Sample Questions for Exam 2
Math 2374, Fall 2003
rogness@math.umn.edu

This collection of sample questions is intended to be much harder than the actual exam. If you understand these problems (which is different than just being able to do them after looking up formulas in the book) you stand a good chance of doing very well on the test.

Note: Because of slight differences in the cutoff point for the exams, there aren't any sample questions for Section 5.4 here. You can find them on "Sample Midterm 3" on the Homework/Help page.

Questions for Section 3.5

1. Find the linear approximation to the given function at the indicated point.

(a) , a=(2,3).
(b) , b=(3,1)

2.  Use your approximations to estimate the value of , and .  Compare these to the true values of the functions at these points.  How good are the approximations?

3.  Find the total derivative for and at the points a and b.  (Note that you should have already done most of the work for this in #1, i.e. finding the Jacobians.)

4. Find a function whose Jacobian matrix is

Questions for Section 3.6

5.  Use the chain rule to find the derivative of at the indicated point.

(a) , , a=(3,2).
(b) g the same as above, , a=(3,1,-1).

6.  Find the partial derivatives of u with respect to s and t for the following functions:

(a) , , .
(b) .

Questions for Section 4.1

This is a difficult section because (1) many students had trouble with it the first time around and (2) there are lots of different types of test questions.  Trying to prepare for each of them would probably be to hard; you're better off learning the definitions of the gradient and the directional derivative and really understanding the relationship between the two. (pp. 214-216)  Also, you should know that gradients are perpendicular to level sets. (pp. 217-218)

7.  Let and , so F=grad(g) wherever g is defined (i.e. wherever ).

(a) At the point (2,1), what is the direction of greatest increase (or "steepest slope") for the function g?

(b) What is the directional derivative in that direction?

(c) Suppose the positive y-axis represents north, and the positive x-axis represents east.  What is a direction vector u in the southeast direction?  (Remember, a direction vector is a unit vector, i.e. it has a length of 1.)  What is the directional derivative of g in the direction of u at the point (2,1)?  If g represents elevation, is this uphill or downhill?

(d) If g represents elevation, in what direction should one go from the point (2,1) to stay at the same height (i.e. neither uphill or downhill).

8.  Find the equation for the line tangent to the curve at the point (2,-1).  (Hint: move everything to the left hand side, and your curve will be a level set of the form f(x,y)=0.  Then remember that gradients are perpendicular to level sets!)

9.  Find the point on the hyperbolic paraboloid at which the tangent plane is parallel to the plane .  (The most common mistake on this problem will be to give a point which is not on the hyperbolic paraboloid.  You should try to avoid this mistake!)

Questions for Section 5.2

10.  Find where and C is parametrized by , .

11.  Evaluate the following integrals:

(a) The line integral of F over C where F is the vector field from problem #7 and C is the unit circle , where .

(b) where C is the unit circle again.

Questions for Section 5.3

12.  Find the area of the region bounded by the parabolas and .

13.  Reverse the order of integration for the following integral:
        

14.  Evaluate .  (You should first reverse the order of integration!)

Questions for Section 6.1

15.  Suppose we are interested in the integral of the vector field over the following curve, which starts at the origin and ends at the point (0,0,10).