A.2 Generalizing to Other Rates of Change

On this screen we'll generalize from thinking about the average rate of change of an object's position with respect to time to more general rates of change, like temperature with respect to time, or vertical position (y) with respect to horizontal position (x).

Quick Recap

Generalizing to Other Rates of Change

The many uses of parentheses in math

Parentheses in mathematics unfortunately have many meanings. One of course is multiplication: for example, 3(6) =3 ∗6 =18.

A different use of parentheses is to indicate a function's input. You're used to seeing, for example, 𝑓(𝑥) =5𝑥2 +3 to indicate that for any inputted value of x, you square it, multiply by 5, and then add 3. We could write instead 𝑓(◻) =5◻2 +3, which is the same function except that we're using ◻ to represent the input value.

We emphasize this because we know that for most students, initially seeing 𝑠=𝑠(𝑡)or𝑓=𝑓(𝑢) just seems weird. After a while you'll come to read the statements as intended: "s is a function of t" and "f is a function of u." (And not "s equals s times t" or "f equals f times u".)

Similarly, we can write 𝑦=𝑦(𝑡)to indicate that position 𝑦 is a function of time 𝑡. We read that statement as "y is a function of t."  This construction means that if we give you a particular time t (like 0.50 s), then you can tell us Calvin's ball's position y at that instant. And as time progresses from one moment to the next, the ball's position continuously varies accordingly – again as a function of time in this formulation.

Indeed, we also considered the rate at which Hannah's position was changing, and the rate at which the position of Calvin's tossed ball was changing. In both cases the rates were time-based: at the moment of interest Hannah was traveling forward at 20 meters per second, and the ball was traveling upward at 5.00 meters per second. We thus say in these scenarios that we are considering the rate with respect to time.

We can of course consider functions that depend on different independent variables. For instance, in a typical standard math problem you're given a function f whose independent input-variable is x, in which case we have 𝑓=𝑓(𝑥),to indicate that the function 𝑓 is a function of 𝑥. In this formulation, the function's output varies as the input variable x takes on different values.

Similarly, we can write 𝑦=𝑦(𝑥),to indicate that 𝑦 is a function of 𝑥.

Let's consider one such physical situation, in which vertical height y is a function of horizontal position x.

Getting Acquainted with the Scenario "Syd bikes uphill"

Syd bikes steadily up the hill shown to the right. Let's use the red dot on the center of his front tire as the indicator of his current position. We'll measure Syd's (the dot's) horizontal and vertical position in meters, again abbreviated "m."

We are interested in Syd's vertical position y as a function of his horizontal position x: 𝑦 =𝑦(𝑥).

Check Question:

𝑑𝑦 refers to a small change in y.
dx refers to a small change in x.

In the preceding topic, we used ds to refer to a small change in Hannah's position s, and dt to refer to a small change in time t. Now generalizing: in Calculus, we use 𝑑◻ to refer to a small change in the quantity ◻, whatever it is. So for instance, as Syd bikes uphill, 𝑑𝑦 refers to a small change in his vertical position y (as in the preceding Topic), and 𝑑𝑥 refers to small change in his horizontal position x.

Rates when the independent variable isn't time

Hence when we think about the rate at which his vertical position changes, we don't want to think in terms of how quickly his vertical position changes as time increases; instead, we must think about it in terms of how much his vertical position changes as his horizontal position increases.

We need to extend our conception of "rate"

We thus need to extend the way we conceive of "rate" so we can correlate small changes in his vertical position with the tiny changes in his horizontal position: 𝑑𝑦=rate∗𝑑𝑥 In words: the small change in vertical position dy equals the appropriate rate multiplied by the small change in horizontal position dx. We're extending our conception of rate as it applies to the familiar "small change in function's output equals rate multiplied by the small change in time" to now more generally "small change in function's output equals rate times small change in [whatever the independent variable is]."

Viewed differently, we can rearrange the preceding equation to see the rate-of-change in Syd's scenario is the quantity 𝑑𝑦𝑑𝑥 : rate =𝑑𝑦𝑑𝑥 The quantity 𝑑𝑦𝑑𝑥 is pronounced "dee-wye dee-ex," rather than, for instance "dee-wye over dee-ex" or "dee-wye divided by dee-ex." Instead, it's just "dee-wye dee-ex." And you can see that we're now considering the rate with respect to (changes in) his horizontal position x: the 𝑑𝑥 in the denominator reminds us that our independent variable is x, and we're looking at what happens when we change x by a small amount.

As we saw earlier, for time-based rates we must specify the particular moment at which a given rate applies. For instance, when providing details about Hannah's trip we stated that her rate of travel at 9:00:00 was 𝑑𝑠𝑑𝑡∣at 9:00:00 =20 meterssecond. At any other moment in time, her rate of travel could be different, and probably is. Similarly, Calvin's tossed-ball was traveling upward at 𝑡 =0.50 s at the rate 𝑑𝑦𝑑𝑡∣at 0.50 s =5.00 meterssecond. A brief time later, at 𝑡 =0.80 s, it is traveling at a slower rate. The key point is that we need to know the relevant rate at the exact time of interest.

Considering now rates with respect to horizontal-position x, we must specify the position x at which the rate applies. We use the same vertical bar to do so, like 𝑑𝑦𝑑𝑥∣at 𝑥=3.00 m =6.00 vertical mhorizontal m.

With this idea in mind, let's continue gaining familiarity with the scenario of Syd's ride.

Now that you're familiar with the various quantities associated with Syd's ride, let's return to the Calculus calculations we introduced in the preceding Topic.

If the calculations in Syd's scenario seem easy to you, great! You've made the switch to thinking about "rate" as it applies to an independent quantity other than time, like "rate with respect to x," which we'll need as we explore many other scenarios. And if considering such a rate doesn't feel natural easy to you yet, please just keep going. With continued exploration rates of all sorts will come to feel more familiar (as is so often true when you're learning something new).

Now, as promised, the scenarios about Syd above have set up our ability to quickly estimate (3.01)2 to within 0.001% . . . as we'll do on the next screen.

The Upshot

1. We write 𝑓 =𝑓(◻) to indicate that f is a function of the independent variable ◻.
2. Functions with independent variables other than time also have rates of change. We talk about the rate of change with respect to [whatever the independent variable is].
3. If we have the function 𝑦 =𝑦(𝑥), then we write the rate at which the function's output 𝑦(𝑥) changes with respect to x at 𝑥 =3.00 m as 𝑑𝑦𝑑𝑥∣at 𝑥=3.00 m Syd's tiny change in vertical position dy when he travels the short horizontal distance dx from 𝑥 =3.00 m is then given by the formulation smallchangeinvertposition=(rateathorizposition𝑥=3.00m)∗(smallchangeinhorizposition)𝑑𝑦=(𝑑𝑦𝑑𝑥∣at 𝑥=3.00 m)⋅𝑑𝑥
4. If we know an object's vertical position 𝑦(𝑥) at a given horizontal location 𝑥, and rate of change with respect to x 𝑑𝑦𝑑𝑥∣at that value of 𝑥, then we can predict its vertical location a short horizontal distance dx away: 𝑦(𝑥+𝑑𝑥)=𝑦(𝑥)+𝑑𝑦⏞¯¯¯¯¯¯⏞¯¯¯¯¯¯⏞ (rate at horiz position 𝑥)∗𝑑𝑥