Exercises

1.

Consider each function listed below on the interval [0,2]. Divide the interval into 2, 4, 8, 16, $\ldots$ equal subintervals and approximate the area under the curve by using each of the left endpoint rule, right endpoint rule and midpoint rule. Use exactly the same procedure on each of the four functions so that you can make comparisons. In this light, how are you going to interpret the directions just given? Which of the three rules gives the best results? Does the answer depend on the number of rectangles used? At the end, summarize your experience in working this exercise.

(a)

f(x) = x(x - 1)(x - 2).

(b)

g(x) = x²(x - 1)(x - 2).

(c)

$h(x) = x(x - 1)(x - 2) + \sin(128\pi x)$.

(d)

$k(x) = x^2(x - 1)(x - 2) + \sin(128\pi x)$.

2.

The following questions refer to the g(x) given in Exercise 1.

(a)

If M₆₄ is used to approximate $$\int^2_0 g(x)dx$$ what accuracy is guaranteed by the error bound formula? State what value of K you chose and explain how you picked it.

(b)

What is a reasonable n to use if 5-place accuracy is desired? That is, if