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- 1.
- Consider the function
*f*(*x*) =*x*on the interval [0,8]. Verify that the`middlesum`command gives the exact value for the area independent of the number of subintervals you specify. Can you explain this? Note that you can use the formula for the area of a triangle to compute that the area is 32. (Hint - look at the plot produced by the`middlebox`command.) - 2.
- Consider the function
*f*(*x*) = 2*x*-7^{3}*x*-9 on the interval [-1,3]. Use the command^{2}`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.) - 3.
- Consider the function
on the interval [-3,3].
- (a)
- If
*M*is used to approximate the area under the curve, what accuracy is guaranteed by the error bound formula? Make sure you state clearly how you obtained the value of_{10}*B*. - (b)
- Use the error bound formula to find the smallest value of
*n*that guarantees that*M*_{n}approximates the area to within 0.00001. That is, find the smallest value of*n*that guarantees that .

- 4.
- Suppose that
*g*(*x*) is a smooth function that is strictly positive and increasing on the interval [*a*,*b*]. Is it always true that, for any n, where*L*_{n}and*R*_{n}are the area approximations with*n*subintervals using the left endpoint rule and the right endpoint rule, respectively? Your answer should be yes. Explain why.

1/21/2000