Exercises

1. Use the Maple int command to work the following problems. Explain your ``setup" for each problem.
   1. Find the area under the graph of on the interval .
   2. Find the average value of the function on the interval [-5,-1].
   3. Find the area between and on the interval .
   4. Find the general antiderivative of

      

2. Let
   1. Approximate I by for n = 20, 60 and 100.
      Approximate I by for n = 20, 60 and 100.
      Approximate I by for n = 20, 60 and 100.
   2. In estimating error, the values M, K and B are often approximated by a number N that is an upper bound of M or K or B. Find such upper bounds of M, K and B for the integrand of I. Note that the Maple command

        > h:=diff(g(x),x,x,x,x);
      

      defines h to be the 4th derivative of a function g. You may also want to make use of the plot command.

   3. If n = 60, what accuracy in estimating I is guaranteed by the trapezoidal error formula? By the Simpson's error formula? By the midpoint error formula? If values from the int command are assumed to be correct, what accuracy was in fact achieved in each case?
   4. Write a paragraph summarizing your experience in this problem. (Feel free to run additional lines of Maple if you need to.) What approximation method seems to be the most accurate? Look at the error formulas for midpoint and trapezoidal approximations and decide, on the basis of these error statements, which method is likely to be more accurate for a given n. How much more accurate is the better method? Include at least one more insight you have gained from this exercise.
3. If is used to approximate the I of Exercise 2, then - according to the error theory - what is a reasonable n to use if 5-place accuracy is desired? That is, if

   

(Note that this exercise is similar to Exercise 3 of Lab 1.)