Difference between revisions of "Taylor polynomial"

Revision as of 17:21, 9 March 2022

The degree- $n$ Taylor polynomial of a function $f(x)$ about $x = a$ is the unique polynomial of degree $n$ whose value and first $n$ derivatives match the value and first $n$ derivatives of $f(x)$ at $x = a$ .

The formula for a degree- $n$ Taylor polynomial of $f(x)$ about $x = a$ is $\sum_{k=0}^{n} \frac{f^{(k)}(a)(x-a)^k}{k!} = f(a) + f'(a)(x - a) + \frac{f''(a)(x-a)^2}{2} + \dots + \frac{f^{(n)}(a)(x-a)^n}{n!}.$ In the formula above, $f^{(k)}$ denotes the order- $k$ derivative of $f$ .

Taylor polynomials are often used to approximate non-polynomial functions that cannot be calculated exactly, such as trigonometric functions, exponential functions, and logarithms.

Derivation of the formula

Special cases

Maclaurin polynomial

A Maclaurin polynomial is a Taylor series with $a = 0$ . Setting $a = 0$ simplifies the appearance of the polynomial somewhat, since every instance of $(x-a)$ in the formula is replaced with $x$ .

For some functions, like $e^x$ and $\sin x$ , Maclaurin polynomials are generally effective across the domain (although using a different $a$ -value might allow greater accuracy for the same choice of degree). However, for functions like $\ln x$ , Maclaurin polynomials cannot be defined because the function and its derivatives are undefined at $x = 0$ . For other functions, Maclaurin polynomials can be defined, but do not in general approximate the function well (see Taylor series), so a value of $a$ closer to the $x$ -value of the desired approximation must be chosen.

Tangent-line approximation

A tangent-line approximation is a first-degree Taylor polynomial, given by $f(a) + f'(a)(x - a)$ . The name "tangent-line approximation" comes from the fact that the graph is a line tangent to the graph of $f(x)$ at $x = a$ . Tangent-line approximations are used in Euler's method and Newton's method.

Error bound

Taylor series

The Taylor series of an infinitely differentiable function $f(x)$ is the infinite series $\sum_{k=0}^{\infty} \frac{f^{(k)}(x-a)}{k!} = f(a) + f'(a)(x - a) + \frac{f''(a)(x-a)^2}{2} + \dots.$ The partial sums of the Taylor series are the Taylor polynomials of $f(x)$ about $x = a$ of each degree.

The Taylor series is the Maclaurin series is the Taylor series chosen with $a = 0$ . The partial sums of the Maclaurin series are the Maclaurin polynomials of $f(x)$ of each degree.

@@ Line 9: / Line 9: @@
 ==Special cases==
 ===Maclaurin polynomial===
+A '''Maclaurin polynomial''' is a Taylor series with <math>a = 0</math>. Setting <math>a = 0</math> simplifies the appearance of the polynomial somewhat, since every instance of <math>(x-a)</math> in the formula is replaced with <math>x</math>.
+For some functions, like <math>e^x</math> and <math>\sin x</math>, Maclaurin polynomials are generally effective across the [[domain]] (although using a different <math>a</math>-value might allow greater accuracy for the same choice of degree). However, for functions like <math>\ln x</math>, Maclaurin polynomials cannot be defined because the function and its derivatives are undefined at <math>x = 0</math>. For other functions, Maclaurin polynomials can be defined, but do not in general approximate the function well (see [[Taylor polynomial#Taylor series|Taylor series]]), so a value of <math>a</math> closer to the <math>x</math>-value of the desired approximation must be chosen.
 ===Tangent-line approximation===
+A tangent-line approximation is a first-degree Taylor polynomial, given by <math>f(a) + f'(a)(x - a)</math>. The name "tangent-line approximation" comes from the fact that the graph is a line [[tangent]] to the graph of <math>f(x)</math> at <math>x = a</math>. Tangent-line approximations are used in [[Differential equations#Approximations|Euler's method]] and [[Newton's method]].
+==Error bound==
-==Error bound==
 ==Taylor series==
-The '''Taylor series''' of an infinitely differentiable function <math>f(x)</math> is the [[Sum#Infinite|infinite series]] <cmath>\sum_{k=0}^{\infty} \frac{f^{(k)}(x-a)}{k!} = f(a) + f'(a)(x - a) + \frac{f''(a)(x-a)^2}{2} + \dots.</cmath> The [[partial sums]] of this series are the Taylor polynomials of <math>f(x)</math> of each degree.
+The '''Taylor series''' of an infinitely differentiable function <math>f(x)</math> is the [[Sum#Infinite|infinite series]] <cmath>\sum_{k=0}^{\infty} \frac{f^{(k)}(x-a)}{k!} = f(a) + f'(a)(x - a) + \frac{f''(a)(x-a)^2}{2} + \dots.</cmath> The [[partial sums]] of the Taylor series are the Taylor polynomials of <math>f(x)</math> about <math>x = a</math> of each degree.
+The '''Taylor series''' is the Maclaurin series is the Taylor series chosen with <math>a = 0</math>. The partial sums of the Maclaurin series are the Maclaurin polynomials of <math>f(x)</math> of each degree.
 ===Convergence===

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