Aggregate X aggregate X G aggregate X Q What shou X Course He X M (no subject X F23M81HV X M (no subject X + V X @ File | C:/Users/14848/Downloads/F23M81HW10.pdf LIA . . . F23M81HW10.pdf 1 / 2 100% + Complete all of the following problems to be turned in before the start of class on the date that it is due. Please follow all of the homework instructions in the syllabus. 1. Use implicit differentiation to find a for each of the following: (a) 1 = 2 y (b) 3x2 + 7xy - xy = 23 (c) vaty =1+x2yz (d) In(x ty) = ery 2. Use your answer from 1.(a) above to find the tangent line to this function at (1, 1). 3. Given f(x, y), find fx(x, y) and fy(x, y) for each of the following: (a) xy2 +xy + xy (b) In(xy) +ex (c) V4-12 -32 y 2 (d) 1+ 22yz (e) , 1 2+ 2 2 4. Given g(K, L) = K3L4 + 2K - 7L, find gk(K, L) and gL(K, L). 5. Given the demand function for good J given by J(p, k, l, Y, T) = 100 - 5p + 2k - 31 + 15YT, find all partial derivatives, Jp, JK, Ju, Jy, and Jr. Note that, in this example, p would be the price of the good J, k and I are prices of two related goods, Y is income, and T is other factors. 1 41OF Rain Q Search ZA CA 12:03 AM 12/11/2023Aggregate X aggregate X G aggregate X Q What shou X Course Hel X M (no subject X F23M81HV X M (no subject X + V X File | C:/Users/14848/Downloads/F23M81HW10.pdf . . . F23M81HW10.pdf 2 / 2 | - 100% + 6. Use Lagrange Multipliers to optimize the function f(x, y) = 4x3 + y under the constraints 2x2 + y2 = 1, x 2 0, and y 2 0. 7. Suppose you are selling widgets of two types. Type A widgets are more expensive, and Type 2 B widgets are cheaper. Demand of both widgets depends upon the price of both. Thus the profit function, in terms of selling a type A widgets and b type B widgets is P(a, b) = a(150 - 5a - b) + b(100 - 26 - a) - (20a + 106 + 100) Find the maximum profit using Lagrange Multipliers, given that you will sell exactly 50 widgets total. 8. Let Q = 4KLi, where K is the level of capital and L is the level of labor of the Oompa Loompas at Willy Wonka's Chocolate Factory. Suppose capital K costs $1 per unit and the 1 41OF Rain Q Search CA 12:03 AM 12/11/2023n Aggregat: X ' n aggregate x S aggregate X G Whatsbo X ' u Course H; X M (nosubjer X @ F23M81t X M (nosubjer X + V _ C71 X c Filelc:/Users/14848/Downloads/F23M81HW10.pdf |> r 3 .i. [I 0 F23 M81 HW10.pdf 7. Suppose you are selling widgets of two types. Type A widgets are more expensive, and Type B widgets are cheaper. Demand of both widgets depends upon the price of both. Thus the prot function, in terms of selling a type A widgets and I) type B widgets is PM, (I) : (1(150 7 5c 7 b) + M100 7 2b 7 a) 7 (20a +1017 + 100) Find the maximum prot using Lagrange Multipliers, given that you will sell exactly 50 widgets total. . Let Q : 4KiLS! where K is the level of capital and L is the level of labor of the Oompa Loompas at Willy Wonka's Chocolate Factory. Suppose capital K costs $1 per unit and the labor costs $3 per Oompa Looinpa's salary. but Willy \Nonka only has $64 in his account. Use the method of Lagrange multipliers to maximize the production level Q. What is that maximum production? Simplify as much as possible. . Suppose bicycle pedals cost $20 each and brake pads cost $5 each. If a is the number of pedals you buy and b is the number of brake pads you buy, and you must spend $1000. use the method of Lagrange Multipliers to determine the quantities which maximize and minimize the utility. U = 802 + ab + 112. and give the maximum and minimum values of U. 12:03 AM a A ' Gmi} 12/11/2023 Q , I .. Rain I. Q searCh