MATH1020 Calculus II	Maple Lab 2
Section 1-4	Due: Tuesday, September 14, 1999.

This lab explores some techniques of integration.

1.

 Consider the function y = f(x) = x³ e^x2.

(a)

 Use Maple to find the antiderivative of this function. What method of integration is Maple probably using?

(b)

Notice that Maple does not add the constant of integration to the answer. Define a function y=F(x) such that F'(x)=f(x) and F(0)=1.

(c)

Plot f(x) and F(x) on the same axes, using an appropriate range. Label (by hand) which function is which. Explain briefly using calculus how you know (that is, do not simply plot the two functions separately).

(d)

Find the values of both x and f(x) where f(x)=F(x). Find these values as accurately as possible.

(e)

(Extra credit (2 points).) Show by hand how Maple arrives at the answer in part 1a.

2.

Consider the function

$$g(x) = \frac{2x^7-5x^5+3x^2+2}{x^5+6x^4+13x^3+12x^2+4x}.$$

(a)

 Find the partial fraction decomposition of g(x). The command you need is

convert (g, parfrac, x);

Where do the various terms come from? In other words, if you were to find this partial fractions decomposition by hand, how would you come by each term? Be brief. (You do not need to actually carry out the decomposition.)

(b)

Integrate the partial fractions decomposition from part 2a. Now, integrate the original function, g(x), and show that the two answers agree. Call the antiderivative given by Maple G(x).

(c)

Plot g(x) and G(x) on the same axes; it is advisable to limit the vertical axis, since g has discontinuities, and an appropriate range is x=-10..10, v=-1000..1000.

(d)

What is the domain of the antiderivative, G(x), as given and plotted by Maple? What is the domain of the actual antiderivative of g(x)?

(e)

Where are the discontinuities of g? For each of the discontinuities, what are the left and right limits of g(x)? You should determine whether they are at $+\infty$, $-\infty$, or at a finite number.