximize production. 35) 36) 37) If f(x, y) = exy, find: (a) fx(x, y) (b) fxx(x, y) (c) fxy(x, y) 37) 38) A manufacturer of widgets has determined that the production function for a weekly production of p thousand gross of widgets is p = 1000 + 20l2k3-5l3-3k4, where l is the number of labor hours per week in thousands and k is the amount of capital in thousands of dollars per week. Determine both of the marginal productivity functions. 39) An open rectangular cardboard box is to have a volume of 4 cubic feet. Find the dimensions of the box so that the amount of cardboard is minimized. 38) 39) 4
40) A sporting goods store determines that the optimal quantity of athletic shoes (in pairs) to 2CM order each month is given by the Wilson lot size formula: Q(C, M, s) =, where C is s the cost (in dollars) of placing an order, M is the number of pairs sold each month, and s is the monthly storage cost (in dollars) per pair of shoes. Find Q. Then find and interpret C Q. C (100, 500, 3) 40) 41) The Cobb-Douglas production function for a company is given by P = 70l1/4 k3/4 where P is the monthly production value when k is the amount of the company's capital investment (in dollars per month) and l is the size of the labor force (in work hours per month). What is the production value when l = 2401 hours and k = $10,000 per month? 41) 42) For x2y + xz + z2 = 4, evaluate z when x = -1, y = 2, z = -1. 42) x 43) Let q A = 50-5p A + 6 p 2 B and q B = 20 p A p -1 B be demand functions, where p A and p B are prices for products A and B, respectively. Find all four marginal demand functions. 43) 44) Use implicit partial differentiation to find z x from ln(xyz) = ey + 79. 44) 5
Answer Key Testname: STUDYQUEST1 1) (a) 160xrs3( x2 + y2) 9 + 40 ye2r+3s-3( x2 + y2) 9 ; (b) 40 ID: INMA11 17.6-1 2) 2P 3 = l k (2401, 10,000) 784 0.0038; 2P 147 = - k2 (2401, 10,000) 160,000-0.0009 ID: INMA11 17.5-18 3) (2xe y + y 2 e x ) 4rs + (x 2 e y + 2ye x ) 2 s ID: INMA11 17.6-5 4) - 5, -1, relative maximum 2 ID: INMA11 17.7-3 5) 2A h w (105, 64) 0.1057; 2A w2 (105, 64) -0.0511 ID: INMA11 17.5-19 6) 4, 16, relative minimum; (-1, 1), neither ID: INMA11 17.7-5 7) Ah (w, h) = 11.339w0.425h-0.275; Ah (105, 64) 26.11; the surface area of a 105-pound person who is 64 inches tall increases by about 26.11 square inches for every 1-inch increase in height. ID: INMA11 17.2-14 8) x = 8, y = 2 ID: INMA11 17.7-16 9) (a) 10) 10 p A 2p B (b) 3 (c) competitive ID: INMA11 17.3-5 Q s = - CM 2s3 ; Q -30.43; When C = 100, M = 500, and s = 3, the optimal quantity decreases by about 30 s (100, 500, 3) pairs per order for each dollar increase in the monthly storage cost per pair. ID: INMA11 17.2-19 6
Answer Key Testname: STUDYQUEST1 11) (a) 2xyz + z2 (b) x2z - z2 (c) x2y - 2yz + 2xz (d) 0 (e) 2xy + 2z ID: INMA11 17.5-1 12) q 1 = 2000; q 2 = 8000 ID: INMA11 17.8-12 13) -e5 ID: INMA11 17.4-11 14) z y = - 3y 2z ; z y (1, 2, -1) = 3 ID: INMA11 17.4-7 15) -48z 3 xe 3y ID: INMA11 17.5-10 16) - 3y 2z ID: INMA11 17.4-1 17) -2 ID: INMA11 17.1-2 18) (1, 2, -2) and (-1, -2, 2) ID: INMA11 17.8-10 19) (a) 1.26L-0.4k0.4 (b) 0.84L0.6k-0.6 ID: INMA11 17.3-3 20) 11.8 ID: INMA11 17.3-2 21) 24xy2z3 ID: INMA11 17.5-12 22) 0 ID: INMA11 17.1-3 7
Answer Key Testname: STUDYQUEST1 23) - xexy 8 ID: INMA11 17.4-6 24) -6 ID: INMA11 17.1-15 25) 2f x2 = 2e3y - y3 x 2 2f y2 = 9x2 e 3y + 6y ln 2x 2f y x = 6xe3y + 3y2 x ID: INMA11 17.5-11 26) 5y5-10x3y2 (x3 + y3) 2 ; 10x 4y - 5xy4 (x3 + y3) 2 ID: INMA11 17.2-11 27) ID: USER-1 28) (a) 0 (b) ±1 ID: INMA11 13.4-1 29) relative maximum when x = 1, relative minimum when x = 5 ID: INMA11 13.4-4 30) relative maximum when x = 60 ID: INMA11 13.4-8 31) B ID: INMA11 12.4-8 32) - 6x + 7y 7x + 2y ID: INMA11 12.4-10 33) dy dx = - 9 4 y + 3 x + 2 ID: INMA11 12.4-18 8
Answer Key Testname: STUDYQUEST1 34) (a) (4xy + 3y)(2r) + (2x2 + 3x + 2y)(-4); (b) -8 ID: INMA11 17.6-2 35) x = 1 ID: INMA11 17.1-5 36) (You do not have to solve this system!) 210 4 k -1/4l1/4-450l = 0 70 4 k 3/4l-3/4-75l = 0 450k + 75l - 60,000 = 0 ID: INMA11 17.8-14 37) (a) yexy (b) y2exy (c) exy(xy + 1) ID: INMA11 17.5-3 38) p l = 40lk 3-15l2; p k = 60l 2k2-12k3 ID: INMA11 17.3-8 39) 2 ft by 2 ft by 1 ft ID: INMA11 17.7-8 40) Q C = M 2Cs ; Q C in the cost of placing an order. ID: INMA11 17.2-17 41) $490,000 ID: INMA11 17.1-17 (100, 500, 3) 0.91; the optimal quantity increases by about 1 pair per order for each dollar increase 42) z x = - z + 2xy 2z + x ; z x (-1, 2, -1) = - 5 3 ID: INMA11 17.4-8 9
Answer Key Testname: STUDYQUEST1 43) q A q A q B 10 q B = -5; = 12p p A p B ; = ; = B p A p B p p A B -20 p A p 2 B ID: INMA11 17.3-7 44) - z x ID: INMA11 17.4-5 10