 This lesson is on linear approximations, or linearization, and also differentials. Remember, linearization and linear approximation are the same thing. Now, what is a linear approximation? The concept of linear approximation is to use a tangent line to approximate the value of the function at a point which is not at the point of tangency. How do we use it? Well, we need a formula. Well, I said use the tangent line, and that's exactly what we're going to look at. What is the tangent line formula? Well, the formula for the tangent line at a point A, F of A, can be written as y is equal to F of A plus F prime of A times x minus A. Now, let's just review a little bit. F of A is your y value at A. F prime of A is your derivative at A, and of course, A is the x coordinate, or x value. So that is the formula you have been using for the equation of a tangent line. Well, what is the linear approximation formula? Let's look at that. And you'll see the formula for the linear approximation, or tangent line approximation, is L equals F of A plus F prime of A times x minus A. And I believe you will see very readily that the only thing that changes is the L. Instead of using y equals, we use L equals. So we really only have the formula for the equation of a tangent line. How do we use this? Well, let's determine the linear approximation of F of x equals the square root of x cubed plus 4 at A equals 2. Well, if you notice from our formula, we need A, which we know to be 2. We need F of A or F of 2, which will equal the square root of 2 cubed 8 plus 4, which is the square root of 12, which we can make into 2 square roots of 3. We also need F prime at A. So let's determine F prime next. F prime of x is equal to 3x squared over 2 square roots of x cubed plus 4. If we evaluate that at A, then we get F prime of A equals 3 times 4 over 2 times. And we know the square root of x cubed plus 4 is equal to 2 square roots of 3. So we can reduce this a little bit and we get F prime of A is equal to 3 over the square root of 3. You can leave it like that. So to make our linearization formula, we have L is equal to F of A, which we know to be 2 square roots of 3 plus F prime of A, which is 3 over the square root of 3 times x minus 2, which is A. And that is the equation of the tangent line, of course, but it's also our linearization formula. Now, let's go a little bit further on this. Yes, we can create these equations, but how do we really use them? What we look at is how the value of the function compares with the value on the tangent line or the linearization. So on our problem, y equals or F of x equals the square root of x cubed plus 4 at x equals 2. We want to first find the value of 2.3. Remember, we figured out our linearization formula 2, but now let's figure out what our value is at 2.3. So if we substitute 2.3 in 4x, we get F of 2.3, which equals the square root of 2.3 cubed plus 4. And we get 4.021 when we round it. What is the value of L of 2.3, the linearization? Well, remember, our linearization formula is L is equal to 2 square root of 3 plus 3 over the square root of 3 times x minus 2. We will fill in 4x 2.3, so now we have 2 root 3 plus 3 over the square root of 3 times 2.3 minus 2, which in essence is 0.3. And we get 3.984 when we round it. Well, how do these compare would be our next question, which is greater? F of 2.3 or L of 2.3, and of course by calculating we can find that F of 2.3 is greater. Well then our final question is of course, why is it greater? And we have to look at the concavity of this graph to figure that out. So let's go back to our calculator, put our function in x cubed plus 4. Let's do a zoom 6 on it. And if you look at x equals 2.3, what is the concavity of that? Well, it is concave up. And if we draw that tangent line, we see that the tangent line is below the curve. So because the graph is concave up, the tangent line is below the curve. That makes the value on the function greater than the value on the tangent line. So the linearization number is lower. Let's go on to another topic that is similar to linearization, but is developing for us some ideas on what we do next, which would be integration. It is called the differential or differentials. So what is a differential? Well, a differential is nothing more than getting our function y is equal to f of x. Doing the derivative dy dx, and of course, that would make it equal to f prime of x. And instead of saying dy dx, we separate the change in y and the change in x. Remember, most of the time we talk about the derivative of y with respect to x. But now we're saying that change in y is equal to f prime of x times that change in x. Now, what does that mean for us at this time? Well, that means we can change all our derivative formulas to their differential form. The first one is y is equal to sine x. Its differential form is dy is equal to cosine x dx. Another one, y is equal to secant x. Its differential form is dy is equal to secant x tan x dx. Again, not much different from what you learned for just derivatives, but we pull that dx to the other side. And you will be seeing this again when we get to integration. What does a differential really look like? Where does it come from graphically? I've created a curve for you and the tangent line to that curve. So at the point of tangency, we'll call that x. A little further away, we'll call that point x plus delta x. That means this length in here is delta x. This length in here is our delta y. And this length here is dy. So we know that delta y, the change in y over the change in x is equal to dy over dx when the limit as delta x approaches zero. And we know these are true from our definition of the derivative. And we also know that this is all equal to f prime of x. So knowing that this is change in y over change in x, small change in y over small change in x, we can say dy is equal to f prime of x dx. So that is how we look at our differential graphically. Let's do an application problem using differentials. Now remember when we use differential, we're looking at a small change. And in this particular problem, we will be looking for the maximum error, which is a small change. The radius of a sphere was measured to be 20 centimeters with a possible error of 0.5 centimeters. Use differentials to estimate the maximum area of the, we're going to do two parts, a, the volume. Well, the volume of a sphere is volume equals 4 thirds pi r cubed. Now the small change in volume would be dv is equal to 4 pi r squared dr. And if we put our numbers in, we get 4 pi r, which is 20 and squared, and then dr is 0.5. And that's equal to 800 pi. Now this looks like a large change, but let's look at how big that volume is in the first place. So let's calculate the volume. So it's 4 thirds pi 20 cubed. That's equal to 10,666 and 2 thirds cubic centimeters. And that's pi. And that's quite large compared to the 800, of course. So this small change definitely is a fairly small change. Let's do one more application problem. Let's look at the surface area of this sphere. Now the surface area formula is s is equal to 4 pi r squared. Small change ds is equal to 8 pi r dr. If we calculate that, we have 8 pi times 20 times 0.5. Don't think we need a calculator for this one because we know that half of 20 is 10 times 8. That'll give us 80 pi. And remember, this is in centimeters as the one above it was in centimeters. Let's look at how large that actually is. So if we calculate the surface area, which is 4 pi 20 squared, so that gives us 1600 pi. 80 pi is a small fraction of 1600 pi. Always try to use our units when we can. This would be 1600 pi centimeters squared because it's surface area. This concludes our lesson on linearization or linear approximations in differentials.