Math for Econ II Homework 1 - Math for Econ II Written...

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Math for Econ II, Written Assignment 1 (28 points) Due Friday, September 12, in recitation Jankowski, Fall 2014 Please write neat solutions for the problems below. Show all your work. If you only write the answer with no work, you will not be given any credit. Write your name and recitation section number (listed on the front page of our NYU Classes site). Staple your homework if you have multiple pages! 1. (5 pts) Let f ( x, y ) = 1 2 ln( x 2 + y 2 ). (a) (2 pts) What is the domain of f ? Very briefly justify your answer. (b) (3 pts) Suppose ( a, b ) is in the domain of f . Show that f xx ( a, b ) + f yy ( a, b ) = 0. 2. (3 pts) Find the domain of the function f ( x, y ) = p 2 x 4 y - 10. 3. (2 pts) Suppose f is a differentiable function whose cross-section for y = 4 is f ( x, 4) = 4 x 2 - x . Find f x (1 , 4). 4. (4 pts) Let f ( x, y ) = 3 y x . (a) (1 pt) Show that the level curve f ( x, y ) = c is given by the curve y = log 3 ( cx ). (assume c > 0) (b) (3 pts) Using the property log 3 ( ab ) = log 3 ( a ) + log 3 ( b ), show the level curves for c = 3 and c = 9 are y = 1 + log 3 ( x ) and y = 2 + log 3 ( x ), respectively. Sketch each of these curves on the same graph. 5. (4 pts) Recall that a Cobb-Douglas production function has the form P = cL α K β with c, α, β > 0 . Economists talk about

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