# 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:

Axes ranges

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

Parameter value

a=
b=
c=

Parameter ranges

a-min=
a-max=
b-min=
b-max=
c-min=
c-max=
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 + $\mathrm{sin}$ $\mathrm{cos}$ $\mathrm{tan}$ ${e}^{x}$ $\mathrm{ln}\left(x\right)$ ${x}^{a}$ ^ $\mathrm{asin}$ $\mathrm{acos}$ $\mathrm{atan}$ ${x}^{2}$ $\sqrt{x}$ $\sqrt{x}$ $\sqrt{x}$ $\frac{\left(\right)}{\left(\right)}$ $\mathrm{sinh}$ $\mathrm{cosh}$ $\frac{ax+c}{bx+c}$ $\frac{a+x}{b+x}$ $\frac{{x}^{2}-{a}^{2}}{{x}^{2}+{a}^{2}}$ $\frac{1}{a+bx}$ $\frac{1+\sqrt{x}}{1-\sqrt{x}}$ $\sqrt{x+a}$ $\sqrt{{e}^{ax}}$ ${e}^{\sqrt{x}}$ $a{e}^{-b{x}^{2}+c}$ $\frac{\mathrm{sin}\left(x\right)}{\mathrm{cos}\left(x\right)}$ $a{x}^{2}+bx+c$ ${e}^{x}\cdot \mathrm{sin}\left(x\right)\cdot \mathrm{cos}\left(x\right)$ $\frac{1}{\sqrt{ax}}$ $a{e}^{bx+c}$ ${e}^{ax}$ ${e}^{a{x}^{2}}$ $\frac{1}{{e}^{ax}}$ $\frac{x}{{e}^{x}}$ $\frac{1}{\mathrm{sin}}$ $\frac{1}{\mathrm{cos}}$ $\frac{1}{\mathrm{tan}}$ $a\cdot \mathrm{sin}\left(bx+c\right)$ $a\cdot \mathrm{cos}\left(bx+c\right)$ $a\cdot \mathrm{tan}\left(bx+c\right)$ $a\cdot {\mathrm{sin}}^{2}\left(bx+c\right)$

### 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:

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

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

Constants

NameDescription
LN2Natural logarithm of 2
LN10Natural logarithm of 10
LOG2EBase 2 logarithm of EULER
LOG10EBase 10 logarithm of EULER
PIRatio of the circumference of a circle to its diameter
SQRT1_2Square root of 1/2
SQRT2Square root of 2

Trigonometric Functions

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

Logarithm and Exponential

FunctionDescription
pow(x, y)xy
sqrt(x)Square root of x
exp(x)ex
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

FunctionDescription
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.

### Screenshot of the diagram

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