 Welcome back to our lecture series Math 12-10, Calculus I for student at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Misseldine. This is the first of actually a three-part lecture, which we're going to introduce the idea of the derivative. Formally speaking, the derivative is the limit of a difference quotient. Now, in this video, the first part, lecture 16, we're going to introduce, well, in this video we'll talk about the tangent line, but in the other videos for this lecture, we'll also talk about the so-called velocity problem. We'll be computing limits of different quotients to see from a geometric or physical point of view why we would care about that. And then in later videos, we'll talk about why these are actually derivatives and therefore why we care about derivatives. Now, to motivate this from a geometric point of view, let's switch over to the graphing calculator. So you see the picture of the unit circle on the screen here graphed using Desmos. Now, for a circle, a so-called tangent line is a line which intersects the circle at a unique point. As you can see illustrated right here, the point in play, the point of intersection between the line and the circle is just a one single point. There's just the one point. We call this a tangent line. All right, this is in contrast to a so-called secant line where a secant line is a line that intersects the circle in two different places, right? So you have one intersection on the top and one intersection here on the right. So a secant line has two points of intersection and a tangent line has a single point of intersection. So in this video, we wanna talk about tangent lines of a curve, that is to say the graph of a function. Intuitively, the tangent line to an arbitrary curve at a point on that curve should touch the curve at just the single point, let's call it P and not at any points nearby as indicated by the curve itself. So let me take a look of what I mean by such a thing. So consider the following curve, right? So this is just a parabola, right? In which case, if we were to talk about a secant line, a secant line of the parabola, it should be a line that essentially hits the curve at two different places. Now I do recognize that with a parabola like this, it could be that curvature of the graph eventually comes down and it could be that this intersects in maybe three or more places. What we mean by a secant line here is that it's intersecting the graph at two points, far away from each other. A tangent line on the other hand is gonna be something that we can see right here that turning off the secant line for a moment. A tangent line ideally should only hit the graph at one point, it's just basically kissing the graph. Again, admittedly the curvature of the graph might cause it to intersect somewhere else, but the idea is locally it's intersecting the graph in only one place right there. So formally stated, what is a tangent line? A tangent line of a graph, which that graph will be given as y equals f of x, the tangent line at a point, we'll call it a comma f of a. So this right here is our point of tangency. So we'll call it P right here, a comma f of a. This is our point of tangency. So our tangent line is gonna touch the function just at this single point P, which a just represents the x-coordinate that we're trying to find the tangent point at. And then f of a because it's a functions graph, the y-coordinate is determined by the function relationship f of a right there. So the tangent line to the graph y equals f of x at the point of tangency a comma f of a is the line through this point whose slope is given as, we'll all lap around the slope in just a second. When it comes to a line, we can always describe a line using the so-called point slope form, point slope form of the line. In which case, this will be given as y minus y1 is equal to m times x minus x1, where in this situation, the coordinates x1, y1, this represents a specific point on the line. As we're trying to construct tangent lines, we will use the point of tangency as these values. So x1 will be a and y1 will be f of a. So we have the point we want for this line. So to finish the equation of line, we need to know the slope. That's the hard part of finding a tangent line. Now, of course, if you have a line in point slope form, it can always be converted to slope intercept form y equals mx plus b. We can find that out later, but we were gonna start with the point slope form. We need to know the slope. And so that's the question at hand right here. How do you find the slope of a tangent line? Well, the slope of the tangent line, which we'll call it m, this is gonna be the limit as b approaches a of f of b minus f of a divided by b minus a. So let's investigate this expression for a moment, okay? So when you look right here, f of b minus f of a over b minus a, this is something we have seen before. This is what we call delta y over delta x with respect to the interval a to b. That is to say, well, I mean, in an expanded form, this is f of b minus f of a over b minus a. This is our so-called average rate of change. Average rate of change. Which the average rate of change, well, I mean, as the name suggests, it tells you on average, how does the function change from x equals a to x equals b? And more important to the question we have about tangent lines right now, the average rate of change measures the so-called slope of the secant line. The line that passes through a comma f of a and the other point b comma f of b. So this blue box right here, now it became a white box, this box is the average rate of change. It's the slope of the secant line that goes through x equals a and x equals b, okay? Now what we are doing in this situation for this slope of the tangent line is we are computing the limit as b approaches a of this slope of a secant line. And we claim that'll be the slope of the tangent line written in a slightly different way. This is equivalent to saying the limit as h approaches zero of f of a plus h minus f of a over h right here. And so the idea here is if h is the difference, right? If h is the difference between b and a, that would imply that b itself is just a plus h. So instead of focusing on two different points, this parameter h is representing the distance between the two points. So you have this fixed point x equals a, the point of tangency, and then you have some variable point, this parameter b that slides. Well, if the distance between the fixed point and the sliding point is h, then you can rewrite in the following way. This is a form we'll use a lot more in the future. In this video, as we're focusing on slopes of tangent lines and secant lines, we'll prefer the more traditional slope formula for this. So to illustrate this point, let's consider the function y equals x squared plus two, which is just the standard parabola that's been shifted to above the x-axis. You can see the x-axis down here. So we just have a good looking parabola right here. And so let's consider finding the tangent line at the point negative one comma three. So in this consideration here, our a value is gonna equal negative one, then f of a, that is f of negative one, is gonna turn out to be three. And you're looking at this point right here on the graph. So if we wanted to find a secant line that goes to that point, well, we'll just find the line. We'll do the algebra for this in just a second. We're gonna take the point negative one comma three, that's our point of tangency, that's our fixed point. And then we can take some other point over here at the moment it's 1.416 and then it's corresponding y-coordinate 4.005, right? You know, we could take any value we want, we could take x equals one, we could take x equals two, which is a little bit off the screen. Again, this point over here on the right, we consider a slider, right? So this b-point can move around. It's the a value that's gonna remain fixed with its corresponding y-coordinate. So as we slide b around, we get all these various different secant lines with all their various slopes. You can see how this changes in real time here on the screen. So as I bring b closer and closer and closer to a, you can see this on the screen. You can start to see that our secant line is approximating a line that's gonna touch it at just one single point, okay? And so we bring it closer and closer and closer. And then at a moment, you notice what just happened, the line disappeared. At the current moment, you'll notice that b is actually equal to negative one. Negative one was our a value, right? So currently b is equal to negative one, a and b are equal to each other. So this is the moment where b is actually touching a and the line disappeared. And I'll show you just a second as we look at the algebraic properties of this thing is that when b touches a, the limit of the secant slope, the limit of the arbitrary change actually becomes the function, well, it becomes the form zero over zero, which is undefined. And so even though we take the form zero over zero, the limit of the secant slope will still be well-defined. I'm just trying to illustrate to you here that as we bring it closer and closer, it starts to approximate something, although we can't allow it to touch because then we'd have to divide by zero and that's not a number, right? And so if we turn off the secant for a moment, what happens when they touch is we actually form the so-called tangent line, this line of the curve that just touched us was one single point. And notice our orange curve, right? When b is far away from a, they don't at all look like each other. But as you bring b closer and closer and closer to the a, you see that these lines become almost identical and they would converge, the secant line converges to the tangent line when b takes the limit towards a right there. So let's consider this exact same problem from a purely algebraic point of view. Let's consider the graph f of x equals x squared plus two, like we saw a moment ago. Let's find the slope and equation of the secant line when x equals, when it goes from one to two, right? So remember, negative one was our fixed value you considered a moment ago. Let's consider some alternative value x equals two, all right? So we first wanna find the slope m. If you're finding the slope of a secant line, this is gonna be the average rate of change delta x over delta y as x ranges from negative one to two, which will have the formula f of two minus f of negative one over two minus negative one for which we're gonna evaluate the function at two. So we get two squared plus two, and then we subtract from that negative one squared plus two. This sits above two minus negative one, which is it's a double negative, you've got two plus one right there. And so we're gonna end up with two squared, which is four, plus two, which is six. You're gonna get negative one squared, which is one, plus two, which is three, and then the denominator turns out, in this case, to be three. So you get six minus three, which is equal to three over three. This is gonna simplify to be one. The slope of the line in this situation is one. So using the point slope form we saw on the previous slide, right? Y minus for y one equals m times x minus x one. Let's use the point of tangency here, x equals negative one, the y corner turned out to be three. We're gonna see here our line looks like y minus three is equal to one times x minus a negative one. Simplifying the right-hand side, you'll notice that times in my one won't do anything. And if you have x minus a negative one, that's gonna be x plus one. So you get three, or y minus three is equal to x plus one. If we add three to both sides of the equation, our equation will then be in slope intercept form. We get y equals x plus four, which is then the equation for the secant line from negative one to two in slope intercept form. Coming back to our picture we saw previously, let's turn off the tangent line. If I slide b over to b two, and I'll zoom out a little bit, you can see that right here. This line, it's y intercept is in fact four. We can see that very clearly. And what about the slope? Notice if I go up one over one, yep, that's another point on the graph. This is the correct function we see right here, the correct secant line. Let's now consider finding the tangent line with its tangent slope as we saw previously. And let's consider the tangent at x equals negative one. So like we saw before, the slope of a tangent line, this is gonna be the limit as b approaches a of this average area change delta y over delta x on the interval a to b are more simply put. We're gonna take the limit as b approaches. In our case our a value is negative one. So we're gonna get the limit as b approaches negative one. We're gonna take f of b minus f of negative one over b minus negative one. And so what's the limit or what's f of b? Well, because of the function it's x squared plus two, we're gonna get b squared plus two. I don't need to compute f of negative one again. I've already done it twice now graphically and numerically here. We saw that's gonna be three. And the denominator, you're gonna get b plus one. And we're taking the limit as b approaches negative one. Notice what happens if we just plug in b equals negative one right here. If you just plug that in, you're gonna end up with negative one squared plus two minus three over negative one plus one for which when you simplify that you end up with zero over zero which zero zero is not a number but we're taking the limit of this difference quotient right now. And therefore perhaps if we simplify the difference quotient before plugging in negative one we can still compute the limit. The limit still can exist in this setting right here. So let's simplify the numerator a little bit. Take the limit as b approaches negative one here. We're gonna get b squared minus one over b plus one. Now I don't like the b plus one of the denominator because if I plug in negative one for b I'm gonna get a zero again, right? But maybe I could factor the numerator to cancel out the b plus one we saw in the denominator. That's something we've seen many times already. After all b squared minus one is a difference of squares. It'll factor as b minus one and b plus one which sits above the b plus one in which case as we know the factors of b plus one on top and bottom cancel out and then we get the simplified limit, the limit as b approaches negative one of b minus one. And in this situation if we plug in negative one we no longer get that zero over zero form we get negative one minus one which is gonna be a negative two. In which case then if we use that for our point slope form we're gonna get y minus three the y-coordinate of the point of tangency equals the slope which is negative two times x plus one right because we get a minus a negative one right there distribute the negative two throughout you're gonna get negative two x minus two add three to both sides then the slope intercept form of our tangent line will be y equals negative two, negative two x plus one. And so this should be the slope intercept form of our tangent line which coming back to our graph let's turn off the C-can line and turn on the tangent line. So what we can see here is I'll zoom out a little bit our what did we claim was the y intercept it was a plus one right because it's y equals negative two plus one sure enough we can see the y intercept here is gonna be a one right zero comma one what about the slope of this thing? Well now I have two points on the graph if we go down right so we go one, two, three, four down, two over so our slope is gonna look like negative four over two which simplifies to be negative two so in slope intercept form we have a slope of negative two and a y intercept of one this is in fact the correct tangent line this illustrates to us that taking the limit of the C-can slope that is the limit of the average rate of change produces the slope of the tangent line