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### Course: AP®︎/College Calculus AB > Unit 4

Lesson 1: Interpreting the meaning of the derivative in context# Interpreting the meaning of the derivative in context

When derivatives are used to describe real-world situations, we need to know how to make sense of them.

## Want to join the conversation?

- why arent the time measurements squared. i.e. kilometers per hour squared? in the practice exam the answers turned out to be unit measurement/time measurement^2(6 votes)
- Hello Jay Frost!

Squared time measurements are generally used for measuring acceleration (change in velocity). For example, a car speeding up at 3m/s per second would be accelerating at a rate of 3m/s^2.

Squared time measurements are also used for rates of change of rates of change. For example, if the rate of change of water leaking from a tank is 10mL/s and the rate increases by 2mL/s per second, the change of the change would be measured as 2mL/s^2.

However, in this video Sal is examining functions of the first rate of change (not the second rate of change!). Therefore, he shouldn't use, for example, m/s^2, as this denotes the rate of change of the rate of change. Sal just wants the first rate of change, and so in this example he should use m/s.

I hope this explanation helped! :)(21 votes)

- Why do we take the derivative with respect to time ?(7 votes)
- Because in the examples what we were measuring (distance and volume) were changing with respect to time. The derivative is basically the rate of change of something so we need to know what it is changing with respect to. Hopefully that makes sense.(14 votes)

- how would you interpret a second derivative in context of the problem?(4 votes)
- Here's an example of an interpretation of a second derivative in a context. If s(t) represents the position of an object at time t, then its second derivative, s''(t), can be interpreted as the object's instantaneous
**acceleration**.

In general, the second derivative of a function can be thought of the instantaneous rate of change of the instantaneous rate of change of the function. In the example given in the previous paragraph, acceleration is the instantaneous rate of change of velocity, which in turn is the instantaneous rate of change of position.(11 votes)

- Why does Sal say after 7 min or after 2 hour should not he instead say at 2nd hour or 7th min?(8 votes)
- How to teach the concepts of derivative and integral calculus using mathematical modelling? Give me very specific realistic examples with data to teach above concepts(6 votes)
- Can we take the derivative of a quantity with respect to any other quantity other than time? If yes, please provide an example.(2 votes)
- to take a derivative you need a function, and time as what you take one with respect to is easy because so many things depend on time. if you have any function though you can take the derivative of it.

a function has an input and output. You can take the derivative with respect to any input a function has. Sometimes an input can be yet another output to a inner input. for example composite functions.

A more definite example would be something like log(sin(x)) sin(x) is the input of the logarithm, but sine also has an input. so you could take the derivative of the whole thing with respect to sin(x) or just x. How you do that is the chain rule.

The derivative is the rate of change (or slope) at a particular point. It is saying, as I change the input the output changes by however much.

Let me know if that doesn't help.(5 votes)

- OK, I love the premise, but am confused. Let's stick with Eddie's 100 KM trip. If he travels for 1 hour, he clearly drives an AVERAGE of 100 k/hr. BUT he passes construction and has to slow to 20KM/hr, he has a couple of pit stops and travels at 0 KM/hr; on the highway, he speeds along at 75 KM/ hour. How do w solve that? Also, I didn't see where the term indicated he traveled 1 hour only. Please help this is ignorant soul?(2 votes)
- In your example and the example in the video, you are not looking at average speed (slope of secant line) but you are looking for the EXACT speed at a particular moment in time. If "D" is a function of distance, and you drove at 20km/hr at 3 hours, then -

D'(3) = 20

Same goes for the other examples you gave. Also, the questions tells you D'(2) =100 (the one in the video) which means the speed at 2 hours is 100km/hr. there is no average speed here, because if you were to go to a different interval, the speed would change(4 votes)

- Calculus is honestly so difficult to understand but Khan Academy has made it so much easier for me. Thankyou.(3 votes)
- Perhaps this is explained elsewhere, but I can't find it at the moment. How would the second derivative be interpreted in context other than the velocity/acceleration examples? :) Thank you in advance(2 votes)
- this is so easy to digest compared to the limit part(1 vote)

## 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.