Taylor polynomial: Difference between revisions

Latest revision as of 14:39, 7 July 2012

Definition

About a general point

Suppose $n$ is a nonnegative integer. Suppose a function $f$ of one variable is defined and at least $n$ times differentiable at a point $x_{0}$ in its domain. The $n^{th}$ Taylor polynomial for a function $f$ at a point $x_{0}$ in the domain is the truncation of the Taylor series to powers up to the $n^{th}$ power. If we denote the polynomial by $P_{n}(f;x_{0})$ , it is given as:

$P_{n}(f;x_{0})=x\mapsto \sum _{k=0}^{n}{\frac {f^{(k)}(x_{0})}{k!}}(x-x_{0})^{k}$

Note that this is a polynomial of degree at most $n$ . The degree is exactly $n$ if and only if $f^{(n)}(x_{0})\neq 0$ .

About the point 0

Suppose a function $f$ of one variable is defined and at least $n$ times differentiable at a point $0$ . The $n^{th}$ Taylor polynomial of $f$ at 0 is:

$P_{n}(f;0)=x\mapsto \sum _{k=0}^{n}{\frac {f^{(k)}(0)}{k!}}x^{k}$

Note that this is a polynomial of degree at most $n$ . The degree is exactly $n$ if and only if $f^{(n)}(0)\neq 0$ .

Intuition

The $n^{th}$ Taylor polynomial, intuitively, is an attempt to be the best local approximation of $f$ about $x_{0}$ among polynomials of degree $\leq n$ .

Particular cases

Value of $n$	$n^{th}$ Taylor polynomial about $x_{0}$	What it means	$n^{th}$ Taylor polynomial about $x_{0}$ case $x_{0}=0$
0	$f(x_{0})$	This is a constant function whose value is the value of $f$ at $x_{0}$ . Clearly, this is the best approximation for $f$ among approximations by constant functions.	$f(0)$
1	$f(x_{0})+f'(x_{0})(x-x_{0})$	The graph of the function is a straight line that equals the tangent line to the graph of $f$ at $(x_{0},f(x_{0}))$ . The tangent line is intuitively the best linear approximation of the graph of the function, so this makes sense.	$f(0)+f'(0)x$ .

@@ Line 3: / Line 3: @@
 ===About a general point===
-Suppose a function <math>f</math> of one variable is defined and at least <math>n</math> times differentiable at a point <math>x_0</math> in its domain. The <math>n^{th}</math> Taylor polynomial for a function <math>f</math> at a point <math>x_0</math> in the domain is the truncation of the [[Taylor series]] to powers up to the <math>n^{th}</math> power. If we denote the polynomial by <math>P_n(f;x_0)</math>, it is given as:
+Suppose <math>n</math> is a nonnegative integer. Suppose a function <math>f</math> of one variable is defined and at least <math>n</math> times differentiable at a point <math>x_0</math> in its domain. The <math>n^{th}</math> Taylor polynomial for a function <math>f</math> at a point <math>x_0</math> in the domain is the truncation of the [[Taylor series]] to powers up to the <math>n^{th}</math> power. If we denote the polynomial by <math>P_n(f;x_0)</math>, it is given as:
 <math>P_n(f;x_0) = x \mapsto \sum_{k=0}^n \frac{f^{(k)}(x_0)}{k!}(x - x_0)^k</math>
@@ Line 16: / Line 16: @@
 Note that this is a polynomial of degree ''at most'' <math>n</math>. The degree is exactly <math>n</math> if and only if <math>f^{(n)}(0) \ne 0</math>.
+===Intuition===
+The <math>n^{th}</math> Taylor polynomial, intuitively, is an attempt to be the ''best local approximation'' of <math>f</math> about <math>x_0</math> among polynomials of degree <math>\le n</math>.
+==Particular cases==
+{| class="sortable" border="1"
+! Value of <math>n</math> !! <math>n^{th}</math> Taylor polynomial about <math>x_0</math> !! What it means !! <math>n^{th}</math> Taylor polynomial about <math>x_0</math> case <math>x_0 = 0</math>
+|-
+| 0 || <math>f(x_0)</math> || This is a constant function whose value is the value of <math>f</math> at <math>x_0</math>. Clearly, this is the best approximation for <math>f</math> among approximations by constant functions. || <math>f(0)</math>
+|-
+| 1 || <math>f(x_0) + f'(x_0)(x - x_0)</math> || The graph of the function is a straight line that equals the tangent line to the graph of <math>f</math> at <math>(x_0,f(x_0))</math>. The tangent line is intuitively the ''best linear approximation'' of the graph of the function, so this makes sense. || <math>f(0) + f'(0)x</math>.
+|}