Difference between revisions of "Taylor polynomial"
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We want the Taylor polynomial to have <math>k</math>-th derivative <math>f^{(k)}(a)</math> at <math>x = a</math>. The [[Derivative/Formulas|Power Rule]] for derivatives gives that the derivative of <math>(x-a)^j</math> is <math>j(x-a)^{j-1}</math> for all positive integers <math>j</math>, and <math>0</math> for <math>j = 0</math> (because when <math>j = 0</math> the function is a constant <math>1</math>). Here the [[Chain Rule]] is used implicitly with the fact that <math>x - a</math> has derivative <math>1</math> for all <math>x</math>. | We want the Taylor polynomial to have <math>k</math>-th derivative <math>f^{(k)}(a)</math> at <math>x = a</math>. The [[Derivative/Formulas|Power Rule]] for derivatives gives that the derivative of <math>(x-a)^j</math> is <math>j(x-a)^{j-1}</math> for all positive integers <math>j</math>, and <math>0</math> for <math>j = 0</math> (because when <math>j = 0</math> the function is a constant <math>1</math>). Here the [[Chain Rule]] is used implicitly with the fact that <math>x - a</math> has derivative <math>1</math> for all <math>x</math>. | ||
− | For <math>m < k</math>, the degree-<math>m</math> term in <math>x - a</math> has <math>k</math>th derivative <math>0</math>, because after <math> | + | For <math>m < k</math>, the degree-<math>m</math> term in <math>x - a</math> has <math>k</math>th derivative <math>0</math>, because after <math>m</math> differentiations the degree of the term will have reached <math>0</math> and then at least one more differentiation ensures that the term is eliminated. |
For <math>m > k</math>, the degree-<math>m</math> term in <math>x - a</math> has <math>k</math>th derivative <math>0</math> at <math>x = a</math>, because the <math>k</math> differentiations leave a term with a positive power of <math>(x - a)</math>, which is zero at <math>x = a</math>. | For <math>m > k</math>, the degree-<math>m</math> term in <math>x - a</math> has <math>k</math>th derivative <math>0</math> at <math>x = a</math>, because the <math>k</math> differentiations leave a term with a positive power of <math>(x - a)</math>, which is zero at <math>x = a</math>. |
Revision as of 16:01, 21 September 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
We want the Taylor polynomial to have -th derivative
at
. The Power Rule for derivatives gives that the derivative of
is
for all positive integers
, and
for
(because when
the function is a constant
). Here the Chain Rule is used implicitly with the fact that
has derivative
for all
.
For , the degree-
term in
has
th derivative
, because after
differentiations the degree of the term will have reached
and then at least one more differentiation ensures that the term is eliminated.
For , the degree-
term in
has
th derivative
at
, because the
differentiations leave a term with a positive power of
, which is zero at
.
The degree- term undergoes
differentiations, leaving a constant term and accumulating all of the factors
for
. As such, its
th derivative is
times its original coefficient for all
, so the coefficient of
should be defined as
.
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
Letting be the degree-
Taylor polynomial of
about
, the Lagrange Error Bound states that
if
is defined and has absolute value at most
on the entire interval
if
or
if
.
The Lagrange Error Bound bounds the true value of both above and below.
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 Maclaurin series is the Taylor series chosen with . The partial sums of the Maclaurin series are the Maclaurin polynomials of
of each degree.
Convergence
If the limit of the Lagrange error bound, given above, is as
goes to infinity, then the Taylor series must converge to the function. Because of the rapidly growing factorial term in the denominator of the error bound, Taylor series often converge for all
. For example, all Taylor series for
,
, and
converge to their respective functions for all
.
However, if the derivatives of successive orders grow fast enough, then the Taylor series may only converge for sufficiently close to
.
Consider the Taylor series of about
. By
and the power rule for derivatives,
for
. Since negative-power functions are strictly decreasing and positive for positive
, the maximum absolute value of
on the interval
is
, so the maximum absolute value of
on
is
. Therefore, for
, the Lagrange error bound evaluates to
If
, then the limit of the above expression as
goes to infinity is
, so the Taylor series must in fact converge to
; but for
, the Taylor series is not guaranteed to converge to
. In fact, the Taylor series for
about
is
which, by the Ratio Test, does not converge at all for
.
It is also possible that a Taylor series converges but not to the value of the function it is meant to approximate. For a well-known example, all of the derivatives of the function
are zero at
, so the Maclaurin series of
converges to
for all
. However,
if
, so the Maclaurin series converges to a different value than
for all
.