]> Haese & Harris - Chapter 20 Notes

20: Introduction to Calculus

Calculus is often called "the mathematics of change." It is only fitting, then, that we begin our tour of the Calculus with the topic of change…

A rate of change (often reduced to just "rate") compares the amounts of two quantities—usually, the amount of change in one quantity compared with a base (unit) change in another quantity. A classic example is speed (kilometers per hour; the change in distance for a change of one time unit).

However, there are two very distinct ways to think about change…

A: Average Rate of Change

You are already familiar with average speed (and thus, average rate of change).

Equation 1 - Average Speed

$average speed = \frac{distance traveled}{time passed} = \frac{change in distance}{change in time} = \frac{final distance - initial distance}{final time - initial time}$

You should also be aware that this process is exactly how we find the slope of a line!

Equation 2 - Slope

$slope = \frac{rise}{run} = \frac{change in y}{change in x} = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$

Slope is the rate of change in y (with respect to x). Average speed is the rate of change in distance (with respect to time).

Those of you who are very observant should now feel a prickly sensation on the back of your neck. If slope and speed are exactly the same, then why aren't those sentences the same?

To answer that question, here's another: must speed be constant? Of course not!

Now, the trick question: must slope be constant?

No!

To use an old saying, you have been led by the nose…you have always talked about slope for lines, and in that case, slope is constant and never changing. There are, however, many more interesting graphs than those of lines…and we can still talk about slope for those graphs! It's just not constant…

Perhaps we should call it average slope…but that might have been confusing, at the time. Better to tell a little white lie then, and fill you in with the Devastating Truth now!

In summary: to find the average rate of change for any two quantities, you just need two values for each quantity. It does not matter whether or not the quantities have linear relationships!

Equation 3 - Average Rate of Change

$avg. rate of change of a (with respect to b) = \frac{change in a}{change in b} = \frac{a_{2} - a_{1}}{b_{2} - b_{1}}$

B: Instantaneous Rate of Change

When the rate is variable, the average rate of change depends heavily on where you take the measurements. Consider the following graph:

An Image

The average rate of change between points A and B is negative (the curve falls more than it rises), and the average rate of change between points A and C is positive (the curve rises more than it falls). However, neither really tells us anything interesting. The problem is that a lot of things can happen to this curve between these points—the points are too far apart to be useful for measurement. If you measure the speed of a roller coaster at the very beginning of the ride, and again at the end of the ride, that average doesn't really mean very much!

The solution is to move the two points close together.

How close, you ask? Arbitrarily close.

What does that mean, you ask?

There is now an answer to that last question—but I must defer for a moment. You see, that very question nearly derailed Calculus in its infancy. For a while, a few great minds simply said it doesn't matter—assume that there is an answer, and all our techniques work!

This was, of course, unacceptable…except for the fact that they were right!

Back to the issue at hand: the average rate of change is not useful if it is measured over an interval where lots of variation in change is possible; thus, we squeeze the interval down to an infinitesimal size (don't ask yet!) to produce an instantaneous rate of change. Since this is over a very small interval, it's like a slope at a single point (the two points are so close together that you can't hardly tell them apart).

Consider the function $f (x) = x^{2}$ around the point $(2, 4)$ —let's consider the average rate of change (average slope?) of that function as the two points converge on $(2, 4)$ . Here is a table of points, and the slopes between them and our target point. We'll watch this graphically in class.

x	y	slope
3	9	5
2.5	6.25	4.5
2.1	4.41	4.1
2.01	4.0401	4.01
2.001	4.004001	4.001
2.0001	4.0004	4.0001

As the points converge on $(2, 4)$ , the slope between the points appears to be converging on 4!

Even though we have not made everything clear with nice definitions, we can see that as the points close in on one another, the average slope appears to be settling down to a single value, which we'll call the instantaneous rate of change, or the slope of the tangent line.

Here's an image of $f (x) = x^{2}$ and a line with slope 4 passing through $(2, 4)$ :

An Image

It looks like this method works pretty well! Let's try it algebraically…the key is to think about the second point in a logical way. Specifically, condition everything on x (since that's the independent variable). Thus, the second point will have an x-coordinate of $2 + h$ , where h is a really small number.

Now, the y-coordinates: one is already known; the other must be found. Fortunately, we chose a simple function! $f (2 + h) = {(2 + h)}^{2} = 4 + 4 h + h^{2}$ .

So, the slope between the two points is $\frac{(4 + 4 h + h^{2}) - 4}{(2 + h) - 2} = \frac{4 h + h^{2}}{h} = \frac{h (4 + h)}{h} = 4 + h$ .

Remember that h is a really small number—the slope, then, will be ever so slightly different from 4. In fact, if you imagine for a moment that h is zero (that we have just one point, rather than two), then you get the slope of the tangent as 4.

Of course, you can't actually let h be zero, because then the slope fraction has zero in the denominator, and the whole thing falls apart…but more on that in the next chapter.

Page last validated 2010-08-15