Page 47 - Calc for the Clueless

Well, that's it. With a little practice you'll be like a pro! We'll do more examples, of course. You should practice
the ones in this book.

Before we do other examples, let us look at the right end of the curve that we drew above.

In the top figure, can we find out how high M is? Yes! Later in this section we will.

In the middle, could this be the right end? Perhaps, but we don't have enough info to know what the end looks
like yet. In some very complicated cases, we might not ever be able to determine what the end is like (exactly).

In the bottom, could this be the right end? No!! There would have to be another intercept after (2,0), and we
know there aren't any.

Example 20—

x intercepts: (8,0), (4,0), and (0,0), which is also the y intercept. Vertical asymptotes: x = 2, x = -3, x =-6.
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Horizontal asymptote: By inspection, the leading term on the top is 3x and the leading term on the bottom is
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2x . Same degree. The asymptote is y = 3/2, the leading coefficient on top over leading coefficient on bottom.
Starting the sketch at (8,0), f(8 ) is positive. The power of (x - 8) is even, so there is no crossing. The sketch
+
starts...

From the sketch, f(4 ) is positive. Since (x - 4) has odd power, there is a crossing at (4,0) heading down to
+
minus infinity at x = 2. The power of (x - 2) is even. So one end at minus infinity means both ends are there,
2
heading up to (0,0). x is an even power, so that there is no crossing, and the sketch continues...
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