 So what else can we do with the tangent line? So let's write the equation of the line tangent to the graph of y equals square root of x at 4, 2. To write the equation of a line, we need a point on the line. Well, this point 4, 2 seems to be good. We also need the slope of the line. And remember, the derivative is the slope of the tangent line. So we'll find the derivative of y equals square root of x. And since our point is at x equals 4, we'll want to evaluate the derivative at x equals 4. It's easiest to do so if we rewrite the derivative without negative or fractional exponents. And so our derivative is going to be one-fourth. And so we know the tangent line passes through the point 4, 2 with slope one-fourth. So its equation is going to be... Historically, two different methods of tangents were invented in the 1630s. One by Rene Descartes and the other by Pierre de Fermat. And there was a bit of a rivalry between the two, and Descartes challenged Fermat to use his method to find the slope of the tangent line to a curve that has since become known as the folium of Descartes. That's this curve x cubed plus y cubed equals something xy. And Fermat said, no problem. Well, actually he was French, so he probably said something like, c'est vrai facile. And we can do the same thing without too much more difficulty. So let's find the derivative. And because this function is defined implicitly, we'll use implicit differentiation. And after all the dust settles, we have this formula for the derivative. Now, our derivative requires the values of both x and y. Fortunately, we have them. And so substituting those in, we find the slope of the tangent line. And so we find that our tangent line has slope negative 1 and passes through the point 2, 2, so the equation will be... So why do we care about tangent lines? Geometrically, if we look at the curve and a line tangent to the curve at a given point, the tangent line is going to very closely match the curve itself. So if we zoom in on the curve, we see that if we're not too far from the point of tangency, the tangent line will be a very good match to the curve. And this leads to the following idea. The tangent line is a good approximation to the curve through the point of tangency. Well, that's a geometric idea. Algebraically, it corresponds to the following useful idea. The line tangent to the graph of y equals f of x at x equals x0 will go through the point x0, f of x0, and have slope f prime of x0. And since we know how to write the equation of a line given a point and the slope, we can write the equation of the tangent line. But if the tangent line approximates the curve, then the equation of the tangent line will be an approximation for the function itself. And consequently, our function will be approximately equal to the equation of the tangent line as long as our x value is close to x0, our point of tangency. This leads to the following idea. Suppose I want to approximate a function value. What I want to do is find an x0 close to my x1, where both the function and the derivative are known exactly. And for our purposes, we'll say that a value is known exactly if it can be found using the basic arithmetic operations, addition, subtraction, multiplication, and division. So for example, 5 plus 3 over 17. Or it has a known rational value, for example, cosine of pi log of 1. Or it has a value that you are given as exact, e or pi. Any other value we'll say is not known exactly. So for example, square root of 2, sine of 5, e squared. Because these can't be found using the basic arithmetic operations, because these don't have a known rational value, and because we're not given these as exact values, these, for our purposes, are not values that we know exactly. Next, because we carefully chose x0 so that we knew the function value and the derivative, then we know the point of tangency, and we know the slope of the tangent line. And so now we can write the equation of the tangent line through the point of tangency, and then use the y value of the tangent line at x1 to approximate the value of the function at x1. So for example, let's use the tangent line to approximate the square root of 4.1, and let's evaluate the accuracy of this approximation. So to proceed, we'll need square root of 4.1 to be the value of some function, and we have our choice of functions. For example, we may let f of x equals, oh I don't know how about, square root of 4 sine 3x plus 0.1. Then, square root of 4.1 is going to be our function evaluated at pi over 6. Or maybe f of x would be log of x, then square root of 4.1 is going to be our function evaluated at e to the power 4.1. And we can use any function we want, so we'll choose the hardest possible function. Or we can use the simplest possible function, if f of x equals square root of x, then square root of 4.1 is going to be our function evaluated at 4.1. Since we want f of 4.1, we want an x value that is close to 4.1, where f of x and f prime of x are known exactly. So let's choose, how about x equals 100? Well, that's a little too far away. How about x equals 4.1? Well, if we knew the value of f of 4.1, we wouldn't bother with the approximation. How about x equals 5? Well, that would be good. That's close, but we don't know the value of f of 5, because that would be the square root of 5, and we don't know how to find that. How about 4? So 4 is close to 4.1. f of 4 is the square root of 4, which is 2, and so we know that exactly. And if we find f prime of x, we find it's equal to, and if x equals 4, this will be, which we know exactly. And so we know the tangent line through x equals 4 has slope 1-4th and passes through the point 4-2. So its equation is going to be, and the tangent line approximates the function. So if I want to find f of 4.1, I'll substitute x equals 4.1 into this equation and get. Now how can we evaluate the accuracy of the approximation? Well, it might be tempting to find square root of 4.1 on your calculator, but here's a useful idea to keep in mind. You don't check an answer by re-solving the problem. In this particular case, finding square root of 4.1 on your calculator is using a different approximation. So how do we know that's correct? And so what we might do is we might fall back on what it means to be the square root. We want a number that, when multiplied by itself, gives us 4.1. So in this case, we might find 2.025 squared and see that it's equal to, and so 2.025 does appear to be very close to the actual square root.