Okay, so one thing that I wanted to do with you all is to say that we can find slopes using different techniques than just changing y over changing x. I mean, it's still the same thing, but we can use a different notation. Let's just put it that way.
Now, let's say that I have a linear function, and I also want to be clear on this. My function doesn't always have to be linear, but this is pleasing to the eye because we're thinking slope. All right, and let's say that I want to find the slope of this line, which I'm defining by f now.
between a and b. Okay so I'm finding the slope between these two points. Well now my y value my x values aren't x1 and x2 they're a and b the way I've defined them and then the y values are the function evaluated at a and the function evaluated at b and so now my change in y is the difference between f of b and f of a And the change in my x is b minus a, subtracting b minus a. And so my slope in this case will be f of b minus f of a, my change in y, over b minus a, which is my change in x. And I believe some of you have seen this before.
Most of you have seen this before. It's just a different way of saying change in y over change in x. All right. Now, excuse my slanty lines here.
I didn't realize how bad those were. Let's say that I'm still starting at point A, F of A. Okay. But now let's say I want to give myself a change in X of a shorter name.
Let's say I call change in X, H. Nice and simple. Now, H is going to be something you're going to see a lot of coming up. That's why I chose it.
All right. That means that to get to my next x value, I have to go a plus this distance h to get to the next x value. So here I am at a plus h, making my y value f evaluated at a plus h. Now, this will be important for when we're finding equations of tangent lines.
When we go through the derivation of how to find the slope. of a tangent line, you'll understand where this is coming from. Okay. All right. Let's go ahead and label our change in Y.
Now this is going to be F evaluated at A plus H minus F evaluated at A. And so now our slope formula is F of A plus H minus F of A all over H. This is also known as your difference quotient. This may be in a slightly different form than what you are used to or what you recognize. You might remember f of x plus delta x minus f of x over delta x.
It's the same. I'm just using different variables h and constant values a because I'm trying to find something specific. A little bit more specific.
So this is just a different way of writing the difference quotient. And again, difference quotient is just another way of writing slope. So hopefully this helps kind of see how, you know, you've got multiple ways to express, you know, that you're trying to find slope. All right.
And again, your function f may not be linear. And we'll talk about, you know, what's going on when you're finding these things for a nonlinear function as we go into the equations of tangent lines. So what we're going to do is the algebra of computing these types of things. All right, so I've got four examples and we're going to start with the first one.
So for all four examples, we're going to use the same function. All right, f of x is x squared minus 5x. For the first example, I want to find f of 4 minus f of 3 over 4 minus 3. So basically, I'm looking at f of 4 minus f of 3 over 4. four minus three. So I'm looking at just slope in this first form here.
Okay. So F of four, we're going to put this all together. We really don't love having you guys partition your workout and having work over here, work over there and having to follow a treasure map.
We'd like everything to flow nice and smoothly throughout the entire process so we can follow your thinking. All right. So this is how we would like you to show your work here, not breaking things off and kind of writing scratch work down. We want to see your thoughts. in motion.
All right. So F of four. Okay.
Let's see here. So F of four. is going to be 4 squared minus 5 times 4. From that, I want to subtract f of 3. So that's going to be 3 squared minus 5 times 3. And that is all over 4 minus 3. And now we're going to play the cleanup game.
Just make this look nice. So 4 squared is 16. 5 times 4 is 20. Minus, I'll go ahead and distribute this negative as I go, which is also why I wanted to show this. We've got to make sure that we have either brackets or parentheses there to show our groups, because this negative must be distributed. 3 squared is 9. And then a negative times a negative, 5 times 3 is a positive 15. And that's all divided by, well, 4 minus 3 is a nice pretty 1. And so now I just have to clean up my numerator. So 16 minus 20 is going to be negative 4. Negative 4 and negative 9 is negative 13. Negative 13 plus 15 is positive 2. So my answer is 2. Nice and clean.
All right. So we're going to go through the same process over here. But notice I didn't give a value for x in the first point. which means I'm going to leave f of x as f of x here to start. So f of x to start, x squared minus 5x minus f of 3. So it's going to be 3 squared minus 5 times 3, all divided by, again, an unknown x minus 3. I can clean up this stuff in the second group of parentheses.
So I have x squared minus 5x minus 9. Again, distribute the negative plus 15, all divided by x minus 3. And then I notice that I can clean up the negative 9 plus 15, which would be positive 6. So I have x squared minus 5x plus 6 all over x minus 3. And then I notice that my numerator looks factorable. Okay. So if I factor my numerator, I get x minus 3 times x plus 2, sorry, x minus 2 divided by x minus 3. So if you remember a couple of days ago, we asked you to make sure that you were practicing your factoring, right?
Negative 3 times negative 2 is positive 6. Negative 3 minus 2 is negative 5. So can I cancel the x minus 3s? Which we talk about this with limits, right? Technically, I can't if I'm just dealing with functions and algebra and no limits.
Unless I want to do that and say specifically that x cannot be 3. Okay, so I can say that this is x minus 2. for x not equal to 3. Now I wanted to do this problem because I'm also going to sort of spin off here and make sure that you understand the difference between the two functions real quick. All right so if I were looking at graphing this rational expression it is the same as this line x minus 2 which I'm going to graph x minus 2 first and then I'm going to graph the rational expression, and I'll show you what I mean. Just give me one second.
I always label my axes. We always label everything. Don't forget.
So x minus 2 is going to have a y intercept at negative 2 and a slope of 1, so it's going to cross the x-axis at positive 2. It's going to look like this. So here is y equals x minus 2, okay? Now this function here, right, is the same as this line except at x equals 3. Give me one second to get these numbers down here.
That's about where 3 would be. So at 3, I would have a whole. So the reason these functions are not the same is because one is just a line and the other is a line with the hole in it. So they're not the same because they don't have the same domain. Okay.
But if you, if you make this statement, you just have to say that X can't be three and then you're safe. All right. So next example, same function.
I'm going to pop this up underneath here. Sorry. One second.
Okay. So same function. Now I want F of three plus H minus F of 3 over h. So now I'm looking at this difference quotient form.
All righty. So f of 3 plus h, that means everywhere I see an x, I'm going to replace it with the quantity 3 plus h. So I need quantity 3 plus h squared minus 5 times the quantity 3 plus h.
And then all of that minus f of 3. which is again, three squared minus five times three. So I'm not just adding H to F, it's I'm taking three plus H within F. So I'm replacing X with the quantity three plus H here.
Okay, and again, that's all over H. So now we're gonna do the cleanup game again. Now we have to be careful because when we're squaring something, it doesn't just mean we square each term. It means that we are multiplying it. by itself.
So I strongly encourage you to write that out. 3 plus h times 3 plus h. All right.
I'm going to distribute through the negative 5, giving me negative 15 minus 5h minus 9 plus 15. Okay. And so the reason why we wrote out that first part there the way we did is because we want to make sure that we distribute properly. Okay, so we are now going to go through the distribution process. All right, don't forget to write over h.
And I know most of you guys will call this FOIL, but that acronym only works for when you multiply a binomial times another binomial. So we have 3 times 3 is 9. All right, then we have 3h and 3h. So 6H. And then we have H times H.
So that's going to be H squared. Oh, you know what's fun about this one? Look at this.
Negative 15 cancels with 15. You're going to notice a lot of things that are supposed to cancel later. We're going to talk more about that within the unit as you do the calculus. Okay, so for right now, we're just practicing. All right, so we still have the negative 5H and the negative 9. Oh, look at that. More stuff cancels.
Playing the cleanup game. And you know what? While I'm at it, I see I have 6H minus 5H. So I can combine those two terms as 1H.
So I have H squared plus H all over H. Oh, I can factor an H out of my numerator. Look at that.
So H times H plus 1 is what I'll have left over all over H. And just like the previous example, I can't just cancel the H's unless I say my answer is H plus 1 for H not equal to 0. If H is 0, I have a problem. So I want to make sure that I am stating the values that H cannot be equal to.
So again, this is just... algebra review. Basic, basic algebra.
It's just a matter of making sure that you remember that when you square a quantity, you multiply it by itself, and then there's some distribution to do there. Always distributing through negatives, making sure parentheses are where they belong, so we get the answers correct over here, right? Because if we're not distributing the negative, things aren't going to work out as nicely as they did for us, all right?
Let's do this again, but this time I'm not giving you a value for x. So this is going to be a tiny bit more complicated, not too bad. So f of x plus h means everywhere I see an x, I'm replacing it with the quantity x plus h.
So I have x plus h quantity squared minus 5 times x plus h. All right, now all of that minus f of x. I'm just going to rewrite f of x.
x squared minus 5x. It's already been defined for me. All divided by... each. So this isn't going to clean up quite as quickly.
Okay. And that's okay. Um, but you know, it's great algebra practice. All right.
Again, I want to go ahead and rewrite that X plus H quantity squared is X plus H times X plus H. We'll go ahead and distribute through our negative five. So we have negative five X minus five H distribute through our negative here. So we have minus X squared plus five X all divided by.
Each. And we're looking at negative 5x and positive 5x. So those two terms do cancel.
We're always looking for things to cancel along the way. All right. Our distribution process here.
X times X. So X squared plus XH plus another XH. So there's two of those. Plus H squared.
We're left with minus 5h here and minus x squared all over h. Don't forget about him. What else cancels?
Let's see. Just x squared minus x squared. And nothing else combines like it did before, unfortunately.
But I will rewrite this in descending order because it's a little bit more pleasing to the eye. So h squared plus 2xh minus 5h all over h. And I do see that I have a common factor of h in every term.
So if I factor out an h, I'm left with h plus 2x minus 5 all divided by h. I can cancel those h's again as long as I say... That H cannot be zero.
This is the type of algebra that you're going to have to do to complete the calculus. But I want to make sure that you all understand that the calculus that you're going to be doing is very minimal compared to the amount of algebra that you're doing. This is going to be the bulk of the grade for that type of problem. Something that you...
already know how to do from high school. So it's a very important thing for you to kind of go back, see the little nuances like parentheses, distributing through negatives, making sure you're multiplying your binomials properly, canceling things as you go, making sure that you're being very careful with these types of things, following the process, making it look nice and smooth, easy for us to follow. And as long as you're able to do this, the way it looks here on this paper, then... The calculus part is easy.