Problem 1. (a) The function z is dened implicitly as a function in a; and y by the equation m4+y4+z4+myz:1. Find 23.; and 2?}. (b) The function z is defined implicitly as a function in a: and y by the equation 8" = atyz. Find 2,1,; and 2y. Problem 2. Let x, 3;) = .133 "(my + e\" be a function of two variables. (a) Explain that f is differentiable at (1,0). (b) Find the equation of the tangent plane at (1,0, 0). (c) Find the first order approximation of f at (1,0). Problem 3. (a) Let P(:c,y) be a point in 2D. Its coordinates are 3 and 4 with an error of at most 0.01. What is the error with which the polar coordinates r and 6' are known? (b) The sides of a rectangular box have been measured to be a = 1, b = 2, and c = 3. To which of these measurements is the volume most sensitive? So, if all a, b, and c are known with an error of 0.01, which one creates the largest error for the volume? Problem 4. (a) Suppose z = f (3:,y) has continuous partial derivatives. Let a: = rcos(6) and y = rsin(6). We want to compute the partial deriva tives of f($(r, 6), y(r, 9)) with respect to r and 6. Explain why we have 8f (9f 5f 87" cos(l9)a 33+ sin(6 )a and, further, 8f _ 1. 3f 8f 89 rs1n() 8m + rcos(6)8y. Finally, show that (2%)2+$(Z)2=(3i) \"3b 5) (b) Let ay): 111(922 +y2) Use (a) to compute (32f 2+()2. (e) Let g(m,y) 2 cm. Use (a) to compute (99) + (99) . 81 6y