Calculus — Estimating Instantaneous Rate of Change from a Table
Corrected lesson transcript


## 0:00 — Estimating Instantaneous Rate from a Table

[0:04] Water is flowing into a pool. Its volume was recorded at five different times.
[0:14] I want to estimate the instantaneous rate of change at the 20 minute mark
[0:32] So this is volume, and this is time.
[0:48] Water is flowing in, so volume is a continuous function. Would this be continuous?
[0:55] Yes. The volume would not instantly jump; it changes steadily.
[1:03] We do not know what the function looks like.
[1:14] It could look something like this, but we do not know. It would probably be a smooth
[1:24] function like that.

## 1:29 — Instantaneous Rate as a Tangent Slope

[1:29] We're trying to find the instantaneous rate of change which is the slope of what?
[1:41] At one point, it is the tangent line.
[1:49] Right at the 20 minute mark.
[1:52] We're trying to guess the slope of that line right here.
[2:01] Why can't we do what we did on Friday?
[2:07] Why can't we just pick values really, really close and find the average rate of change?
[2:19] It's asking for the instantaneous rate of change.
[2:23] It's definitely asking for that.
[2:25] But last class, we did this, like the average rate of change.

## 2:36 — Why the Formula-Based Method Is Unavailable

[2:36] For example, we used the average rate of change [f(2.99) − f(3)] / (2.99 − 3).
[2:47] Then we kept using numbers
[2:49] that get closer and closer to the target.
[2:51] Remember this?
[2:52] OK.
[2:53] So why can't we do that for this problem?
[3:06] And what do we not have here?
[3:09] We have no equation
[3:12] To use that method, we would need to be given
[3:17] a function formula such as f(x) = …
[3:21] But we do not know what the formula is.
[3:25] So here we can't do this method where you keep plugging in numbers closer and closer
[3:32] to that number.
[3:33] Does this make sense?
[3:35] Okay.
[3:36] So we need some other way to estimate it.

## 3:42 — Use Secant Slopes on Both Sides

[3:42] So we find the average rate of change between these two, and that would give us the slope
[3:50] of that line.
[3:52] And then do the same thing between 20 and 25, which would give us the slope of that
[4:04] line. Then what should we do with the two slopes? Average them.
[4:20] One possible—but less reliable—idea is to draw the tangent line we expect and then
[4:32] use the known point plus a second estimated point. That would still be a guess,
[4:39] so it is better to use the table data. That is what we will
[4:42] do. First, find the average rate of change on the interval [15, 20].

## 4:58 — Compute the Neighboring Average Rates

[4:58] AROC on [15, 20] = (3,500 − 2,500) / (20 − 15).
[5:07] That is 200.
[5:11] Is that 200?
[5:14] Okay.
[5:15] What are the units?
[5:17] The volume values are measured in gallons, and the time values are
[5:26] measured in minutes, so the units are gallons per minute.
[5:35] The slope of the green line is 200 gallons per minute.
[5:48] Is that 300?
[5:56] The slope of the orange line is 300 gallons per minute.

## 6:18 — Average the Two Rate Estimates

[6:18] Okay. We found the first slope, 200 gallons per minute, and then
[6:47] the orange slope, 300 gallons per minute. Our best estimate
[6:58] is the average of those two slopes.
[7:15] We are estimating the instantaneous rate of change at one single point in time—20 minutes—not over an interval
[7:25] of time. At that exact moment, our best estimate is (200 + 300) / 2 = 250 gallons per minute.

## 7:38 — When to Use This Method

[7:38] This method is not very complicated.
[7:48] So when are we going to do this method?
[7:49] Use this method when only a table or graph of values is given.
[7:53] Use it when no function formula is available. If a function or equation is given, use
[8:02] the method from Friday: find average rates over increasingly small intervals. Does that make sense?
[8:11] Okay