微积分Midterm1.pdf

Name: Math 2263 Midterm 1 Show all work for each problem so that partial credit may be given. Answers without supporting work will receive no credit. Clearly indicate your answers. Make sure that you have 6 problems on your test. A scien- tific calculator may be used. 1 1. The height of a hill is given by the function . (a) Locate all critical points for this function, and classify each critical point as a local maximum, local minimum, or saddle point. (b) Consider the graph . What is the name of this quadric surface? (Hint: Complete the square to rewrite the equation for the surface.) (c) A raindrop falls at the point and rolls downhill in the direc- tion of steepest descent. Give a unit vector u describing this direc- tion. 2 2. Determine whether the function is continuous at . if if 3 3. Consider the planes and . (a) Find the angle between these planes. (b) Find symmetric equations for the line where these planes inter- sect. 4 4. Suppose we have a function , where and Given that and find when .