1. Let f(x) = 1/(1+x) for x   -1. What is the nth derivative, f(n)(x)?

A. n!(1+x)n+1

B. -n!/(1+x)n+1

C. n!/(1+x)n+1

D. -n!/(1+x)n

E. (-1)nn!/(1+x)n+1

solution

 

2. If f(x) = (x e3x)/(1 + x) for x   1, then f(1) = ?

A. 3e3/4

B. 5e3/4

C. 7e3/4

D. 9e3/4

E. 4e3

solution

 

3. If u = (x2+9)1/2 and v = 3x2-2x, then what is du/dv as a function of x?

A. 1/(4(3x-1)(x2+9)1/2) for x   1/3

B. (3x-1)/(2(x2+9)1/2)

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

D. x/(2(3x-1)(x2+9)1/2) for x   1/3

solution


4. The volume (in gallons) of water in a tank after t hours is given by f(t) = 600 sin2(Pi*t/12) for 0 ú t ú 6. What is the rate of flow of water into the tank, in gallons per hour?

A.100(Pi)sin(Pi*t/12)

B.100(Pi)cos(Pi*t/12)sin(Pi*t/12)

C.1200cos(Pi*t/12)sin(Pi*t/12)

D.50(Pi)cos2(Pi*t/12)

E.600cos2(Pi*t/12)
solution


5. What is the slope of the line tangent to the curve y3 - x2y + 6 = 0 at the point (1,-2)?

A. -(2/5)

B. -(4/11)

C. 4/11

D. 11/4

E. 8

solution


6. If f(x) = xex/sin(x) for 0 < x < p , then f(x) =

A. ex/cosx

B. ex(x+1)/cosx

C.ex[sinx+xcosx/sin2x]

D.ex[x(sinx+cosx)+sinx]/sin2x

E.ex[x(sinx-cosx)+sinx]/sin2x

 
solution


7. What is the slope of the tangent line to the curve x3 + y3 - 3xy = 13 at the point (2,-1)?

A. -5

B. -4

C. -1/5

D. 4

E. 5

solution

 

8. Let f(x) = x2 ln x - (x3 + 3)/2x for x > 0. Then f(x) =

A. 2 - (3x2/2)

B. 2x ln x + x - (3x2/2)

C. 2x ln x + (3/2x2)

D. 2x ln x + 2x - (3/2x2)

E. 2 - x + (3/2x2)

solution

 

9. Let f(x) = 2sin x. Then f"(0) =

A. -ln 2

B. -(ln 2)2

C. 0

D. (ln 2)2

E. ln 2

solution

 

10. Let f(x) = sin(2x + 1) and g(x) = x3 + 3 for all real x. Which of the following is equal to the derivative of the composite function f[g(x)]?

A. sin(2x3 + 7)

B. cos(2x3 + 7)

C. 6 sin(2x3 + 7)

D. 6 cos(2x3 + 7)

E. 6x2 cos(2x3 + 7)

solution

 

11. The equation y2 exy = 9 e-3 x2 defines y as a differentiable function of x. What is the value of dy/dx for x = -1, y =3?

A. -15/2

B. -9/2

C. -3/2

D. 6

E. 15

solution

 

12. A point moves along a number line so that its position at time t >= 0 is s(t) = 2t3 - 15t2 + 36t - 10. What is the position of the point when it first changes direction?

A. -63

B. 0

C. 17

D. 18

E. 98

solution

 

13. What is the derivative of f(x) = sin(xx) for x > 0?

A. xx(1 + ln x) cos(xx)

B. xx cos(xx)

C. xx ln x cos(xx)

D. cos(xx)

E. cos(xx ln x)

solution

 

14. What is the derivative of f(x) = 4x(x2 + 1)3 ?

A.4(x2 + 1)2(x2 + 3x + 1)

B. 4(x2 + 1)2(7x2 + 1)

C. 8x2(x2 + 1)3

D. 12x(x2 + 1)2

E. 24x2(x2 + 1)2

solution

 

15. Which of the following functions is continuous everywhere, but has at least one point where it is not differentiable?

A. tan x

B. |x|/x

C. sin x

D. e-x

E. x1/3

solution

 

16. What is the slope of the line tangent to the circle x2 + y2 = 45 at the point (6,-3)?

A. -6

B. -1/2

C. 1/6

D. 2

E. 11/2

solution