Calculus 1  Hour Exam 3  (continued)   -   Math 221     Fall, 2010

 

 

 

 

 

 

 

 

5.

 

 

 

Let  L4  be the Riemann sum for  f(x)  =  ex  on  [0,2] corresponding to a regular

partition of  [0,2] into four subintervals with the sample points being the left

endpoints.   Write out   L4  as an explicit sum.  You cannot simplify the answer.

 

                                                               Answer:  L4  =  [ e0 + e1/2 + e + e3/2 ] (1/2)

 

                   2

Is   L4   or   ∫ ex  dx    smaller and  why ?

                  0

                                                                                                                   2

                                     Answer:   Since  ex  is increasing on [0,2],  L4  <  ∫ ex  dx   

                                                                                                                  0

 

Let  h(x)  =  5  for  2 ≤  x  ≤  13 .   It is true that all Riemann sums for  h  have the

same value.  What is this common value?  You need not justify your answer.

 

                                  Answer:  55