Calculus 3 (Math-UA-123) Fall 2016 Homework 4 Due: Thursday, October 6 at the start of class Please give complete, well-written solutions to the following exercises. 1. Match each set of level curves with the appropriate graph of function. Briefly explain your choices. 1. 2. 3. 4. 5. 6. 1 Calculus 3 (Math-UA-123) Fall 2016 2. For each of the following functions: (i) find the function's domain, (ii) find the function's range, and (iii) sketch several of its level curves. (How many level curves? At least three for each function; generally, sketch however many is sufficient to get a sense of the shape of the graph of f ). (Not to be graded: use a computer algebra system, such as Wolfram Alpha, to sketch the surface z = f (x, y). Does this surface match the level curves that you sketched?) (a) f (x, y) = 2yx x+y+1 (b) f (x, y) = 3 |x| 4|y| p (c) f (x, y) = x2 y 2 16 3. (a) Use the squeeze theorem to evaluate \u0012 lim tan(x) sin (x,y)(0,0) (b) State whether function f (x, y) = tan(x) sin plain why or why not. 1 |x| + |y| \u0010 1 |x|+|y| \u0011 \u0013 . is continuous at (0, 0). Ex- 4. Consider the function f (x, y) = xa y b x2 +y 2 (x, y) 6= (0, 0) 0 (x, y) = (0, 0) where a, b are nonnegative integers. For each of the following values of a and b, determine if the function f is continuous at (0, 0). (a) a = 1, b = 5 (b) a = 0, b = 1 (Not to be graded: For what values of a and b is the function f continuous at (0, 0)?) 2