 Hi everyone, we are going to take a look at a couple of examples of how we can use linearization to do some basic arithmetic. Our first one we're going to look at is we want to figure out what 1.97 cubed is, and we can actually use a little bit of calculus to do that. Certainly, you could sit there and do 1.97 times 1.97 times 1.97, try doing that without a calculator. I don't think you'd like to. So let's look at how we can use the linearization process that we've been talking about to find a pretty good estimation of this value. The first thing we need to determine is the function that's going to be the basis for our linearization. Recall what the linearization equation states. It states that l of x is going to be equal to f of a plus f prime of a times the quantity x minus a. And we've talked about that's really simply a variation on the point slope form of the equation of a line. Since our equation is based upon a particular function, we need to determine a function to use for this. Now since we're cubing a value, it might make sense to use x cubed as our function. We know we're going to have to find the derivative of that, so let's go ahead and do that. That simply is 3x squared. We will need to evaluate that at a particular x value. Now we want to do 1.97 cubed. And if you think back to the whole idea behind linearization, it's the idea that we're using a value close to the targeted x. So since we're targeting 1.97 and we want something close to that, that will be easy to cube. Let's make our a value b2. We need then to do f prime of 2 and that gives us 12. Another thing we're going to need is the actual function value at 2. That simply is going to be 8, 2 cubed. So now we have everything to substitute into our linearization equation. Our linearization then states that L of x will be f of 2, so that was 8, plus f prime of 2, that was 12 times the quantity x minus 2. If you distribute and simplify that, you find that we have a linearization equation of 12x minus 16. Now remember the whole idea behind linearization, it's to allow us to use a simple linear equation to obtain a more complicated value. We saw that in earlier videos that you've watched. If we want then to approximate 1.97 cubed, notice I said approximate, we can do that by substituting 1.97 in place of our x in the linearization equation. So we'd want to do 12 times 1.97, take away 16, when you do that you should get 7.64. So that would be a pretty close approximation for the true value of 1.97 cubed. Go ahead and try it on your calculator, do actually 1.97 cubed and you should notice your answer is pretty close to 7.64. Let's take a look at another example. Now we want to determine square root of 24. We definitely can estimate it, we know it's going to be in between 4 and 5 because 4 squared of course is 16, 5 squared is 25. So it's probably going to be closer to 5 than it would be 4 by the time we're done. Once again we're going to use linearization to do that. As we saw in the last example, we need a function that is going to serve as the basis in our linearization as well as the a value that we will use in the linearization as well. Because we're trying to do square root of 24, let's use the function square root of x. And we need an a value to use, we want that a value to be close to 24 and something that would be nice to take the square root of. Let's use 25. We need to find our derivative which simply is going to be 1 half x to the negative 1 half. We can rewrite that as 1 over 2 square root of x. We need to evaluate that derivative at 25. When you do that you get 1 tenth. Of course we need the original function value at 25, that's simply going to be 5. As we substitute into our linearization we get that L of x is equal to f of 25, remember that's simply 5, plus the derivative value, 1 tenth, times the quantity x minus the a value of 25, distribute and simplify and you should obtain 1 tenth x plus 2.5. We can then use this to approximate the true value of square root of 24. You would have 1 tenth times 24 plus 2.5, that ends up to be 2.4 plus 2.5 and we get 4.9. Remember how we had said earlier that our guess is that it would be closer to 5 than it would be 4. If you go ahead and actually do this on your calculator, again you should find that the true value of square root of 24 is pretty close to 4.9.