1. If u = ex/y, x = 2r - s, and y = r + 2s for r + 2s non-zero, then, in terms of x and y, the partial derivative of u with respect to r is,

 

A. ((2-x)/y)ex/y

B. 3ex/y

C. (2/y)ex/y

D. ((2y-x)/y2)ex/y

E. ((2y-x)/y)ex/y

solution

2. If u = xy z for x, y, and z > 0, then

A. 3u

B. u(1 + z + z2/y)

C. u(1 +y ln z +z ln y)

D. u(1 + z2/y + y ln z)

E. u(1 + z + z ln y)

solution

 

3. Let u(x,y,z) = 1/(x2 + y2 + z2)1/2 for x, y, z not all zero. Then

A. 0

B. 1

C. -(x+y+z)/(x2+y2+z2)1/2

D. (9/2)(x2 + y2 + z2)-5/2

E. (x2 + y2 + z2)-2

solution

 

4. Let u(x,y) = (a2x - by2)1/2 for a2x - by2 > 0. Which of the following is equal to

A.

B.

C.

D.

E.

solution

 

5. Let f(x,y) = (x2/y) + (y2/x) for x > 0 and y > 0. Then

A. (2y/x2) - (2x/y2)

B. (2x2/y3) + (2/x)

C. (2x/y2) - (2y/x2)

D. (2/y) + (2y2/x3)

E. -(2x/y2)-(2y/x2)

solution

 

6. Let f(x,y) = xey/(xy + 3) for xy > 0. What is

A. ey/y

B. -3ey/(xy + 3)2

C. 3ey/(xy + 3)2

D.(2xyey + 3ey)/(xy + 3)2

E. (ey - xyey)/(xy + 3)

solution

 

7.

A. -xy2cos(xy)-2ysin(xy)

B. -x2ysin(xy)-2ysin(xy)

C. -x2ysin(xy)-2xcos(xy)

D. -x2ysin(xy)

E. -xy2cos(xy)

solution

 

8. Let f(x,y) = 3y2 ln(x3 + 4) + 2y/x for x > 0 and y any real number. What value of y minimizes the partial derivative of f with respect to x, when x = 2?

A. -1

B. -1/3

C. 0

D. 1/12

E. 1

solution