MATH 201 Exam 2, Fall 2010

Math 201 - Exam 2 (Fall 10) T. Tlas ================================================ • Please answer question 4 on the same sheet of paper on which it is written. Questions 2 and 3 have an extra sheet and question 1 has two extra sheets for you to write your answer on them. Any part of your answer written on the wrong page will not be graded. • When finished leave your work on your desk for it to be collected by the proctors. • There are 4 problems in total. Some questions have several parts to them. Make sure that you attempt them all. • This is a closed book exam and no calculators are allowed. ================================================ Name : ID # : Section : ================================================ Q1 Q2 Q3 Q4 TOTAL Problem 1 Consider the function of two independent variables f(x, y) = (x2 + y2)2 − 2x y i- (10 points) What is the domain of the function? Is it bounded? Open? Closed? Give brief justifications to your answers. ii- (10 points) Sketch the level curve of this function that passes through (0, 0). iii- (10 points) Find all the critical points of this function. iv- (10 points) Find the plane tangent to the graph of this function for (x, y) = (1, 1). Hint: You might find polar coordinates useful in part (ii). ================================================ ADDITIONAL SHEET FOR PROBLEM 1 ANSWER ADDITIONAL SHEET FOR PROBLEM 1 ANSWER Problem 2 (10 points each) Investigate which of the functions below possesses a limit as (x, y) → (0, 0). If the function has a limit find it, if not prove that it doesn’t exist. i- f(x, y) = (x y)2 + (x y)4 x2 + y2 + 1 ii- g(x, y) = x y y − x2 − x3 iii- h(x, y) = x y y − x − x3 ================================================ ADDITIONAL SHEET FOR PROBLEM 2 ANSWER Problem 3 (20 points) Which point(s) on the graph of the function f(x, y) = xy + 2 is nearest to the origin? In other words find the point(s) on the graph of this function whose distance to the origin is minimum. Hint: Distance is minimized if and only if the square of the distance is minimized. ================================================ ADDITIONAL SHEET FOR PROBLEM 3 ANSWER Problem 4 (10 points) Assume f(x, y) is a function whose range is [0, ∞], and that for every c, the level curve corresponding to c is a square centered at (0, 0) whose sides are parallel to the axes and have length equal to c. Find all the points where this function takes its maximum and minimum values on the circle x2 + y2 = 1, as well as the actual values of the maximum and the minimum on this circle. ================================================