# 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.

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Taylor series:
Number of Taylor elements:
Elements:
f(x):

Axes ranges

x-min=
x-max=
y-min=
y-max=

Parameter value

a=
b=
c=

Parameter ranges

a-min=
b-min=
c-min=
a-max=
b-max=
c-max=

Development point

x0=

f(x)=

cl
ok
Pos1
End
7
8
9
/
x
4
5
6
*
a
b
c
1
2
3
-
π
(
)
0
.
+
sin
cos
tan
ex
ln
xa
a/x
^
asin
acos
atan
x2
√x
ax
a/(x+b)
|x|
sinh
cosh
a⋅x+c / b⋅x+c
a+x / b+x
x2-a2/ x2+b2
a / x+b
1+√x / 1-√y
exsin(x)cos(x)
x+a
ea⋅x
a⋅x2+b⋅x+c
$\mathrm{sin}\left(\pi x+\frac{\pi }{4}\right)$
$\mathrm{cos}\left(\pi x+\frac{\pi }{4}\right)$
$\mathrm{tan}\left(\pi x+\frac{\pi }{4}\right)$
${\mathrm{sin}}^{2}\left(\pi x+\frac{\pi }{4}\right)$
${\mathrm{cos}}^{2}\left(\pi x+\frac{\pi }{4}\right)$
${\mathrm{tan}}^{2}\left(\pi x+\frac{\pi }{4}\right)$
$\frac{1}{\mathrm{sin}\left(x\right)}$
$\frac{1}{\mathrm{cos}\left(x\right)}$
$\frac{1}{\mathrm{tan}\left(x\right)}$
$\frac{\mathrm{sin}\left(x\right)}{\mathrm{cos}\left(\pi \cdot x\right)}$
$\mathrm{sin}\left(\mathrm{cos}\left(x\right)\right)$
${e}^{x}\mathrm{sin}\left(x\right)\mathrm{cos}\left(x\right)$
FunctionDescription
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
pow(a, b)Power ab
sqrt(x)Square root of x
exp(x)e-function
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 ...

### Derivatives for the Taylor polynomial

Here are the derivations for the Taylor series elements:

### What is a Taylor development?

The Taylor series is a method of mathematics to approximate a function by a finite sum of powers of one variable. It is named after the mathematician Brook Taylor, who developed it in the 18th century. A Taylor series describes a function f(x) around a given point a, and consists of a sum of powers of (x-a) with coefficients representing the derivatives of the original function at that point a.

A general form of a Taylor series is:

f(x) = f(a) + f'(a)(x-a) + (f''(a)/2!) (x-a)^2 + ... + (f^(n)(a)/n!) (x-a)^n + ...

The Taylor series allows to approximate a function in a neighborhood of the point a by considering only a limited number of derivatives. The higher the number of derivatives considered, the more accurate the approximation. The Taylor series has many applications in mathematics, physics, engineering, financial mathematics, and many other fields. It is used to simplify complex functions, to solve problems analytically, and to find numerical solutions to differential equations.

### 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:

$fn(x)= ∑ k = 0 n f(k)(x0) k! ( x-x0 )k$

### Screenshot of the diagram

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