MFE2-hw2 - n 14.4 Math for Econ II Written Assignment 2(28...

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Unformatted text preview: n 14.4 Math for Econ II, Written Assignment 2 (28 points) Due Friday, February 14th 14-4w27 tion. Assume 29. grad f = (2x + 3e )i + 3xe j New York University h the function ins14-4w28-29 Please write neat solutions for the problems below. Show all your work. If you only write the answer with no work, In Exercises 30–31, find grad f from the differential. you will not be given any credit. y 14-4w28 + 100L 14-4w29 ins14-4w30-35 πr2 h = K 0.3 L0.7 2 y 30. df = 2xdx + 10ydy • Write your name and recitation section number. 31. df = (x + 1)yex dx + xex dy • Staple your homework if you have multiple pages! In Exercises 32–37, use the contour diagram of f in Fig1. (5 pts) Use the contour diagram of f in Figure below to decide if the specified directional derivative is positive, ure 14.34 to decide if the specified directional derivative is positive, negative, or approximately zero. negative, or approximately zero. y 2 3 2 6 (x/y) 8 ln(x + y ) 2 4 1 y x 6 8 ins14-4w30-35fig 4 −2 2 −1 t. −3 −3 −2 −1 1 2 3 ! Figure 14.34 14-4w30 14-4w31 14-4w34 14-4w35 14-4w32 tive fu (1, 2) 14-4w33 ins14-4w36-43 3x − 4y sin(2x − y) 14-4w38 32. At point (−2,thein direction i .2), in direction i. (a) At 2), point (−2, 33. At point (0, −2), in direction −2), in direction j. (b) At the point (0, j . 34. At point (0, −2), in direction i 1), j . direction i + j. (c) At the point (−1, + 2 in 35. At point (0, −2), in direction i 1), j . direction −i + j. (d) At the point (−1, − 2 in 36. At point (−1,thein direction −2), in direction i − 2j. (e) At 1), point (0, i + j . 37. At point (−1, 1), in direction −i + j . 2. (3 pts) Using the limit definition of the directional derivative, find Du f (1, 3) if In Exercises 38–45, use the contour diagram of f in Figf (x, y) = 2xy 2 , ure 14.34 to find the approximate direction of the gradient vector at the given point. 3. (4 pts) Let f (x, y) = 14-4w39 38. (−2, 0) 39. (0,14-4w36 −2) x u= −1 1 √ ,√ . 2 2 . Find the directional derivative of f at (1, 2) in the direction of 3, 5 . 2 2 40. x + y (2,14-4w37 0) 41. (0, 2) he directional 4. 14-4w43 Find14-4w40equation of the tangent plane of the following functions at the given points: (4 pts) the 14-4w42 42. (−2, 2) 43. (−2, −2) 44. (2,14-4w41 45. (2, −2) 2) n of v . the gradient. = (3, 4) and tional deriva- 1 from P = 3j . Give five (a) (2 pts) (b) (2 pts) f (x, y) = xexy at the point (2, 0, 2) g(x, y) = 1 + x ln(xy − 5) at the point (2, 3, 1) 5. (4 pts) Find the differential of the following functions. v 1 + uv −L2 −M 2 (b) (2 pts) P (K, L, M ) decimal places in your answer. = KM e (a) (2 pts) f (u, v) = (b) Use P and Q to approximate the directional deriva√ 6. (5 pts) (x, y) directional derivative of v (x, y) at (2, 1) in the direction going from (2, 1) toward the point (1, 3) tive of f The = x + y in the direction of f . √ (c) is −2/ 5, and the for the directional derivative (2, 1) in the direction going from (2, 1) toward the point (5, 5) is 1. Give the exact value directional derivative at Compute fx (2,part and fy (2, 1). you estimated in 1) (b). 7. (3 pts) 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. ...
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