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Interpreting the meaning of the derivative in context

AP Calc: CHA‑3 (EU), CHA‑3.A (LO), CHA‑3.A.1 (EK), CHA‑3.A.2 (EK), CHA‑3.A.3 (EK)

Video transcript

- [Instructor] We are told that Eddie drove from New York City to Philadelphia. The function d gives the total distance Eddie has driven in kilometers t hours after he left. What is the best interpretation for the following statement? D prime of two is equal to 100. So pause this video, and I encourage you to write it out. What do you think this means, and be sure to include the appropriate units. And now let's do this together. If d is equal to the distance, the distance driven, then to get d prime, you're taking the derivative with respect to time. So one way to think about it is, it is the rate of change of d. So we could view this as d prime is going to give you the instantaneous, instantaneous, instantaneous rate. And they are both functions of t. So one way to view d prime of two is equal to 100, that would mean, well, what is our time now? Well, that is our t, and that's in hours. So two hours, actually let me color code it. So two hours after leaving, after leaving, Eddie, Eddie, drove, drove, and this means, so it'll be grammatically correct. Drove at an instantaneous, instantaneous, instantaneous rate of, and let me use a different color now for this part, of 100, and what are the units? Well, the distance was given in kilometers, and now we're gonna be thinking about kilometers per unit time. Kilometers per hour, so this is 100 kilometers, kilometers, per hour. So that's the interpretation there. Let's do another example. Here we are told a tank is being drained of water. The function v gives the volume of liquid in the tank, in liters, after t minutes. What is the best interpretation for the following statement? The slope of the line tangent to the graph of v at t equals seven is equal to negative three. So pause this video again and try to do what we just did with the previous example. Write out that interpretation, make sure to get the units right. All right, so let's just remind ourselves what's going on. V is going to give us the volume, as a function of time. Volume is in liters, and time is in minutes. And so they're talking about the slope of the tangent line to the graph, the slope of the tangent line to the graph of v, that's just v prime. So if you take the derivative with respect to time, that's going to give you v prime, and these are all functions of t. These are all functions of t. And they say at t equals seven, it's equal to negative three. So this, which is the same thing as the slope of tangent line, slope of tangent, tangent line. And they tell us that v prime of at time equals seven minutes, our rate of change of volume with respect to time is equal to negative three. And so you could say if we were to write out, this means that after, after seven minutes, seven minutes, the tank is being drained at an instantaneous, instantaneous, that's why we need that calculus for that instantaneous rate. An instantaneous rate of, now you might be tempted to say it's being drained at an instantaneous rate of negative three liters per minute. But remember, the negative three just shows that the volume is decreasing. So one way to think about it is this negative is already being accounted for when you're saying it's being drained. If this was positive, that means it is being filled. So it is being drained at an instantaneous rate of three liters per minute. Three liters per minute. And how did I know the units were liters per minute? Well, the volume function is in terms of liters, and the time is in terms of minutes. And then I'm taking the derivative with respect to time, so now it's gonna be liters per minute, and we are done.