Math 267 Third Exam Spring 2014
1. Given f ( x, y )  16  4 x 2  y 2 . (Please sketch each separately.)


x2 y 2

 1 You can sketch this.
a. Find and sketch the domain of f. Answer:  x, y  :
4
16


b. Identify and sketch the graph of f. Answer: Top half of the hyperboloid of one sheet…
c. Describe and sketch the level curve f ( x, y )  2 . “Answer”: hyperbola on the xy-plane…
2. Evaluate each limit or else show that the limit does not exist.
tan  x 2  y 2 
xz
lim
a.
=1
b.
DNE
lim
2
2
2
(
x
,
y
,
z
)

(0,0,0)
( x , y ) (0,0)
x  2y 2  z 2
x y
3. Given f ( x, y )  x 2 y  6y 2  3 x 2 , P (1,2) . Find
a. the instantaneous rate of change of f at P in the direction of the x-axis
b. the instantaneous rate of change of f at P in the direction of the y-axis
c. the gradient of f at P
-2
-23
2, 23
d. the directional derivative of f at P in the direction of the vector u that makes an angle of 60 with
the positive x-axis (please give an exact answer, i.e. no decimal approximations)
e. the increment f if ( x, y ) varies from P (1,2) to Q(1.1, 1.9)
f. the differential df if ( x, y ) varies from P (1,2) to Q(1.1, 1.9)
g. all local minima, local maxima, and saddle points of f
f  0,0   0 LMAX ,  6,3, f  6,3   SP,  6,3, f  6,3   SP
2  23 3
2
2.009
2.1
4. Find w r if w  x cos y  y sin x, x  rs 2 , y  r  s .
Answer: cos  r  s    r  s  cos  rs 2   s 2  rs 2 sin  r  s   sin  rs 2 
5. Find equations for the tangent plane and normal line to the surface x 2 y 3 z 4  xyz  2 at P0 (2,1, 1) .
TP: 3 x  10y  14z  30
NL: x  2  3t, y  1  10t , z  1  14t
6. Use partial derivatives to find dy dx if x e y  sin  xy   y  ln2  0 .
Answer: 
e y  y cos  xy 
xe y  x cos  xy   1
7. (a) Find a formula for the approximate error in the quotient x y of two numbers, due to small errors in the
x
y
4.02
(b) Use the result in (a) to evaluate
.
0.9
numbers. Hint: Find df for f  x, y  
Hint: Use nice values of x and y!
8. The two legs of a right triangle, a and b, are increasing at rates of 0.6 and 0.4 cm per second, respectively.
At what rate is the area of the triangle changing when a and b are 12 and 7 cm ? Answer: 4.5 sq. cm/sec
9. The temperature of a region in space is given by T ( x, y , z )  x 2 yz 3 . Find the maximum rate of increase in
temperature at (2,1, 1) and find a unit vector in that direction.
Answers: 4 11;

1
11
,
1
11
,
3
11
10. An open rectangular box containing 18 cubic inches is to be constructed so that the base material costs 3
cents per square inch, the front face costs 2 cents per square inch, and the sides and back each costs 1 cent
per square inch. Use the Lagrange multipliers technique to find the dimensions of the box for which the cost of
construction will be a minimum. Answer: 2 by 3 by 3