(Some) Answers to Sample Test 2
Math 2374, Fall 2003
rogness@math.umn.edu

By popular demand, here are some answers to the questions on the practice test.  In some cases I've indicated where you can find the answer; "5.3. #42," for example, means this is exercise #42 in section 5.3, and the answer is in the back of the book.  Not all of the answers are here.  One warning: if you look at the answers before working out the problems, the sample test will be worthless.

Remember that you can find sample problems for section 5.4 on the third sample midterm; problem #1 on that sample exam is example 5.4.4, and #2 is problem 25 in section 5.4, so the answer is in the back of the book.

#1.

(a) Lf(x,y)=13+4(x-2)+6(y-3)
(b) Same idea, although the Jacobian is now 2x2 instead of 1x2, so your answer will be a vector valued function instead of a plane as in (a).

#2. Do Lf(2.1,3) and Lg(2.9,0.99) and compare to f(2.1,3) and g(2.9,0.99).

#3.  Multiply the Jobian by a coordinate column vector, such as [x y] (in a column).

#4. You can check your answer by finding the Jacobian of your answer and see if it matches.

#5 (a) 3.6 #23

#6 (b) 3.6 #15

#7.  (a) The gradient points in the direction of the greatest increase, so the direction we're interested in is the gradient of g at (2,1), i.e. .  So one way to state the answer is to say, "in the direction of the vector (-1/5, 2/5)."  (Another way is to give a unit vector in the direction of (-1/5, 2/5).

(b) The easiest way to find a directional derivative is to take the dot product of the unit vector in that direction with the gradient.  But because of how the formula works out, the directional derivative in the direction of the steepest increase -- i.e. in the direction of the gradient -- is simply the length of the gradient itself.  In this case,

.

(c)  Southeast is in the direction of the vector (1,-1).  A unit vector in this direction is:

or ).

The directional derivative in this direction at the point (2,1) is the dot product of with the gradient of g at (2,1), i.e. F(2,1):

.

(d) In the direction of (2,1) or -(2,1).

#9.  If I recall correctly, I believe the answer is (1,3/2,5/4), but you should check this.

#10. 5.2 #3.  The answer is approximately -1201.7.  This question was harder than I intended. so you should probably set up the integral and then ask your calculator or Mathematica to numerically evaluate the integral.  If you get -1201.7, you (apparently) set up the integral correctly.

#11. (a) and (b) are both .

#12. Example 5.3.7

#13. 5.3 #42

#14. Example 5.3.8

#15. (a) show that the Jacobian is symmetric

(b) and (c) are both zero; one possible potential function is .  This is found in Example 6.1.1.

#16.  Similar to 6.2 #1

#17.  6.2 #7