Approximating Functions by Taylor Polynomials

mes (67)in DTube • 4 days ago

In this video, I demonstrate how Taylor polynomials can be used to approximate functions and how to evaluate the accuracy of these approximations by examining the absolute magnitude of the error. The first-degree Taylor polynomial uses only the first derivative of the function and is equivalent to basic linearization or linear approximation. By increasing the number of terms from the Taylor series, we obtain a closer approximation to the function. I illustrate this concept by approximating the exponential function at x = 0. It's important to note that the further we move from x = 0, the more terms are required for the approximation to converge accurately. We can determine the degree of the Taylor polynomial needed to achieve a specific accuracy by estimating the error using methods such as graphing, the Alternating Series Estimation Theorem, or Taylor's Inequality.

Timestamps:

Approximating Functions by Taylor Polynomials: 0:00
First-degree Taylor Polynomial is a linear approximation: 3:23
Derivatives of f match T at a: 4:35
Comparing the exponential function vs its Taylor polynomials: 6:15
- Graphing the exponential function vs Taylor polynomials: 9:36
- Table of Taylor Approximations of the exponential function: 11:59
Taylor Polynomials converge slower at further values from a: 14:39
Absolute value of remainder to specify desired degree of polynomials: 15:15
Taylor's Inequality for determining size of remainder: 16:23
Recap on the Alternating Series Test and Estimation Theorem: 18:03

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