Sample Questions for Exam 1
Math 2374, Spring 2002
rogness@math.umn.edu

1. Find the equation for the following piece of a single cone.  Note that the vertex is at (0,0,200), and the bottom is on the circle in the xy-plane.

2.  (a) Find the equation for a plane containing the points (1,2,3), (2,4,9), and (-1,3,-4).

(b)  Find the parametric equation for the plane .

(c)  Find the cartesian equation for the plane .  

(d) Which, if any, of the following vectors is normal to the plane used in part (c)?
        (0,1,1), (2,2,2), (-3,-3,2), (0,1,0).

[Note: you should be able to answer this question either before or after finishing part (c), i.e. you should be able to do it using the information in either the cartesian or the parametric equations to the plane.]

3.  Suppose we have the following linear transformations:





(a) Find T(S()), if possible.  If it is not possible, explain why.

(b) Find S(T()), if possible.  If it is not possible, explain why.

4.  (a) using the limit defintiion of partial derivatives, find the value of (0,0) for the function

(b) Using any valid method, find a general formula for at any point away from

5.  (a) Plot the level curves for c=0,1,4,9 of the function

(b) Plot the level surfaces for c=1,4 of the function

6. Given a three dimensional vector , show that T()=(.) is a linear transformation.  (Hint: write the right hand side using matrix notation and use the fact that k=k for a scalar k.)  [This is problem 22 in section 2.3]

7. Parametrize the intersection of the cylinder and the plane .

8.  Give a parametrization for the line segment from the point (1,0,0) to the point (2,3,-4).  How would you change this to give a parametrization for the entire line containing these two points?

9.  Working in , consider the points A=(0,0,1,0), B=(1,0,2,0), and C=(3,2,0,-1).

(a) Find the vectors and .

(b) Find the angle between and .

(c) Find the area of the parallelogram spanned by and .  Warning: the cross product is only defined in three dimensions, so you cannot use the cross product to answer this question.  Draw a few good pictures and use a little trigonometry.  Remember the area of a triangle is (1/2)(base)(height), and the area of a parallelogram is (base)(height).