Difference between revisions of "Taylor polynomial"
(Basic facts; will add details later.) |
(Added details for "Special cases" and Maclaurin series information) |
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==Special cases== | ==Special cases== | ||
===Maclaurin polynomial=== | ===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=== | ===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== | ||
+ | |||
− | |||
==Taylor series== | ==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 | + | 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=== | ===Convergence=== |
Revision as of 18:21, 9 March 2022
The degree- Taylor polynomial of a function
about
is the unique polynomial of degree
whose value and first
derivatives match the value and first
derivatives of
at
.
The formula for a degree- Taylor polynomial of
about
is
In the formula above,
denotes the order-
derivative of
.
Taylor polynomials are often used to approximate non-polynomial functions that cannot be calculated exactly, such as trigonometric functions, exponential functions, and logarithms.
Contents
Derivation of the formula
Special cases
Maclaurin polynomial
A Maclaurin polynomial is a Taylor series with . Setting
simplifies the appearance of the polynomial somewhat, since every instance of
in the formula is replaced with
.
For some functions, like and
, Maclaurin polynomials are generally effective across the domain (although using a different
-value might allow greater accuracy for the same choice of degree). However, for functions like
, Maclaurin polynomials cannot be defined because the function and its derivatives are undefined at
. For other functions, Maclaurin polynomials can be defined, but do not in general approximate the function well (see Taylor series), so a value of
closer to the
-value of the desired approximation must be chosen.
Tangent-line approximation
A tangent-line approximation is a first-degree Taylor polynomial, given by . The name "tangent-line approximation" comes from the fact that the graph is a line tangent to the graph of
at
. 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 is the infinite series
The partial sums of the Taylor series are the Taylor polynomials of
about
of each degree.
The Taylor series is the Maclaurin series is the Taylor series chosen with . The partial sums of the Maclaurin series are the Maclaurin polynomials of
of each degree.