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Tangent Lines - Slope of a Graph at a Point
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Data Science Math Skills
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Measuring Rates of Change
This module begins a very gentle introduction to the calculus concept of the derivative. The first lesson, "This is About the Derivative Stuff," will give basic definitions, work a few examples, and show you how to apply these concepts to the real-world problem of optimization. We then turn to exponents and logarithms, and explain the rules and notation for these math tools. Finally we learn about the rate of change of continuous growth, and the special constant known as “e” that captures this concept in a single number—near 2.718.
Conheça os instrutores
Daniel EggerExecutive in Residence and Director, Center for Quantitative Modeling
Pratt School of Engineering, Duke University
Paul BendichAssistant research professor of Mathematics; Associate Director for Curricular Engagement at the Information Initiative at Duke
Mathematics
Okay. Welcome back everyone.
This is the beginning of a three-video series on tangent lines.
And honestly, the entire point of the first video is that I want
you to understand everything that's written on this screen.
I'm gonna go through what's on this screen and take
a few more screens to really describe all the different concepts.
The key idea here is, for the moment,
let's imagine we just have the graph of a function,
y = f(x), and we're watching this graph here in green.
Now suppose we take a particular point x = a.
So right here is x = a.
And we want to ask the question: how fast is f(x) changing at x = a?
I want you to appreciate that's a really actually tricky question
that we take for granted all the time.
It's like saying right here on this point in the road,
I'm moving 55 miles per hour.
That doesn't mean in the next hour I will have moved
55 miles or it will take me one hour to move the next 55 miles.
It says right now I'm moving 55 mph.
It's a really tricky idea.
Though we often use calculus to describe as instantaneous rate of change.
The cool geometric picture is if you look at this red line I've drawn right here,
that's what's called the tangent line to the graph of the function at x = a.
That's a key geometric concept.
If you think back, way back, several videos ago,
the only thing we know how to take slopes of are lines;
we understand how to take the slope of a line.
This is gonna be a line,
and its slope is actually gonna be
that instantaneous rate of change. That's what we've written here.
This is the tangent line to the graph of the function at the point,
and its slope gives us the rate of the change there.
If you take nothing else out of the video, that's a key idea.
Even though the graph itself is not a line,
it's a curve – at each point,
I can draw a line that's tangent and its slope is
what we call that instantaneous rate of change.
That's also called the derivative of the function at that point,
and that's this little symbol here: f'(a).
Now how do you actually compute f'(a)?
Understand that's really tricky because if you want to compute a slope of a line,
we need two points on the line – we really only have this one point, this red point here.
That's this tricky formula here.
Don't be scared of the limit – we're gonna unpack that as we go through the screens,
but those are the key ideas.
To work up that though,
let me start with a much, much simpler example.
So suppose right here,
we have the graph y = 3x.
And for the moment, let's take a really fanciful example.
Any business majors in here,
please don't be offended, I know this is simple.
I'm gonna pretend this model's the revenue versus the price of an item.
So there's a particular item I'm gonna sell.
I want to sell it for x dollars – that's
the price – and then I want to see how much revenue comes in.
Okay, so y is it unrealistic,
you'll notice it's just the graph of y = 3x.
What this is saying is that as I increase the price,
the revenue goes up forevermore.
So obviously, what I should do is set the price to be
one million bajillion trillion dollars.
Obviously, that's why it's unrealistic.
But for the moment,
let's suppose that right now my business has set
the price of the item as a dollars – it's gonna cost you a dollars for the item.
And I want to ask the question,
if I increase the price,
will the revenue go up?
Obviously, it will, but I want to know how much.
What's the rate of change at this point?
It's a really simple question with a line.
Let's see why that's true.
Let's go up here to the point on the graph.
So that's the point (a, 3a).
Now suppose I increase the price of my unit by $1,
so here's a+1. Hit the graph.
What are the coordinates of that point?
That was our a+1 and 3(a+1).
Okay. Now let's figure out the slope of this line between the two.
The slope of that line segment,
I think about calculating that,
is going to be the rise – the difference in the y values,
so 3(a+1) - 3a – divided by the run – the difference in the x values,
so a+1 - a.
And if you work out all that algebra –
that's not really the point of this video – you're gonna get three.
Okay. The key idea here is that there's nothing special about increasing a $1.
If I increase $2,
that rise-over-run calculation will be
exactly the same – that's what's special about a line.
The slope of the line doesn't really matter which line segment you take.
So in other words, if I increase $1,
the revenue will go up three times $1.
If I increase $2,
the revenue will go up three times $2.
That's really what the slope of the line is saying.
If you remember back to the slope of line videos,
that was really the key point of what the slope of the line means.
Okay. That's great. Let's return to a much more realistic model.
So if you look at the same picture,
now notice it's the same picture as we had in the very first except I've added y
= f(x) and I said that I'm graphing price versus revenue.
So this green curve,
I pointed to you that's a more realistic price versus revenue curve.
What this is saying is normally when I raise the price,
my revenue goes up because I get more dollars for each unit I sell.
However, eventually, I keep raising the price and people get mad at me and say,
"I'm not really gonna buy your product.
It's not worth that much money, buddy."
And so the revenue starts to go now.
Okay. So that's essentially what this graph here means.
Now let's ask the question again.
If I'm at a dollars and I want to raise my price,
does my revenue go up?
And if so, by how much? Does it go down?
And if so, by how much?
In other words, what's the instantaneous rate of
change of the revenue at this price point?
A very key idea here is the following.
What makes this different from a line?
So the answer to that question depends on my price point.
So for example, suppose my price point was
right here – at B – and let's look right up there.
Without even drawing the tangent line,
you can probably guess that if I increase the price a little bit from B,
my revenue would go up but by not nearly as much as it would if I increase it from a.
And it gets even worse over here at c. If that's my price point and I increase the price,
my revenue is actually gonna go down.
Right? So it's a bad idea that I put it there.
Business majors among you might say it's a bad idea to have
that price at c in the first place – that's another matter.
But the key takeaway,
the key mathematical takeaway here is that
the slope of the tangent line changes depending on where you are on the curve.
Notice that's not the case when my green curve was a line,
but it is the case here.
Okay. So, the answer to the question,
'By how much will
the revenue increase if I increase my price a little bit from the price point a?'
The answer to that is the slope of this red line.
That's what we call the derivative of the function at the price value x=a.
So the thing we're after is we're after something called f'(a) –that's
the derivative of the function
f at the point x=a.
Great, so it has a name, but now we need to calculate it.
Okay. So I imagine you're watching the video on slopes of lines.
A fair question to ask you would be – and if you did some of the quiz questions,
you'll know we asked you this question – given two points,
calculate the slope of the line segment between them.
That's a fair question; I think you're all pretty good at doing that by now.
Here's an unfair question: calculate the slope of that red line,
and all I'm gonna do is give you the coordinates of that red dot.
So I'm gonna tell you what a is, I'm gonna tell you what f(a) is,
go nuts and give me the slope.
That's not fair. You have a right to be angry at me for asking that question.
So let me ask you another question which you can do,
although the answer won't give you the answer you want.
What I'm gonna do is I'm gonna take another point of
the line and I'm gonna draw a little line segment between them.
So, let's think about what all those points are.
Let's say here is the point a+h.
h is sort of a notional symbol but it stands for a little bit.
So in other words, right here,
this is h. Let's think about what the coordinates of that point are.
First of all, the coordinates of this point right here,
we know what those are – those are (a, f(a)).
And we know what the coordinates of this point are – they're (a+h, f(a+h)).
Okay, fine.
So, I know how to calculate the slope of that line segment. What is it?
Well, what it is is the difference in the y values – what we call the rise,
which is f(a+h) - f(a) – over the difference in the x values – which is the run,
which is a+h - a,
but looking at this picture, that difference is just h,
so let's put it there.
Alright. So, let's pause here for a second.
I don't know the slope of the tangent line but I would like to know it.
I do know the slope of this little line segment assuming you give me values for this,
but if you give me values for the function,
I can give you values here,
so I know how to make that a number.
Here's where calculus comes in.
I can write an =, I can write a limit sign,
and I can say h goes to zero.
Okay. Believe it or not, in the next video,
we'll calculate explicitly what this means.
But for now, the really conceptual point is it
says actually I don't really want this point to be here,
I don't want that point to exist at all.
I want h to be zero because if I move h towards zero,
this little line segment snaps toward pointing in the same direction as the tangent line.
So as I move h towards zero,
the slope of that little line segment – imagine it moving along,
snap, snap, snap – snaps to the slope I care about.
That's why I put a limit as h goes to zero here.
It's not h equal zero,
but h gets closer and closer to zero.
Okay. That's the point of this lesson.
In the next video, we'll calculate an explicit example with an actual function,
and we'll show you how this formula lets you describe that slope. Thanks for listening.
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