Sample Questions for Exam 2
Math 2374, Spring 2002
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.

Comments on 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)

1.  Let g(x, y) = Arctan(y/x) and F(x, y) = (-y/(x^2 + y^2), x/(x^2 + y^2)), so F=grad(g) wherever g is defined (i.e. wherever x≠0).

(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).

2.  Find the equation for the line tangent to the curve x^3 + y^3 = (-7/2) xy 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!)

3.  Find the point on the hyperbolic paraboloid z - x^2 + y^2 = 0 at which the tangent plane is parallel to the plane -2x + 3y + z = 11.  (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!)

4.  Find ∫_Cu dL where u(x, y) = 16x - y^2 + 49 and C is parametrized by f(t) = (t^2 + 1, 4t + 7), 0≤t≤3.

5.  Evaluate the following integrals:

(a) The line integral of F over C where F is the vector field from problem #1 and C is the unit circle f(t) = (cos t, sin t), where 0≤t≤2π.

(b) ∫_C (x - y - x (x^2 + y^2)) dx + (x + y - y (x^2 + y^2)) dy where C is the unit circle again.


6.  Find the area of the region bounded by the parabolas y = x^2/4and y = 5 - x^2.

7.  Reverse the order of integration for the following integral:
        ∫_ (-1)^1∫_ (y - 3)^y^2f (x, y) dx dy

8.  Evaluate ∫_0^1∫_x^1e^y^2dy dx.  (You should first reverse the order of integration!)


9.  Suppose we are interested in the integral of the vector field F(x, y, z) = (yz, xz, xy) over the following curve, which starts at the origin and ends at the point (0,0,10).

[Graphics:HTMLFiles/index_19.gif]

Because the curve is messy, evaluating integrals over this curve by direct calculation would be hard.  We'll look for a better way:

(a) Verify that F is path independent (or conservative, which means the same thing).

(b) Find ∫_CF · dx by (i) replacing C with a nicer curve that starts at (0,0,0) and ends at (0,0,10), and then (ii) evaluating the integral directly.  Why can we replace C with a different curve?  Why do the endpoints of the new curve matter?

(c) Find ∫_CF · dx by (i) finding a potential function for F, i.e. a function g(x, y, z) such that F is the gradient of G, and (ii) evaluating g(0,0,10)-g(0,0,0).  Why is this a valid method for finding the integral?

[Note: if your answers for (b) and (c) didn't agree, go back and find your mistake.]

10.  Anytime a vector field is the gradient of a function g, it is path independent.  You should know that integrals of path independent vector fields over closed curves are zero.  (Explain why!  Hint: if F is path independent, you could find a potential function and do something similar to 9c.)  The vector field F in problem #1 was defined as the gradient of a function g, but in 5(a) you should have calculated that its integral over the unit circle is 2π.  Why isn't it zero?

11.  Use Green's Theorem to find ∮_C 3xy dx + (x - y^2) dy where C is the triangle with vertices (0,0), (2,0) and (0,1).

12.  Use Green's Theorem to find ∮_C (x^2 - y) dx + (x - y^2) dy where C is the boundary of the region bounded by the curves y = x^2 and y = 6x + 7.

13.  Evaluate the integral ∫∫∫_SfdV, where f(x, y, z) = xy + yz and S is the "unit tetrahedron" bounded by the coordinate planes and the plane x+y+z=1

14. Consider the integral ∫_ (-1)^0∫_0^1∫_1^2f(x, y, z) dy dz dx.  Change the order of integration to dx dy dz and dz dx dy.

15. Consider the integral ∫_0^1∫_ (1 - x^2)^(1/2)^1∫_0^1f(x, y, z) dz dy dx.  Change the order of integration to dx dy dz and dz dx dy.

16. Consider the integral ∫_CF · dx in two dimensions with F(x,y)=(F_1(x, y), F_2(x, y)).  (We could also write the integral as  ∫_CF · dx = ∫_CF_1dx + F_2dy.  Use whichever form  you feel more comfortable with.)

(a) What conditions on the curve C and/or the vector field F do you need to use the Fundamental Theorem of Calculus for Line Integrals to evaluate the integral?

(b) What conditions on the curve C and/or the vector field F do you need to use Green's Theorem to evaluate the integral?

Created by Mathematica  (March 23, 2004)