# Exercises

1. 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 .
1. 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 .
2. 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.)
3. 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.
2. Consider the function

on the interval .
1. 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 .

2. 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 .