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Calculus I                                                          Quiz #6                            October 22, 2004

Name____________________                   R.  Hammack                              Score ______
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(1) Find derivatives of the following functions.
  
(a)  f(x) = 3x^4 + π x^2 - x + 4
       f ' (x) = 12x^3 + 2π x - 1


(b)   f(x) = x^2/(x + 1)

         f ' (x) = (2x(x + 1) - x^2(1))/(x + 1)^2 = (x^2 + 2x)/(x + 1)^2


(c)  f(x) = 3^(1/2) + 1/x^3

        f(x) = 3^(1/2) + x^(-3)

        f ' (x) = 0 - 3x^(-4) = -3/x^4
        

(2)   If y = x^(1/2), find (d^2y)/dx^2.

        y = x^(1/2)
        
      dy/dx = 1/2x^(-1/2)
      
      (d^2y)/dx^2 = -1/4x^(-3/2) = -1/(4x^(1/2)^3)

(3)  Suppose functions f and g and their derivatives obey the following table. x f(x) f ' (x) g(x) g ' (x) 0 0 1 ... 2 1 2 2 -2 3 2 1 -2 -1 5
If    h(x) = 3f(x) + f(x) g(x),    find h ' (2).

h ' (x) = 3f ' (x) + f ' (x) g(x) + f(x) g ' (x)  h ' (2) = 3f ' (2) + f ' (2) g(2) + ... p;           = 3 (-2) + (-2) (-1) + (1) (5) = 1

(3)   Find the values of x  at which the slope of the tangent to the graph of y = 2x^3 - 3x^2 + 4  is horizontal.
y ' = 6x^2 - 6x
Finding the values of x for which the tangent line is horizontal (slope 0) amounts to solving
0 = 6x^2 - 6x 0 = 6x(x - 1)
Answer: x = 0 and x = 1