 Hello and welcome to a screencast about using a tangent line. So now that you know how to find an equation of a tangent line, how do you use it? And the question of the day is how do you use it to estimate a function value? You notice the word estimate is in bold as well as very, as well as caps. So that's the important part here. We're not finding an exact value, we're doing an estimation. So from the last video you recall the equation of our tangent line to a differentiable function f of x at the point x equals a is y equals f prime of a times the quantity x minus a plus f of a. So given the equation of a tangent line to a differentiable function, I'm not going to tell you what it is, call it f of x. At a point x equals negative 2 is this function. y equals 10 times the quantity x plus 2 minus 16. Which if you watched the last video should look familiar. And we want to estimate f of negative 1.9 and f of negative 2.05. Now you'll notice that these two values that we're going to be estimating are pretty darn close to the value here for a, which is negative 2, okay? The further away you get, the worse your estimation is going to be. But because I stayed pretty close to that value, our estimation is going to be pretty good. Okay, so all we're going to do then is we're just going to take the value that we're given here inside of this function and we're going to go ahead and plug it into our tangent line. So this is going to be approximately equal to 10 times the quantity of negative 1.9 plus 2 minus 16. So once you crunch out that algebra, you should get a value of, let's see, negative 15. So if you did watch the previous video, you notice that this f of x function was actually a quadratic. So if you were to try to square negative 1.9, that's certainly not something you can do in your head or fairly quickly. I mean, you could have called a calculator and do that. But with these numbers that we have here, you could definitely do this one pretty quickly in your head. And you get a pretty good approximation to it. All right, why don't you go and try f of negative 2.5, or sorry, negative 2.05, pause the video and come back and check to make sure you get the same answer that I do. All right, so what you're going to do is again, you're going to go ahead and plug that value in. So we're going to have 10 times the quantity negative 2.05 plus 2 minus 16. And that's going to give us a value of negative 16.5. So again, this is a good way to show why we do an equation of a tangent line is because we now take some algebra that can otherwise be kind of funky or difficult or challenging and we make it into just a line, which is about the easiest algebra you can do. Great, thank you for watching.