 In this video, we're going to apply the remainder estimation theorem, or Lagrange error bound, to estimate the magnitude of the error in approximating natural log of 1.5 using a third degree Taylor polynomial about x equals 0. To do this, let's consider the function, natural log of x, 1 plus x, on the interval from 0 to 0.5. Since we're using the Taylor polynomial centered about x equals 0, and we're approximating the value of natural log of 1 plus 0.5. Since we're looking at the third degree Taylor polynomial, we want to find the fourth derivative of f to help identify a value for m in the theorem. Pause this video and verify that this derivative is negative 6 divided by 1 plus x to the fourth. On the interval 0 to 0.5, the absolute value of this is largest when x equals 0. So the absolute value of the fourth derivative of f on the interval from 0 to 0.5 is less than or equal to 6. So let's let m equal 6. According to the theorem then, since we were interested in the error associated with the third degree Taylor polynomial, according to the theorem, this error is bounded above by 6 over 4 factorial times x minus 0 to the fourth. Now on the interval from 0 to 0.5, the error in using this third degree Taylor polynomial to approximate natural log of 1.5 is at most 6 divided by 4 factorial times 0.5 minus 0 to the fourth, because that will give us my largest value for this error. And that's equal to 1 over 64, which is equal to 0.15625. And there is the maximum error associated with using the third degree Taylor polynomial to approximate natural log of 1.5.