Taylor Series Example

In this segment, we will take an example of the Taylor series. So, here's an example we're asked to find the value of f(6), the function at 6, when we are given the values of the function at 4, the first derivative second derivative, third derivative, and all the other higher order derivatives are 0. And we want to find out what the value of the function is at six. Let's go and see how we can go about doing that.

So, we have this as our Taylor series. Hey, give me the value of the function at a single point and all its derivatives at that particular point and I can find the value of the function at any other point, of course with the fine print that the derivatives exist in a continuous from the point where you are to the point where you are going to. And since this is the case now what we have is that x is 4. But we want to find the value at 6, or x plus h. This will be 6, and since h is 4 ,sorry x is 4, we will get h = 2. So, x is 4 h is 2. Let's go and see what happens now.

We have the value of the function at 4 plus 2 is the value of the function of 4, plus the derivative of the function at 4, plus the second derivative of the function at 4, divided by 2 factorial, times 2 squared, plus the third derivative of the function at 4, divided by 3 factorial times 2 cubed, and then we have, 0, 0, 0 onwards because all the higher order derivative terms after the third derivative they're all given to be zero. So, if we simplify this this is what we're going to get. So, we have that. That is just by this is 2 here, that's 2 here, this 2 squared by 2 factorial, that's 2, this is 8, and that is 6. So, that’s what is all here. So now what we're going to do is we're going to substitute the values which are given to us which will give us the value of the function at 4 is given as 1.35. The value of the first derivative of the value of the second derivative is given as 74. The value of the second derivative is given as 30. And then the value of the third derivative is given as 6. So, if we do the calculations, we will get 341 as the value of the function at six. And that's all we're asked to do. And in fact, this is the exact value of the function at six because we have used the value of the function in all its derivatives at the point x = 4, in this case; so, we have the exact value of the function at 6. And that is the end of this segment.