4)
3)
2)
NAME__________________________________________________________________________________________________
MAYNARD
1)
MATH 277
Sketch the level curves for y f(x, y)
TEST 3 FUNCTIONS OF SEVERAL VARIABLES SAMPLE
 x
2 x
 y
2
, for c
 
1/2,

1,

3/2,

2 x
Discuss the continuity of the function and evaluate the limit of f(x, y) (if it exists) as (x, y)

(0, 0) along the specified paths. f(x, y) = xy Path 1: y = 0;
(x 2

y 2 ) Path 2: y = x
Find the four second partial derivatives of z = x 2

2xy

3y 2
Electrical power P is given by P = E 2 /R, where E is voltage and R is resistance.
Approximate the maximum percentage error in calculating power if 200 volts is applied to a 4000-ohm resistor and the possible percentage errors in measuring E and R are
2% and 3%, respectively.

5)
6)
7)
8)
Find
 w/
 s and
 w/
 t using the Chain Rule when w = x 2 + y 2 and x = s + t , y = s − t.
Differentiate implicitly to find the first partial derivatives of z when e xz

xy = 0 .
Let

be the angle between equal sides of an isosceles triangle and let x be the length of these sides. If x is increasing at 1/2 meter per hour and

is increasing at

/90 radian per hour, find the rate of increase of the area when x = 6 and

=

/4.
Find the directional derivative of h(x, y, z) = ln(x

y

z) at the point P(1, 0, 0) in the direction of Q(4, 3, 1)

11)
12)
9)
10) g(x, y) = ln 3
 x 2

y 2 and the maximum value of the directional derivative at the point (1, 2).
Use the gradient to find a unit normal vector to the graph of 4x 2

y = 6 at the point (2, 10) .
Find the path followed by a heat-seeking particle placed at point P(10, 10) on a metal plate with a temperature field given by
T(x, y) = 400

2x 2

y 2
Find an equation of the tangent plane to the surface given by xy 2

3x

z 2 = 4 at the point (2, 1,

2).

16)
15)
14)
13) Find an equation of the tangent plane and find symmetric equations of the normal line to the surface given by z = arctan (y/x) at the point (1, 1,

/4).
Find symmetric equations of the tangent line to the curve of intersection of the surfaces given by x 2 + z 2 = 25 and y 2 + z 2 = 25 at the point (3, 3, 4).
Examine f(x,y) = −5x 2 + 4xy − y 2 + 16x + 10 for relative extrema and saddle points.
Examine h(x,y) = x 2 − 3xy − y 2 for relative extrema and saddle points.

20)
19)
18)
17) Determine whether there is a relative maximum, a relative minimum, a saddle point, or insufficient information to determine the nature of the function f(x, y) at the critical point (a, b). f xx
(a, b) = 9, f yy
(a, b) = 4, f xy
(a, b) = 6
Use Lagrange multipliers to maximize f(x, y) = 2x + 2xy + y , x, y > 0 , subject to the constraint 2x + y = 100.
Use Lagrange multipliers to minimize f(x, y, z) = x 2 + y 2 + z 2 , x, y, z > 0 , subject to the constraint x + y + z − 6 = 0.
Use Lagrange multipliers to find the dimensions of a rectangular box of maximum volume that can be inscribed (with edges parallel to the coordinate axes) in the ellipsoid
x 2 + y 2 + z 2 = 1
a 2 b 2 c 2

ANSWERS
1)
2) Path 1: limit = 0
Path 2: limit = 1/2
The function is not continuous
at (0, 0) since the limit does not
exist at that point and the function
is undefined at that point.
 2
3)
 x z
2


2 y 2 z


2
6

 x
2 z
 y


 2 z y
 x
 
2
4) 7%
5)
 w/
 s = 4s and
 w/
 t = 4t
 z
6)
 x
 z
 y

 
 e ze
 xz xz xe
 xz y
7)
8)

2 (15
10
7

19
19
 
) m 2 /h
9)
3 x

2
2
 y
2

 x i
 y j

, 2
15
5
10)
257 
16 i
 j

257
11) y 2 = 10x
12) x + y + z = 1
13) x − y + 2z = π/2
14) x − 1 = y − 1 = z − π/4
1 −1 2 x − 3 = y − 3 = z − 4
4 4 −3
15) Relative maximum at (8, 16, 74)
16) Saddle point at (0, 0, 0)
17) Insufficient information
18) f(25, 50) = 2600
19) f(2, 2, 2) = 12
20)
3

3

3