Calculus 1  Hour Exam 3  (continued) -   Math 220     Fall, 2011

 

 

 

 

 

 

6.

 

Fill in the missing information for the following theorem.

 

Rolle’s Theorem:  Let  f  be a function that satisfies the following three hypotheses:

 

1.        F  is continuous on the closed interval [a,b]

2.        F is differentiable on the open interval (a,b)

3.        _____________________________________

 

Then there is a number  c  in (a,b) such that  ___________________________.

 

Answer:  3.  f(a) = f(b)        such that  f’(x) = 0

 

 

 

 

 

 

 

 

 

7.

 

 

 

 

 

 

 

If Newton’s Method is used to approximate a solution to the equation  f(x) = 0, then

it generates a sequence of approximations  x1, x2, x3, . . . Which one of the following

relations correctly shows how xn can be used to determine the next approximation

xn+1 ?

 

a.        xn+1 = [xn + f’(xn)] / f(xn)           b. xn+1 = xn + f’(xn) / f(xn)

 

      c.  xn+1 = [xn + f(xn)] / f’(xn)             d. xn+1 = xn + f(xn) / f’(xn)

 

      e.  xn+1 = [xn - f’(xn)]/ f(xn)               f.  xn+1 = xn – f’(xn) / f(xn)

 

      g.  xn+1 = [xn - f(xn) / f’(xn)               h.  xn+1 = xn - f(xn) / f’(xn)

 

Answer:  h

 

 

 

 

 

 

8.

 

 

Fill in the missing information to show that the given definite integral can be expressed

as the limit of a Riemann sum.  The only variables appearing in your limit should be

n and k.  You do not need to evaluate this limit.

 

  6                                      n

  ∫ (x5 + 8)4 dx  =   lim       Σ   [                                                 ]

 2                         n→∞   k=1

 

Answer:         (  ( ( 2 + k (4/n) )5 + 8 ) (4/n)

 

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