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|Section 1 | Section 2 | Section 3 |||Section 4 | Section 5 |||Section 6 | Section 7 | Section 8 | Section 9 | Section 10 | Section 11 | Section 12 |||Section 13 | Section 14 | Section 15 | Section 16 | Section 17 | Section 18 |||Section 19 |||Section 20 | Section 21 | Section 22 |||Section 23 |||Answers|
In Calculus, one of the first topics we discuss is the rate of change of a function.
What is a rate of change? Rate of change of a function is the same as the slope of a function. Recall that slope is defined as:
In section 8 you graphed both linear and quadratic functions. Let's look at these two functions to see how their rates of change are different.
First let's consider a linear function. Think about the function y = 3x-1.
If you select two points on this graph, such as (2,5) and (7,20) what is the rate of change or slope between these two points? What if you pick two other points?
Did you notice in a linear function the slope or rate of change of y was always the same when the x-values changed by the same quantity. In the linear example, y=3x-1, the y values increased by 3 every time the x-values were increased by 1. The change in y in comparison with the change in x is called slope.
When a slope is equal to 3, as in our example, , when . Why does this happen with the equation y=3x-1?
The follow two windows show us a graph (zoom 4 Decimal window) and a set of points associated with the line y=3x-1.
Recall from geometry that two points are always collinear, but when a third point is collinear with two others, the slope between any two points will be the same. It does not matter which two points you select on the line, the ratio
is always the same or in this case it always equals 3. In the equation y=3x-1, the x-values are always multiplied by 3 and then 1 is subtracted from this product. Because the x-values are always multiplied by 3 (and 1 is subtracted), the y-values will always be separated by 3 when there is a change is 1 in the x-value. If x changed by 2, then the jump in the y-values would be 6, or 2x3. All the 1 does is vertically shift the y-values down by 1.
So in linear functions, the slope or rate of change of y with respect to x is always the same. Since we are used to writing straight lines in slope intercept form (y=mx+b), the slope will always be the m value in the form. This is because x is always multiplied by the m.
When a function is not linear, such as our quadratic example, the rate of change of y changes as the x-values were changed by the same quantity. (quadratic example) In other words, the rate of change of y changes depending upon the x values used. This means that the quantity
changes for different values of x used.
Did you notice in the quadratic example that the x-values always changed by 1 and the y-values changed by different amounts?
When we study examples which are not linear, one type of rate of change we might consider is the average rate of change of the function. This is done by selecting two ordered pairs and on the function and determining
This would be the average rate of change for the function between these two points.
|Using the quadratic data (1,2), (4,3), and (7,6) from Section 20|
|We can find this by typing and then press ENTER. So if the x-values were time and the y-values were position, the slope would represent the average velocity from time 1 to time 4.|
|Now change this equation to calculator the average rate of change of y for the second two ordered pairs. Press 2nd ENTER to recall the last line. Then cursor to the numbers that need to changed. Re-press ENTER to calculate a new value.|
What do these two average rates of change tell
you? First notice that the change in time is always 3, 1 to 4 and 4
to 7. Second, you notice they are not equal. And third, you
notice that between time 1 and 4 the average rate of change was 1/3.
From time 4 to 7 the average rate of change increased to 1, The
slope seems to be getting steeper as we move to larger x values. We
calculated these two average rates of change using the three points which
we started with.
If we now switch to using the equation
which passes through these three points we can do some further study. Let's study the average rate of change for two points near (4,3) to see how the function is changing in the vicinity of (4,3)? Type a similar equation to the previous one but this time use Y1 instead of L2 and two values of x near 4. Here is an example:
( You can get the symbols Y1 by pressing Vars, (to Y-Vars), 1. Function, and select 1. Y1. )
This is the average rate of change of Y1 between x = 4 and x = 4.1.
Try calculating the average rate of change between (4,3) and the following points to the right of 4 for the function
|x-value||y-value||Average rate of change of y|
Notice that the average rates of change are different between (4,3) and the points near it. This means that the slope is changing all the time. But as you get closer to (4,2) from the right the slopes appear to be getting smaller an approaching a number around .6xxxx.
Check Your Understanding
1. Try calculating the average rate of change of y for x-values to the left of (4,3) for the function .
|x-value||y-value||average rate of change of y|
2.What observations can you make about the average rates of change of y as you move toward (4,3) from the right on the graph ? Toward the point (4,3) from the left? What does this tell you about the behavior of the function? Check your statements out by viewing a graph of quadratic function.
3. As you study the average rates of change in the last two tables, what happens to the average rates of change of y with respect to x as the second point gets closer and closer to 4, either from the left of the right? Do the average rates of change of y with respect to x seem to be approaching one particular value? What do you think we could call this number or ratio?
4. Now let's compare that to what happens to the rate of change in the linear example. What happens to the rate of change of y with respect to x as you approach the point (3,8)? Set up tables as you did with the quadratic function. Are you surprised with the results?
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© Rahn, 2000