Calculus 1  Hour Exam 3   -   Math 221     Fall, 2010

 

 

 

 

 

 

 

1.

 

 

 

A farmer wants to construct two identical adjoining rectangular pens each with are of

50 square feet.  What should be the length and width of the entire pen be so that the

least amount of fence is used?  Show your reasoning.

                  

 

Answer:    20 / √6   by   5 √6

 

 

 

 

 

 

 

2.

 

The function,  f(x)  is given by     y  =  x / ( x2  +  12 )2  

 

It has absolute extrema at  x  =  ± 2  and inflection points at  x =  0,  ±√12.    

 

                                      √12

Evaluate the integral     ∫  [ x / ( x2  +  12 )2 ]  dx             Answer:  1/48

                                     0

                                              

                                      √12

Evaluate the integral     ∫  [ x / ( x2  +  12 )2 ]  dx             Answer:  0

                                 -  √12

                                              

 

 

 

3.

 

 

Evaluate the integral     ∫  [ ( 1 – t ) / (1 + t2 ) ]  dt            

                   

                                                                         Answer:   tan-1 (t)  - (1/2) ln ( 1 + t2 )  +  C                

 

Evaluate the integral     ∫  [  sin (3π/x) / x2 ]  dt            Answer:     cos (3π/x) / 3π/x   +  C

 

 

 

 

 

 

 

 

4.

 

 

                     x

Let   g(x)  =  ∫  [ ( t2 - 1 ) / (1 + t2 ) ]  dt           On what interval(s) is  g: 

                    0

 

a.                               Increasing?                       Answer:  ( - ∞, -1 )    and  ( 1 , + ∞ )

b.                              Concave downward?       Answer:   ( - ∞, 0 )

 

                     cos(x)

Let   H(x)  =     ∫  [ ( √ (1 + u4 ) ]  du           Find H’(x).  Show your reasoning.

                        0

 

Answer:   H’(x)  =  ( -sin x) √( 1 + cos4 x )

 

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