Mathematics

Problem Set 6

Due Date: November 16 This problem set is graded and worth 5% of your final

grade Problem 1: For each F(x, y) = 0, (i) find dy

dx (ii) find the points for which dy

dx is not

defined:

(a) 2y + 3x = 0

(b) 2x2 + y2 = 0

(c) x2y + ey + xy = 0

(d) ln y + ex + 5xy2 = 0.

Problem 2: For each of the given equations F(x, y) = 0, is an implicit function y = f(x) defined around the point (x, y) = (0, 2). (a) x5 − 2xey−2 + 3y ln(x + 1) − y2 = −4 (b) 2x3 + 4xy − y4 + 16 = 0 If your answer is affirmative, find dy/dx and evaluate it at the said point. Problem 3: Let F(x, y) = 4x2 + 9y2. Draw the level set (or curve) at F(x, y) = 16. Find all points (x, y) on the said level set where lines tangent to the level set at (x, y) have slope -0.5. Problem 4: A consumer consumes x units of good 1 and y units of good 2 and his prefer- ences are represented by the utility function U(x, y) = x0.25y0.75. Suppose the consumer 16 units of good 1 and 81 units of good 2 which is optimal.

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(a) What is the consumer’s current level of utility?

(b) Suppose that the consumer’s income increased slightly and she wants to increase his utility the most. What should be the ratio of the changes of consumption goods.

(c) Suppose that the consumer now wants to increase the consumption of good 1 by one unit. Use calculus to estimate the corresponding change in good 2 consumption that would keep her utility at its current level.

Problem 5: One solution of the system

y1 + 2e y2−1 + 3 ln y3 + x1x2 = 3

y1y2y3 + x1x2 = 2

y1y3 + 5y2y3 − 5y2 + 8y3 + ln x1 + 5 ln x2 = 9

is y1 = 1, y2 = 1, y3 = 1, x1 = 1 and x2 = 1. Take y1, y2 and y3 as functions of x1 and x2. Calculate the partial derivatives of y1, y2 and y3 at the said solutions. Use calculus to estimate the corresponding y1, y2, and y3 when x1 = 1.1 and x2 = 0.9.

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