QUESTION 1 Let f(r, y) = V212 - y2. (a). Find the directional derivative of f at the point (-3,3) in the direction of the vector . (b). In which direction does f increase most quickly at the point (-3,3)? What is this maximal rate of increase? QUESTION 2 Is there a point on the sphere a2 + y' + 22 = 9 at which the tangent plane is parallel to the plane 2x + 2y - z = 10? If so, find it. QUESTION 3 Suppose that the temperature at a point (z, y, 2) is given by T(x, y, z) = 80 + coS(TI) - 12 - 22. (a). Find the rate of change of the temperature at the point (1, -2, 1) in the direc- tion of the vector . (b). At the point (1, -2, 1), in which direction does the temperature increase most quickly? What is that maximum rate of change of temperature? QUESTION 4 Find equations for the tangent plane and normal line to the surface a' + y' = 2 sin(z) at the point (1, -1, 3x/2). QUESTION 5 Consider the cone 22 = x2 -y' and the sphere r2 +y' + 22 = 18. They intersect at the point (-3,3,0), among others. Find the angle (or the cosine of the angle) between these surfaces at the point (-3,3, 0). Hint: the angle between two surfaces is the same as the angle between their normal lines