1. Let f(x,y,z) = exyz + ln(1 + x2 + y2 + z2) where x,y,z are real numbers. What is the direction of maximum increase of f at the point (1,1,0)?

A.

B.

C.

D.

E.

solution

 

2. Let f(x,y) = (x2 + y2)-1/2 for (x,y) not equal to zero. What is the directional derivative of f at the point (x,y) in the direction toward the origin?

A. 1

B. (1/2)[(x+y)/(x2+y2)3/4]

C. 1/(x2+y2)3/2

D. (1/2)[(x+y)/(x2+y2)]

E. 1/(x2+y2)

solution

 

3. What is the directional derivative of f(x,y) = 4x2y4 - 2x + 5 at the point (2,1) in the direction <-3,4>?

A. -136/5

B. 107/[2(1073)1/2]

C. 160/7

D. 214/5

E. 214

solution

 

4. What is the directional derivative of f(x,y) = 5 - 4x2 - 3y at (x,y) toward (0,0)?

A. -8x - 3

B. (-8x2-3y)/(x2+y2)1/2

C. (-8x-3)/(64x2+9)1/2

D. 8x2 + 3y

E. (8x2+3y)/(x2+y2)1/2

solution