1. Limits and Differentiation
Limits are the most fundamental ingredient of calculus. Learn how they are defined, how they are found (even under extreme conditions!), and how they relate to. In this section we define the derivative, give various notations for the derivative and Proof of Trig Limits · Proofs of Derivative Applications Facts · Proof of Various . In the first section of the Limits chapter we saw that the computation of the The next theorem shows us a very nice relationship between. The fact that in your example the limit and the derivative are equal is only a You can understand the relation between a limit and a derivative if you look at their.
So, we are going to have to do some work. In this case that means multiplying everything out and distributing the minus sign through on the second term.
After that we can compute the limit. However, outside of that it will work in exactly the same manner as the previous examples.
Also note that we wrote the fraction a much more compact manner to help us with the work.
1. Limits and Differentiation
So, we will need to simplify things a little. In this case we will need to combine the two terms in the numerator into a single rational expression as follows.
So, upon canceling the h we can evaluate the limit and get the derivative. You do remember rationalization from an Algebra class right? In an Algebra class you probably only rationalized the denominator, but you can also rationalize numerators. Remember that in rationalizing the numerator in this case we multiply both the numerator and denominator by the numerator except we change the sign between the two terms.
So, cancel the h and evaluate the limit. So, plug into the definition and simplify. So let's figure out what the change in why over the change in x is for this particular case. So the change in y is equal to what? Well, let's just take, you can take this guy as being the first point, or that guy as being the first point.
But since this guy has a larger x and a larger y, let's start with him. The change in y between that guy and that guy is this distance, right here. So let me draw a little triangle.
That distance right there is a change in y. Or I could just transfer it to the y-axis. This is the change in y. That is your change in y, that distance. So what is that distance? It's f of b minus f of a. So it equals f of b minus f of a. That is your change in y.
Now what is your change in x The slope is change in y over change in x. So what our change in x? Remember, we're taking this to be the first point, so we took its y minus the other point's y. So to be consistent, we're going to have to take this point x minus this point x. So this point's x-coordinate is b. So it's going to be b minus a.
And just like that, if you knew the equation of this line, or if you had the coordinates of these 2 points, you would just plug them in right here and you would get your slope. And that comes straight out of your Algebra 1 class. And let me just, just to make sure it's concrete for you, if this was the point 2, 3, and let's say that this, up here, was the point 5, 7, then if we wanted to find the slope of this line, we would do 7 minus 3, that would be our change in y, this would be 7 and this would be 3, and then we do that over 5 minus 2.
Because this would be a 5, and this would be a 2, and so this would be your change in x. So 7 minus 3 is 4, and 5 minus 2 is 3. Now let's see if we can generalize this. And this is what the new concept that we're going to be learning as we delve into calculus.
Let's see if we can generalize this somehow to a curve. So let's say I have a curve. We have to have a curve before we can generalize it to a curve. Let me scroll down a little.
Well, actually, I want to leave this up here, show you the similarity.
Let's say I have, I'll keep it pretty general right now. Let's say I have a curve. I'll make it a familiar-looking curve. Let's say it's the curve y is equal to x squared, which looks something like that.
And I want to find the slope. Let's say I want to find the slope at some point. And actually, before even talking about it, let's even think about what it means to find the slope of a curve.
Here, the slope was the same the whole time, right? But on a curve your slope is changing.
How do derivatives relate to limits?
And just to get an intuition for that means, is, what's the slope over here? Your slope over here is the slope of the tangent line. The line just barely touches it. That's the slope over there. It's a negative slope. Then over here, your slope is still negative, but it's a little bit less negative. It goes like that.
I don't know if I did that, drew that. Let me do it in a different color. Let me do it in purple. So over here, your slope is slightly less negative. It's a slightly less downward-sloping line. And then when you go over here, at the 0 point, right here, your slope is pretty much flat, because the horizontal line, y equals 0, is tangent to this curve. And then as you go to more positive x's, then your slope starts increasing. I'm trying to draw a tangent line. And here it's increasing even more, it's increased even more.
So your slope is changing the entire time, and this is kind of the big change that happens when you go from a line to a curve. A line, your slope is the same the entire time. You could take any two points of a line, take the change in y over the change in x, and you get the slope for the entire line. But as you can see already, it's going to be a little bit more nuanced when we do it for a curve. Because it depends what point we're talking about.
We can't just say, what is the slope for this curve? The slope is different at every point along the curve. If we go up here, it's going to be even steeper. It's going to look something like that. So let's try a bit of an experiment. And I know how this experiment turns out, so it won't be too much of a risk. Let me draw better than that.
So that is my y-axis, and that's my x-axis. Let's call this, we can call this y, or we can call this the f of x axis. And let me draw my curve again. And I'll just draw it in the positive coordinate, like that. And what if I want to find the slope right there?
What can I do? Well, based on our definition of a slope, we need 2 points to find a slope, right? Here, I don't know how to find the slope with 1 point. So let's just call this point right here, that's going to be x.
We're going to be general. This is going to be our point x. But to find our slope, according to our traditional algebra 1 definition of a slope, we need 2 points. So let's get another point in here.
Let's just take a slightly larger version of this x.
Derivative - Wikipedia
So let's say, we want to take, actually, let's do it even further out, just because it's going to get messy otherwise. So let's say we have this point right here. And the difference, it's just h bigger than x. Or actually, instead of saying h bigger, let's just, well let me just say h bigger.
So this is x plus h. That's what that point is right there. So what going to be their corresponding y-coordinates on the curve? Well, this is the curve of y is equal to f of x.
So this point right here is going to be f of our particular x right here. And maybe to show you that I'm taking a particular x, maybe I'll do a little 0 here. This is x naught, this is x naught plus h. This is f of x naught.
And then what is this going to be up here, this point up here, that point up here? Its y-coordinate is going to be f of f of this x-coordinate, which I shifted over a little bit.
So what is a slope going to be between these two points that are relatively close to each other? Remember, this isn't going to be the slope just at this point.
This is the slope of the line between these two points. And if I were to actually draw it out, it would actually be a secant line between, to the curve. So it would intersect the curve twice, once at this point, once at this point. You can't see it.