QUESTION 1 Let f(x, y) = V9 -x2 -92. a). Calculate its first partial derivatives fx and fy. What is the domain of the function fx? (b). Calculate its second partial derivatives fex, fyy, fry, and fyx- QUESTION 2 (a). For which real number a is there a two-variable function f(z, y) such that fx = 3x2 + ary + y? and fy = 2xy + 212? (Hint: use Clairaut's theorem about fry and fur.) (b). (Optional; extra credit, 5 points.) For the value of a you found above, find a function f(x, y) such that fx = 3x2 + ary + y and fy = 2xy + 2x2. QUESTION 3 Let z be defined implicitly as a function of a and y by the equation y Find 2z/Or and Oz/dy at the point (2, 2, 2). QUESTION 4 Let f(x, y) = cos(22 -4y?). Find an equation for the tangent plane to the graph of f at the point (-2, 1, 1). QUESTION 5 Let f(x, y) = V4x2 + yz. (a). Explain why f is differentiable at the point where (r, y) = (2,3). (b). Find a linear approximation for f(r, y) near (2, 3), and use it to approximate the number v4(1.95)2 + 3.052. QUESTION 6 Find a point on the surface z = tan(y) + In(x) where the tangent plane is parallel to the plane r + 2y -2z =0. (Hint: the surface z = tan(y) + In(x) is the graph of the function f(x,y) = tan(y) + In(x).)QUESTION 7 A can has the shape of a cylinder, with base radius r and height h. Its volume is given by V (a, b, h) = arch. (a). Find the differential dV in terms of dr and dh. (b). A cylindrical can is measured to have height h = 7 centimeters and radius r = 4 cm. The error for each of these measurements is at most 0.1 cm. Use differentials to estimate the maximal error in measuring the volume of the can. QUESTION 8 Suppose there is a surface S whose equation we don't know, but we want to find an equation for the tangent plane to S at the point (0, 1, 1). We do know that the curves with vector equations ri(t) = and r2(s) = lie on the surface S. (a). Find equations for the tangent lines to ri(t) and to r2(s) at the point (0, 1, 1). (b). Find an equation for the tangent plane to the surface S at the point (0, 1, 1). (Hint: the tangent plane contains both of the lines that you found in part (a).) QUESTION 9 Let f (x, y) = 208(1) (a). Find the domain of f. (b). Find fx and fy. Where is f differentiable? (c). Find a linear approximation to f(x, y) near the point where (r, y) = (1, w/4). QUESTION 10 Let f(x, y) = [felt dt. (a). Find f, and fy. (Hint: use the fundamental theorem of calculus.) (b). Find an equation for the tangent plane to the graph of f at the point (2, 2, 0)