In a moment, we will look at a simple example involving the function f(x) = 100 x

Transcription

1 Rates of Change Calculus is the study of the way that functions change. There are two types of rates of change: 1. Average rate of change. Instantaneous rate of change In a moment, we will look at a simple example involving the function f(x) = 100 x, graphed below. Graph of f(x) = 100 x Before we think about the example, let s try to understand what rate of change means. Keep in mind that the independent variable x, which is measured along the horizontal axis, controls the values for the dependent variable y, which is measured along the vertical axis. In a sense, x controls the shape of the graph, and changing x causes the height (y-values) of the curve to change. We would like to understand exactly how changing x affects the y-values of the curve. For example, we could move from x =.5 to x = 1.5 and ask how this affects the height of the curve: 1

2 Change in x causes a change in y. In the following example, we will answer some specific questions about the rate of change of f(x). Example. On a bus ride to New York, you get a bit bored and decide to write a function that describes your distance traveled as a function of time. You come up with the equation f(x) = 100 x, whose graph is shown below: Graph of f(x) = 100 x From the graph, we see that D(1) 70; in other words, you traveled about 70 miles in the first hour. Again inspecting the graph, we can see that you traveled the remaining 30 miles in the second hour. As mentioned above, we want to think about the way that this function changes. Since it took hours to drive the entire 100 miles, you could describe the change by saying that your distance changed by 50 miles per hour. However, this is a bit misleading we already know that the function changed by about 70 miles (not 50!) in the first hour, and only 30 in the second hour. To be more precise about what we mean, we should say that the average rate of change of the function f(x) from x = 0 to x = is 50 (miles per hour).

3 As a side note, think for a moment about the line joining the endpoints of our curve: Secant line joining (0, f(0)) to (, f()). Notice that the slope of this line is exactly 50 in other words, computing the average rate of change of the function from x = 0 to x = gave us the same answer as if we had computed the slope of the secant line joining the points (0, f(0)) and (, f()). This is an important relationship to keep in mind. As mentioned before, we have computed the average rate of change of f(x) over a two hour time period, but your speed certainly was not 50 miles per hour at each point of the trip. In fact, you traveled much faster the first half of the trip than you did the second half. This leads to a natural question: what was the rate of change of our function f(x) at a specific point in time, say at x = 1? We call this the instantaneous rate of change of f(x) at x = 1. 3

4 This question is quite difficult, and answering it will occupy a good bit of our time in this class. However, it turns out that there is a similarity here to our previous discussion of average rate of change: the instantaneous rate of change of f(x) at x = 1 is precisely the slope of a tangent line to the curve at x = 1: Slope of the tangent line at x = 1 is the instantaneous rate of change of f at x = 1. So answering the question comes down to finding the slope of the line above. But we only know one point on this line, the point (1, f(1)). To compute its slope, we must know more information about the line. It turns out the we actually do not have enough information to compute this line s slope. However, we can try to approximate its slope using information we already know. Compare the original tangent line to the secant line that passes through the points (1, f(1)) and (, f()): 4

5 It appears that the slopes are fairly close together, so if we can find the slope of the secant line (which is easy to compute), then we can use this number as an approximation for the slope of the tangent line. Since slope is rise run, the slope of the secant line joining the points (1, f(1)) and (, f()) is f() f(1) = = So the instantaneous rate of change of f(x) at x = 1 is about 9 miles per hour. Let s keep track of the data that we have just computed using the table below. x slope of secant 9 In the first slot, we wrote down the x coordinate of the second point of the secant line starting at (1, f(1)); in the second slot, we wrote down the slope of the secant line joining (1, f(1)) and (, f()). Now we may not be quite satisfied with this answer it is only an approximation, and perhaps not a very good one. How can we make the approximation better? One idea that presents itself quickly is to try again compute the slope of another secant line, hopefully one whose slope is closer to the slope of the tangent line. Let s see what happens if we slide the second point of the secant we used earlier back along the curve a bit closer to (1, f(1)), say to (3/, f(3/)): 5

6 We can use this new point to build a new secant line, graphed in orange below: Notice that the new secant line appears to have slope much closer to that of the tangent line. 6

7 Let s compute its slope, rise run : f(3/) f(1) 3/ 1 = / 1 = ( ) = 00 ( ) 3. Let s add this new data to our chart: x slope Now we could continue this process ad naseum: the closer we slide the second point of the secant line towards (1, f(1)), the closer the slope of the secant is to the slope of the tangent: 7

8 We ll do one more example, choosing t quite close to 1, say t = 1.1, which gives us the secant line graphed below in light blue: Since slope = rise run, we get f(1.1) f(1) = = 10 ( ) = 1000 ( )

9 for the slope of the secant line passing through (1, f(1)) and (1.1, f(1.1)). Continuing our table (and adding on some extra slopes that I won t compute in class), we have x slope 9 3/ From the table, it appears reasonable to guess that the instantaneous rate of change of the function f(x) at x = 1 is about In other words, if your bus driver had looked at his speedometer at exactly one hour into the trip, it would have read very close to miles per hour. This number is still an approximation for the instantaneous rate of change of f(x) at x = 1, but in future sections we will learn how to compute instantaneous rates of change precisely. To be able to do this, we will need to make the idea of sliding one point towards another more concrete, which we will do in the next section when we discuss limits. Let s summarize the ideas we ve seen so far. There are two types of rates of change, and each type is related to the slope of a particular line: 1. Average rate of change: the average rate of change of the function f(x) from x = a to x = b is the slope of the secant line joining the points (a, f(a)) and (b, f(b)).. Instantaneous rate of change: the instantaneous rate of change of the function f(x) at x = a is the slope of the line tangent to the curve f(x) at the point (a, f(a)). 9

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