 So one of the more useful features of the derivative is it allows us to do what's called a linear approximation. And this is part of a more general topic called a spline. Roughly speaking, a spline is some sort of polynomial function that I can use to approximate some other function, possibly including another polynomial. And we do this because sometimes the other function that we want to work with is too complicated or it's too messy, or possibly we don't even know the actual values of it, but polynomials are very nice and well-behaved functions. Now because this is an approximation, there are going to be certain restrictions on its use, but we'll discuss that at a later point. So for example, a common approximation in physics is that if I run into sine of x, it is convenient and usually reasonably good to approximate sine of x with just x. And so here I have a messy function, sine of x, and here I have a much easier function to work with x. If I'm a chemist, I might take something like square root 1 plus x squared, and an approximation is that square root 1 plus x squared is approximately equal to 1 plus 1 half x squared. And these approximations are not limited to things that rely on facts and evidence and logic and reason. We often hear things like that from politics, and where a politician might use an approximation like, well, one person said this, so I guess 10 million people are in favor of it. Well, we can't actually explain where that last approximation comes from, but the ones that rely on evidence actually originate out of calculus. And where this comes from is what's called the tangent line approximation. And so this goes back to the idea that the slope of the secant line has as its limit the slope of the tangent line. So I suppose I have some function f of x, where f of a is some value, we know as precisely as we need to, and we might consider our graph of y equals f of x, and because I know the value of f of a, that corresponds to some point a f of a that I can locate precisely. So it's someplace along the graph there. And assuming that I can actually find these things, I have a function, I have a point, I can talk about what the tangent line to the graph at that point looks like. And in general, that tangent line is going to look very similar to the curve. Here's an important point to note. If I don't stray too far away from the point of tangency, that tangent line is pretty good. The farther away I get from the point of tangency, the worse that tangent line looks as an approximation. But as long as I stay close, it is a reasonable approximation. And because I know the tangent line passes through a very specific point, a f of a, and it has a slope that's equal to the derivative of my function at a, well, I have a point, I have a slope, and so that lets me write the equation of the tangent line. And so here I'm just going to represent that equation with l of x. So I have that equation of the tangent line. Now suppose I want to find f of b. Suppose I want to find my function at some other value. Well, again, if I think about the point b f of b, that's some other point along the curve. Maybe it's up here someplace. And so now I want to approximate f of b. Well, that's my y-coordinate. That's the height of that point above the x-axis. Well, I'd like to know where that is. And where I can use a tangent line is that the point b l of b, that point on the tangent line for the same x value, there's some point on the tangent line. And if my point b isn't too far away from my point a, if b and a are close together, the heights of these two points, f of b here and l of b there, are going to almost be the same. And so that means that I can use l of b, my line, to approximate f of b, my function. And so what we can say is that the equation of the tangent line, this l of x, is a linear approximation to the function that's f of x, near x equals a. For example, let's take a look at this. Find a linear approximation to square root of x, near x equal to 4, and then I want to use that to approximate square root of 4.1. So here's two things to note about this. I have a linear approximation to a function, but I have to specify where that linear approximation is going to be centered. And so here it's centered near x equals 4. Why would I do that? Well, there's two reasons why I'd pick that point. One, square root of 4 is a value that I know exactly. Square root of 4 is equal to 2. And the second is that when I approximate something, I want to pick a value that's close by. So here I want to approximate square root of 4.1, so I'm going to pick something close by that has an exact square root that I can find. And that's actually the hardest part of writing any linear approximation is determining where we're going to center it, because the rest of it is just finding the equation of a tangent line. So first off, I can write the equation of the line tangent to the graph of y equals square root of x at x equals 4. Let's go ahead and graph y equals square root of x, first of all. I'm centering it around x equal to 4, so I'll find where that point is located. At x equal to 4, y is square root of x, y is equal to 2, so I have the point x equals 4, y equals 2, some place on the graph. And if I want to write the equation of the tangent line, I have a point on the line, so the other thing I need is I'm going to need the slope of the tangent line. Well, actually that's the derivative at x equal to 4. If you only want to learn one thing in calculus, you'll probably fail the course. But if you learn many things in calculus, one of the things you do want to learn is that the derivative and the slope of the tangent line are equivalent objects, that the derivative will tell you what the slope of the tangent line is. So let's go ahead and find the derivative, y equals square root of x. I'll do a little bit of algebra, that's x to the power of 1 half, and I now have to differentiate x to the n. So I'll differentiate it, and since I do need to evaluate the derivative at a point, it helps to rewrite it so I don't have any negative or fractional exponents. And so at x equals 4, my derivative, I'll substitute that x value in, and after all the dust settles, I get derivative equals a quarter. Derivative is slope. So now I have the slope of the tangent line, I have a point on the tangent line, and now I have everything I need to write the equation of the tangent line. So, pick your favorite method of writing the equation of a line. Here's one from pre-calculus, or from even earlier, algebra. This is our point-slope method of writing the equation of the line, and I'll just substitute in the coordinates of my point and the known slope. And that gives me my equation of that tangent line. And again, the reason that this is actually useful is that the function, square root of x, is going to be approximated by the linear equation, the equation of our tangent line, one-quarter quantity x minus 4 plus 2. So if I want to find an approximation to square root of 4.1, I'll let x equal 4.1. Left-hand side, I have no idea what that is, so the right-hand side is just a polynomial, and I can evaluate that using just addition, subtraction, multiplication, and occasionally division by some number. And so, after all the dust settles, I get an approximation 2.025 as my square root of 4.1. Now, the natural question to ask at this point is how good of an approximation is that? And one possibility is I could find the square root of 4.1. Philosophically, though, there's kind of an objection to that. If I could find the square root of 4.1, why am I bothering to get an approximation to it? The idea here is that I'm finding a linear approximation because I don't actually have a good method of finding the value otherwise. So, how do I know that square root of 4.1 is actually close to 2.025? And so one thing we might do, remember the square root of a number is the thing when squared that gives us the radicand. So what I might do is I might want to check out what 2.025 squared is. And that is just an arithmetic operation I can do using multiplication. And I can get an exact value, 4.100625, which is actually pretty close to my desired value, 4.1. And so that says the square of 2.025 is almost 4.1, which suggests that the square root of 4.1 is reasonably close to 2.025.