Exercises

1.

Use the Maple int command to work the following problems.

(a)

Find the average value of $f(x) = x \sin(x^2)$ on $[0,\pi]$.

(b)

Find the general antiderivative of $h(x) = x^3 \cos(x)$.

2.

Let $I = \int^5_1(x^3 +6x^2 - 10x + 5)dx$

(a)

Approximate I by each of T_n, S_n and M_n for n = 2,4,10.

(b)

Use the int command to evaluate I.

(c)

Discuss whether the approximation methods of (a) seem to yield good approximations. Which of the three gives the best results? Why does this method give the best result for this particular problem? (Hint:Think about its error bound statement.).

3.

Use Simpson's rule, the trapezoidal rule and the midpoint rule to approximate the given integrals to four decimal places. For each method give the number of subintervals needed and justify the number you give. Be sure to actually give the four place approximation. Which method seems to give the best approximation?

(a)

$\int^{10}_0 \sqrt{1 + x^3}dx$

(b)

$\int^\pi_0 x \sin(x)dx$

4.

Use Simpson's Rule to approximate $I = \int^1_{-1} 15x^2(x^2 - 1)dx$. Use $n = 2,4,8, \ldots$. Consider I-S_n for $n = 2,4,8, \ldots$. What is the first value of n for which the error is the maximum possible, e.g. for which the error is actually as large as permitted by the error bound statement for Simpson's rule? Based on your investigation, make a guess as to the value of I - S_n, expressed as a function of n.