Download Test - FloridaMAO

Multivariable Calculus
FAMAT State Convention 2009
The answer choice E. NOTA denotes that “None of These Answers” are correct. DNE stands for “Does Not Exist”. The
domain and range of functions are assumed to be either the real numbers or the appropriate subset of the real numbers.

1. Evaluate the following limit: lim lim
x 1 h 0

A) 2
B) 
1
4y
2. If x3  y 2  z  6 xyz  1 , what is
A)
3x 2  6 yz  2 y
1  6 xy
B)
2 x 2 ln( y  h)  2 x 2 ln( y) 

h

2
C)
D) 4 ln y
y
E) NOTA
z z
 ?
x y
3x 2  2 xz
1  6 xy
C)
3x 2  6 yz  6 xz
1  6 xz
D) 1
E) NOTA
4 f
3. Consider the function f ( x, y, z ) and u ( x, y, z )  a b c , where a  b  c  4 (a,b,c are all whole
x y z
numbers). What is the maximum number of distinct values of u (1, 2,3) ?
A) 64
B) 20
C) 32
D) 15
E) NOTA
For questions 4 and 5, consider the volume V bounded in the 3-dimensional coordinate system by the
graphs x 2  y 2  9, x  z  5 , and z  0 .
4. Find the z coordinate of the center of gravity, given that the volume bound is V  45 units 3 .
A)
113
45
B)
119
45
C)
13
5
D) 3
E) NOTA

5. Find the radius of gyration k z with respect to the z-axis  I z 
A)
2
2
B)
3
2
C)
3 2
2


D)
3
 ( x
2
V

2
 y 2 )dV   k z  V  .

E) NOTA
6. Find the distance between the point (1, 2, 0) and the tangent plane to the
surface 2 x 2  4 y 2  z 2  9 at the point (0, 2, 5) .
A) 0
B)
17 89
89
C)
7 89
89
D) 5 87
E) NOTA
3
1
For questions 7 and 8, consider the production function P( K , L)  2 K 4 L 4 , where P is production, L is
labor, and K is capital.
7. If K remains constant, find C such that PL ( K , C)  2  PL ( K , 2)  , where PL ( K , L) is the partial
derivative of the production function with respect to L.
1
2
1
A) 3
B) 3
C) e
D)
e
2
2
E) NOTA
8. Find the marginal increase in production with respect to capital when K = 16 and L = 81.
A)
6
4
B)
5 6
8
C)
3 6
2
D)
3
E) NOTA
Multivariable Calculus
FAMAT State Convention 2009
9. The electric field vector at a distance d away from a charged particle has magnitude
of E  k 
q
directed away from positive charges (where k is a constant, q is magnitude of
d2
the charge). Find the magnitude of the vertical component of the electric field at a point 2
meters above the center of a charged solid disk that is 2 meters in radius given E   dE .
Take the charge density of the disk to be   0.1 C
A)

30
k
B)

40
k
C)
 2
20
k
D)
2
400
m2
k
.
E) NOTA
For questions 10 and 11, consider f ( x, y)  x 2 y 3  4 y .
10. Find the directional derivative of f ( x, y ) at the point (2, 1) in the direction of the vector 2,1 .
A) 1
B) 
16 5
5
C)
16 5
5
D) 0
E) NOTA
11. If the directional derivative of f ( x, y ) at the point (2, 1) in the direction of the unit vector a, b is a
minimum, find
a
.
b
A) 0
B) 
1
2
C) 2
D) 1
E) NOTA
12. Using the linearization method of approximation at the point (2,1,8) , estimate f (2.05,.95) for the
function z  f ( x, y)  x 2  3xy  2 y 2 with x  .05, y  .05 .
A) 7.75
B) 7.90
C) 8.05
D) 7.80
E) NOTA
13. Given that f  F and f  2 xy  y 2 z 2  2, x 2  2 z  2 xyz 2  3, 2 y  2 xy 2 z  z ,
find F (1,1,1) if F (0,1, 0)  4 .
A)
9
2
B)
1
2
C) 4
D) 3
E) NOTA
14. Let A be the number of saddle points, B the number of local minimums, and C the number of local
maximums for the function f ( x, y)  x 4  y 4  4 xy  1 . Find A  B  C .
A) 2
B) 1
C) 1
D) 0
E) NOTA
15. Find the equation of the best fit line for the points A(2,1), B(0,3), C (1, 4) . The best fit
line Ŷ  0  1 X is determined by the values of  0 , 1 that minimize
3
 Y   
i 1
i
 1 X i   .
2
0
Multivariable Calculus
15
A) Yˆ   X 
4
FAMAT State Convention 2009
B) Yˆ   X  3 C) Yˆ   X 
7
2
D) Yˆ   X 
11
3
E) NOTA
y (1)  0
16. Consider the third order differential equation y   xy  y  y and the initial conditions y ( 1)  1 .
y(1)  1
Using Euler’s method with x  0.5 , find y(0) .
A) 1.5
B) 2.5
C) 3.5
D) 2
E) NOTA
17. Given that f ( x, y )  x 2  y 2 , which of the following critical points occurs at ( x, y )  (0, 0) ?
A) Local minimum
C) Local maximum
E) NOTA
B) Impossible to determine
D) Saddle point
42 x
18. Which of the following integrals represents a reversal of the order of integration of

2
4 y
2 x 4
A)

 xdxdy
x
2
B)
  xdxdy
2
2 y
4
42 x
C)

2
xdydx ?
x
2 x 2
ydydx
x
D)
  xdxdy
x
E) NOTA
4
19. The density of a certain Martian sphere 1 meter in radius is given by the
r kg
where r is the distance to the center of the sphere. Find the mass of
m3
2
the sphere (in kilograms).

4
3
A)
B) 
C)
D)
E) NOTA
3
32
2
equation  (r ) 
20. Given that the surface area of a rectangular box with no lid is 48 in 2 , find the maximum volume of
the box (answers are in in3 ).
A) 16 2
B) 20
C) 32
D) 24
E) NOTA
21. Use differentials to estimate the volume of steel used to construct a uniform cylindrical steel container
which can hold a column of water 3 inches in radius and 10 inches in height. The thickness of the
steel is
1
inch (all answers are in in3 ).
2
A) 46
B) 39
C) 42
D)
143
4
E) NOTA
Multivariable Calculus
FAMAT State Convention 2009
xy
.
( x , y ) (0,0) x  y 2
B) 0
C) 
22. Evaluate the following limit
A) Does not exist
lim
2
E) NOTA
1 2
4 2 32
. Find the length of the arc of the function
t  2t ,
t
2
3
23. Consider the vector function r (t ) 
from t  0 to t  4 .
A) 16
B) 12
D) Cannot be determined
C) 10
D) 8
E) NOTA
 x1  x2  1
24. Find the maximum value of the sum x1  2 x2 subject to the constraints: x2  2
and x1 , x2  0 .
x1  x2  4
A) 4
B) 6
C) 8
D) 10
E) NOTA
25. Find the curvature of the parametric vector function r (t )  3sin t ,3cos t .
A)
3
3
B)
1
2
26. Evaluate the integral
C) 1
D) 3
E) NOTA
 xyds , where C is the upper right quarter of the ellipse defined by the
C
equation
A) 3
x2 y 2

 1.
16 9
B) 2+ 3
C)
148
7
D)
2  3
7
E) NOTA
For questions 27-29, consider the vector field defined by F ( x, y, z )  ( xyz )iˆ   x 2 z 3  ˆj   y 2 z  kˆ .
27. Find curl F at the point ( x, y, z )  (1, 2,3) .
A) 15iˆ  2 ˆj  57kˆ
C) 39iˆ  2 ˆj  57kˆ
B) 15iˆ  2 ˆj  51kˆ
D) 39iˆ  2 ˆj  57kˆ
28. Find div F at the point ( x, y, z )  (1, 2,3) .
A) 12
B) 6
C) 10
29. Find div curl F at the point ( x, y, z )  (1, 2,3) .
A) 0
B) Undefined
C) 12
E) NOTA
D) 2
E) NOTA
D) 1
E) NOTA
30. Evaluate the discriminant of the curve f ( x, y)  x 2 y  xy 3 at the point (1, 2) .
A) 52
B) 148
C) 100
D) 0
E) NOTA