Midterm Exam 1.(10pts) Let P(1, 2, 3) and Q(4, -7, 9) be two points. Find the point R that is two-thirds of the way from P to Q on the segment PQ. Work: Answer: 2.(10pts) The angle between two intersecting planes is defined to be the angle between their normal vectors. Find the angle between the planes x - 2y + z = 0 and 2x + 3y - 2z = 0. Find the parametric equations of the line of intersection of the two planes above. Work: Answers: 3.(10pts) Find two different unit vectors that are perpendicular to both of the following vectors: 2,-1, 4 and -3, 2, 5 . Work: Answers: 4.(10pts) Find the distance from the point P(3, -1, 4) and the line whose parametric equations are: x = -2 + 3t, y = -2t, and z = 1+ 4t. Work: Answer: 5.(10pts) Find the curvature of the curve curve is curvature maximum? Work: Answers: . At what point on that 6.(10pts) An object is in motion according to the conditions: . Find the velocity and position functions for this motion. What is the maximum height (z value) that this object reaches? Work: Answers: x2 7.(10pts) Let f (x, y) = 2 2 . The domain for this function is all points (x, y) except (0, x +y 0), and it is continuous for all (x, y) in its domain. Can f(x, y) be made continuous at (0, 0) by defining f(0, 0) to be a specific value? Explain! Draw the level curves for this function for function values: 0, 0.5 & 1. Work: Answers: z z with z defined implicitly as a function of x and y and x y in the equation 3x 2 z - x 2 y 2 + 2z3 + 3yz = 5 8.(10pts) Find the partials Work: Answers: 9.(10pts) Suppose that a mountain climber is at the point (500, 300, 3390) on the surface of a mountain described by the function h(x, y) = 4000 - 0.001x 2 - 0.004y2 . In what direction would the climber have the steepest up slope? What is that slope? Work: Answers: 10.(10pts) Find the equation of the plane tangent to the surface z = x 3 - 3xy + y3 at the point P(1, 2, 3) on the surface. Work