Interpreting the meaning of the derivative in context | AP Calculus AB | Khan Academy

We're 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' ) of 2 is equal to 100?

So, pause this video and I encourage you to write it out. What do you think this means? Be sure to include the appropriate units.

All right, now let's do this together. If ( d ) is equal to the distance driven, then to get ( d' ), 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 ).

We could view this as ( d' ) giving you the instantaneous rate, and they are both functions of ( t ). One way to interpret ( d' ) of 2 is equal to 100 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, Eddie drove. This means, let me be grammatically correct, drove at an instantaneous rate of, and let me use a different color now for this part, 100.

And what are the units? Well, the distance was given in kilometers, and now we're going to be thinking about kilometers per unit time: kilometers per hour. So this is 100 kilometers per hour.

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 = 7 ) is equal to negative 3?

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.

So if 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' ). If you take the derivative with respect to time, that's going to give you ( v' ), and these are all functions of ( t ).

They say at ( t = 7 ), it's equal to negative 3. This, which is the same thing as the slope of the tangent line, tells us that ( v' ) at time equals 7 minutes, our rate of change of volume with respect to time is equal to negative 3.

You could say, if we were to write it out, this means that after seven minutes, the tank is being drained at an instantaneous rate. That's why we need that calculus for that instantaneous rate.

At an instantaneous rate of, now you might be tempted to say it's being drained at an instantaneous rate of negative 3 liters per minute, but remember the negative 3 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 3 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.

Then, I'm taking the derivative with respect to time, so now it's going to be liters per minute. And we are done.