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
In and Recitation grad f from the differential.
ﬁned. have multiple pages. 3 5
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
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
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
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 .
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
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.
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)
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)
14-4w42 42. (−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√
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
(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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