Exercises

1. Consider the function on the interval [-1,3]. Use the command leftsum to approximate the definite integral

   

   to two decimal places. Then explain why the leftsum and rightsum commands give the same numerical values for the same number of subintervals. (Hint - use the leftbox and rightbox commands to see what is going on.)

2. Suppose that the velocity of a particle moving in one dimension is given by

   

   Find the average value of v(t) on the following time intervals.

   a. [0,5]

   b. [0,10]

   c. [5,10]

3. Let the continuous function v(t) represent the velocity of a particle moving in one dimension. Let a and b be real numbers in the domain of v and let c be (a+b)/2. That is, c is exactly halfway between a and b. Show that the following equation is always satisfied

   

   where is the average value of v on the interval [a,b], is the average value of v on the interval [a,c], and is the average value of v on the interval [c.b]. Note - this is not a Maple problem. This problem is a thinking problem. I would suggest leaving some space on your worksheet and writing your answer in by hand. Part of using Maple is learning when not to use it.