(a) Approximate f by a Taylor polynomial with degree n at...

(a) Approximate $f$ by a Taylor polynomial with degree $n$ at the number a. (b) Use Taylor's Inequality to estimate the accuracy of the approximation $f(x) \approx T_{n}(x)$ when $x$ lies in the given interval. (c) Check your result in part (b) by graphing $\left|R_{n}(x)\right|$ $$f(x)=e^{x^{2}}, \quad a=0, \quad n=3, \quad 0 \leqslant x \leqslant 0.1$$

(a) Approximate $f$ by a Taylor polynomial with degree $n$ at the
number a.
(b) Use Taylor's Inequality to estimate the accuracy of the approximation $f(x) \approx T_{n}(x)$ when $x$ lies in the given interval.
(c) Check your result in part (b) by graphing $\left|R_{n}(x)\right|$
$$f(x)=e^{x^{2}}, \quad a=0, \quad n=3, \quad 0 \leqslant x \leqslant 0.1$$

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Key Concepts

Taylor Polynomial

A Taylor polynomial is an approximation of a function around a given point using a finite sum of its derivatives evaluated at that point. It provides a polynomial that represents the function locally, making use of the formula f(x) ? ? (f^(k)(a)/k!) (x - a)^k, where k runs from 0 to n. This concept is fundamental in approximating functions using simpler polynomial forms for analysis or computation.

Taylor's Inequality

Taylor's Inequality is a tool used to estimate the remainder or error term when approximating a function by a Taylor polynomial. It provides an explicit bound on the error between the actual function and its polynomial approximation, often depending on the next derivative beyond the highest used in the polynomial and ensuring that the approximation is within a predictable range over an interval.

Remainder Term and Error Analysis

The remainder term, often expressed in the Lagrange form, quantifies the error in approximating a function by its Taylor polynomial. By analyzing the remainder, one can assess the accuracy of the polynomial approximation. This approach is especially useful when validating approximation results through additional methods like graphing the absolute error over a specified interval to visually assess its behavior.

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