Online calculator Taylor series

The Taylor series is used in calculus to represent a smooth function in the neighborhood of a point by a power series which is the limit of Taylor polynomials. This series expansion is called Taylor expansion. Series and development are named after the British mathematician Brook Taylor.

The calculator can be used to perform a Taylor series expansion on a function. The point around which the polynomial is developed can be moved on the graph. The recalculation is done after selecting the 'Update' button. In the definition of the function the parameters a, b and c can be used and varied by means of the sliders.

Scale screen:

Digits:

Taylor series:

Number of Taylor elements:

Elements:

Offset

Stretch

f(x):

Development point

x₀=

f(x) =

clear	ok					Pos1	Ende
7	8	9		$/$			$x$
4	5	6		*		(	)
1	2	3		-	a	b	c
0	.		π	+	$\sin$	$\cos$	$\tan$
$e^{x}$	$\ln (x)$	$x^{a}$		^	$asin$	$acos$	$atan$
$x^{2}$	$\sqrt{x}$	$\sqrt[3]{x}$	$\sqrt[4]{x}$	$\frac{()}{()}$		$\sinh$	$\cosh$
$\frac{a x + c}{b x + c}$	$\frac{a + x}{b + x}$	$\frac{x^{2} - a^{2}}{x^{2} + a^{2}}$	$\frac{1}{a + b x}$	$\frac{1 + \sqrt{x}}{1 - \sqrt{x}}$	$\sqrt{x + a}$	$\sqrt{e^{a x}}$	$e^{\sqrt{x}}$
$a e^{- b x^{2} + c}$		$\frac{\sin (x)}{\cos (x)}$	$a x^{2} + b x + c$		$e^{x} \cdot \sin (x) \cdot \cos (x)$		$\frac{1}{\sqrt{a x}}$
$a e^{b x + c}$	$e^{a x}$	$e^{a x^{2}}$	$\frac{1}{e^{a x}}$	$\frac{x}{e^{x}}$	$\frac{1}{\sin}$	$\frac{1}{\cos}$	$\frac{1}{\tan}$
$a \cdot \sin (b x + c)$		$a \cdot \cos (b x + c)$		$a \cdot \tan (b x + c)$		$a \cdot \sin^{2} (b x + c)$

Derivatives for the Taylor polynomial

Here are the derivations for the Taylor series elements:

Definition of the Taylor series

If a function f(x) is differentiable enough times, it can be approximated by an nth order polynomial.

The Taylor series is:

$f_{n} (x) = \sum_{k = 0}^{n} \frac{f^{(k)} (x_{0})}{k!} {(x - x_{0})}^{k}$

Usable expressions in the definition of the functions f, g and h

Constants

Name	Description
LN2	Natural logarithm of 2
LN10	Natural logarithm of 10
LOG2E	Base 2 logarithm of EULER
LOG10E	Base 10 logarithm of EULER
PI	Ratio of the circumference of a circle to its diameter
SQRT1_2	Square root of 1/2
SQRT2	Square root of 2

Trigonometric Functions

Function	Description
sin(x)	sine of x
cos(x)	Cosine of x
tan(x)	Tangent of x
asin(x)	arcsine
acos(x)	arccosine of x
atan(x)	arctangent of x
atan2(y, x)	Returns the arctangent of the quotient of its arguments.
cosh(x)	Hyperbolic cosine of x
sinh(x)	Hyperbolic sine of x

Logarithm and Exponential

Function	Description
pow(x, y)	x^y
sqrt(x)	Square root of x
exp(x)	e^x
log(x), ln(x)	Natural logarithm
log(x, b)	Logarithm to base b
log2(x), lb(x)	Logarithm to base 2
log10(x), ld(x)	Logarithm to base 10

More functions

Function	Description
ceil(x)	Get smallest integer n with n > x.
abs(x)	Absolute value of x
max(a, b, c, ...)	Maximum value of all given values.
min(a, b, c, ...)	Minimum value of all given values.
random(max = 1)	Generate a random number between 0 and max.
round(v)	Returns the value of a number rounded to the nearest integer.
floor(x)	Returns the biggest integer n with n < x.
factorial(n)	Calculates n!
trunc(v, p = 0)	Truncate v after the p-th decimal.
V(s)	Returns the value of the given element, e.g. sliders and angles.

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