HW2 - 785 14.4 GRADIENTS AND DIRECTIONAL DERIVATIVES IN THE...

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Unformatted text preview: 785 14.4 GRADIENTS AND DIRECTIONAL DERIVATIVES IN THE PLANE Math for Econ II – Homework Assignment 2 New York University Due in Recitation, Friday, February 19th rcises and Problems for Section 14.4 cises • Please write neat solutions for the problems below. Show all your work. If you only write the answer with no work 14-4w27 xercises 1–14, find the gradient of the function. Assume 29. grad f = (2x + 3ey )i + 3xey j you will not be given any credit. variables are restricted to a domain on which the function • Do not forget to write your name Exercises 30–31, findsection number on your work and to staple your work, if you ins14-4w28-29 In and Recitation grad f from the differential. fined. have multiple pages. 3 5 x 2 f (x, y) = 14-4w28 30. df = 2xdx + 10ydy 14-4w29 31. df = (x + 1)yex dx + xex dy 14-4w3 2. Q = 50K + 100L − 4 y6 7 f (m, n) = m + 1. (3 pts)zFind y unit vectors which are perpendicular to 8, −3 . 4. = xe all ins14-4w30-35 In Exercises 32–37, Figure below to decide Fig2. (5 pts) Use the contour diagram of f (x, y) inuse the contour diagram of fifinthe specified directional derivative is 14-4w8 6. f (r, h) = πr 2 h f (α, β) = 5α2 + β ure 14.34 to decide if the specified directional derivative is positive, negative, or approximately zero. n2 14-4w4 z = (x + y)ey 14-4w6 f (r, θ) = r sin θ 14-miscw32 14-4w12 2 y 2 10. f (x, y) = ln(x + y ) 3 12. z = tan−1 (x/y) 2 xercises 15–22, find the gradient at the point. −1 f (x, y) = x2 y + 7xy 3 , at (1, 2) −2 2 1 x 2 f (α, β) = 4 y 2α + 3β 14-4w14 14. z = x e 2α − 3β x+y 8 z = sin(x/y) positive, negative, or approximately zero. 8. f (K, L) = K 0.3 L0.7 6 2 f (m, n) = 5m + 3n , at (5, 2) f (r, h) = 2πrh + πr2 , at (2, 3) f (x, y) = esin y , at (0, π) 8 4 −3 −3 −2 −1 ins14-4w30-35fig 6 4 2 1 2 3 ! Figure 14.34 √ 14-4w30 32. At point f (x, y) = sin (x2 ) + cos(a)atAt2π , 0)point (−2, 2), in direction i. (−2, 2), in direction i . y, ( the 14-4w31 33. At point (b) 1) f (x, y) = ln(x2 + xy), at (4,At the point (0, −2), in direction j.(0, −2), in direction j . 14-4w34 34. At point (0, −2), in direction i + 2j . (c) At the point (−1, 1), in direction i + j. f (x, y) = 1/(x2 + y 2 ), at (−1, 3) √ 14-4w35 35. At point (d) 1) i + j. f (x, y) = tan x + y, at (0, At the point (−1, 1), in direction −(0, −2), in direction i − 2j . 14-4w32 36. At point (−1, (e) At the point (0, −2), in direction i − 2j.1), in direction i + j . xercises 23–26, find the directional derivative fu (1, 2) 14-4w33 37. At point (−1, 1), in direction −i + j . y he function f with u 3. (3i pts)j Let f (x, y) = = (3 − 4 )/5. . Find the directional derivative of f at (2, 1) in the direction of −3, −5 . 2 + y2 x ins14-4w36-43 In Exercises 38–45, use the contour diagram of f in Figf (x, y) = xy + y 3 14-4w22 pts)fFind the equation of the 14.34 to find the for g(x, y) = 1 + x ln(xy − 5) at the point (2, 3, 1) ure tangent plane approximate direction of the gradient 4. (3 24. (x, y) = 3x − 4y vector at the given point. 2 2 14-4w25 f (x, y) = x − y 5. (5 26. f (x, y)directional y) pts) The = sin(2x − derivative of f (x, y) at (2, 1) in the direction going from (2, 1) toward the point (1, 3) √ 2 is −2/ 5, and the directional derivative at (2, 1) in the direction going from (2, 1) toward the point (5, 5) is 1. 14-4w38 38. (−2, 0) 14-4w39 39. (0,14-4w36 40. (2,14-4w37 41. (0, 2) −2) 0) Compute fx (2, 1) and fy (2, 1). If f (x, y) = x y and v = 4i − 3j , find the directional 14-4w43 43. (−2, −2) 44. (2,14-4w41 45. (2, −2) 14-4w40 14-4w42 42. (−2, 2) 2) derivative at the point (2, 6) in the direction of v .of the following functions. 6. Find the differential y xercises 28–29, find the differential df from(x, y) = (a) (2 pts) f the gradient. grad f = y i + xj 1 + xy (b) (2 pts) P (K, L, M ) = KM e−L 2 −M 2 7. (4 pts) An international organization must decide how to spend the $2000 they have been allotted for famine relief in a remote area. They expect to divide the money between buying rice at $5/sack and beans at $10/sack. Let f (P ) = 15 and f (Q) = number, P ,= (3, 4) andwho would be fed if theyyour answer. of rice and y sacks of beans is given by decimal places in buy x sacks The 20 where P of people blems Q = (3.03, 3.96). Approximate the directional derivative of f at P in the direction of Q. (b) Use P and Q to approximate the directional deriva√ 2 2 tive of f (x, y) = x + y in x ydirection of v . P = x + 2y + the 8 . (c) Give the exact value for the ·directional derivative 2 10 (a) Give Q, the point at a distance of 0.1 from P = you that can part (b). What is the maximum number of peopleestimated in be fed, and how should the organsization allocate its money? (4, 5) in the direction of v = −i + 3j . Give five You may assume that the maximum value occurs at a solution given by the Lagrange multiplier method. 8. (Review of MFE 1) A firm produces and sells two commodities. When the firm produces x tons of the first commodity, it is able to sell the commodity at a price of 96 − 4x dollars per ton. When the firm produces y tons of the second commodity, it is able to sell the commodity at a price of 84 − 2y dollars per ton. The cost of producing and selling x tons of the first commodity and y tons of the second commodity is C(x, y) = 2x2 + 2xy + y 2 . (a) (1 point) Find the firm’s profit function π(x, y). (b) (2 pts) Find all critical points of π. Classify each critical point as a local maximum, a local minimum, or a saddle point. Extra problems–not to be handed in: 1. Suppose that f (x, y) is a differentiable function and (a, b) is a critical point of f . Is it true that the directional derivative of f at (a, b) in every direction must be zero? If your answer is yes, show that it must be true. If your answer is no, give an example of a differentiable f , a critical point (a, b), and a unit vector u such that Du f (a, b) = 0. 2. (More Lagrange multiplier practice) Given x units of capital and y units of labor, a company produces P (x, y) = 10x1/2 y 1/3 units of its product. Each unit of capital costs $2000, and each unit of labor costs $4000. Use Lagrange multipliers to find the values of x and y which maximize the company’s production if the company spends $20, 000 total on capital and labor. You may assume the maximum occurs at a point where the Lagrange multipliers condition is satisfied. ...
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