Example 1: Approximating Cube Root Function by a 2nd Degree Taylor Polynomial

mes (67)in DTube • 3 days ago

In this video I approximate the cube root function x^(1/3) by using a 2nd degree Taylor polynomial at a = 8. The first step is to obtain the first 2 derivatives of f at x = 8, and then the third derivative to be used for the error estimation. Since the resulting Taylor approximation is not alternating, we can't use the alternating series estimation theorem but instead we can use Taylor's Inequality. For the interval between x = 7 and x = 9, the error is less than 0.0004, which is very accurate. Double checking our result using the Desmos graphing calculator shows that the error is even more accurate: less than 0.0003.

Timestamps:

Approximate cube root function near x = 8: 0:00
Solution to (a): First 3 derivatives of f: 0:23
2nd degree Taylor polynomial at a = 8: 5:51
Taylor approximation of cube root function: 7:12
Solution to (b): Taylor series is not alternating: 7:50
Taylors's Inequality with n = 2 and a = 8: 12:59
Plugging in all values for Taylor's Inequality for the remainder: 17:27
- Remainder is less than 0.0004: 18:50
Double-checking by graphing with Desmos: https://www.desmos.com/calculator/grpvrbzkqp: 19:37
Graphing out the remainder function: https://www.desmos.com/calculator/wvdwj2uval: 20:25
- Graph shows the error is less than 0.0003 when x is between 7 and 9: 22:20