- Suppose that the the velocity of an object traveling in one
dimension is given by
for
. Let
be the position of the object at time , assuming that .
- As explained in your text, the position of the object at is equal to the area under the velocity curve from to . Use the midpoint rule with subdivision to approximate the position of the object at .
- Let be the right endpoint rule and the left endpoint
rule approximations with subdivisions to the position of the
object at . Explain why the following relation holds for any
value of .

(Hint - look at the results of`leftbox`and`rightbox`commands.) - Evaluate and . Based on your results, do you think your approximation using the midpoint rule with subintervals was within of the exact value for the area? Explain why or why not.

- Consider the function

on the interval .- Use the error bound formula to find the smallest value of that
guarantees that approximates the area to within . That
is, find the smallest value of that guarantees that
.
- The value of given by the error bound is usually
conservative. That is, in practice the desired accuracy can be
achieved with a smaller value of . Given that

find the smallest value of such that .

- Use the error bound formula to find the smallest value of that
guarantees that approximates the area to within . That
is, find the smallest value of that guarantees that
.

2001-03-19