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• Question #1: The function given by:

  is differentiable at x=0. What is the value of f(n-m)?

  (a) 2+e

  (b) 3+e²

  (c) e²

  (d) 2e

  (e) e³

  Solution: Since f(x) is differentiable at x = 1, it must be continuous there, which means that

  

  

  f(0) = n+1

  n + 1 = 4, and n = 3

  Now we can determine m from the fact that f(x) is differentiable at x = 0.

  The right-hand branch slope is
  
  and f’(0) = 2.
  The left-hand branch slope is f’(0) = m.

  m = 2

  At this point we can calculate n – m = 1

  f(1) = 3+e²

• Question #2: If dy/dx = sin(x³), what is d²/dx²?
  (a) 3x²cos(x³)(b) -3x²cos(x³)

  (c) x²cos(3²)

  (d) -x²cos(3²)

  (e) cos(x³)

  Answer:
  u(x) = x³

  du/dx = 3x²

  y(u) = sin(u)

  dy/du = cos(u)

  dy/dx = (dy/du)(du/dx) = cos(x³)·3x²

  dy/dx = 3x²cos(x³)

• Question #3:

  (a) -1

  (b) 1/5

  (c) 1

  (d) -1/3

  (e) 1/3

  Answer:

  

• Question #4: What is the value of a, if:

(a) ¶

(b) 1

(c) 1 + ¶

(d) √2

(e) 1 + e

Answer: u = 2 + sin(ax)

du/dx = a·cos(ax)

ln(3/2) = (1/a)ln(3/2) and a = 1.

• Question #5: Which of the following are antiderivatives of f(x) = 2^x?(a) 2^x/ln(2) + ln(2)

  (b) 2^2x/ln(2) + ln(2)

  (c) x²/ln(2) + 1/ln(2)

  (d) x + 2ln(2)

  (e) 2^x + ln(2)

  Answer:

  The easiest way to solve this problem is to calculate the derivative of all 5 possible answers. We find that answer (a) is correct:

  d(2^x/ln(2) + ln(2))/dx = ln(2)2^x/ln(2) + 0 =2^x

 

 

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