Taylor Polynomial Calculator

Calculator inputs

Function family

Choose a supported function for exact derivatives.

Parameter p

Used in px for exponential, trigonometric, logarithmic, and rational forms.

Polynomial coefficients

Enter c0, c1, c2... for c0 + c1x + c2x².

Expansion center a

This is the point where derivatives are evaluated.

Evaluation point x

The calculator compares exact and approximate values here.

Polynomial order n

Higher orders usually improve local accuracy near the center.

Plot minimum

Left side of the graph interval.

Plot maximum

Right side of the graph interval.

Graph samples

More samples create smoother curves.

Plotly graph

The graph compares the exact function and its Taylor approximation over your chosen interval. The vertical dashed line marks the expansion center.

Example data table

Function	Order n	x	Example approximation	Use case
e^x	5	1	2.716667	Maclaurin estimate of exponential growth
sin(x)	7	0.8	0.717356	Local trigonometric approximation
ln(1 + x)	4	0.5	0.401042	Series estimate for logarithms
1 / (1 - x)	6	0.4	1.649344	Geometric series comparison

Formula used

T_n(x) = Σ [ f^(k)(a) / k! ] (x - a)^k, for k = 0 to n

The calculator builds the Taylor polynomial from derivatives evaluated at the center a. Each coefficient equals f^(k)(a) / k!, then multiplies the shifted power (x − a)^k. The final approximation is the sum of all terms through order n.

How to use this calculator

Select a supported function family or choose the custom polynomial option.
Enter the parameter p when the selected function uses one.
For a custom polynomial, type coefficients in ascending powers.
Set the expansion center a and the evaluation point x.
Choose the order n of the Taylor polynomial.
Define the graph interval and sample count for plotting.
Press the calculate button to view the summary, coefficient table, and graph.
Use the CSV or PDF buttons to export the result table.

FAQs

1) What does a Taylor polynomial approximate?

It approximates a smooth function near a chosen center using derivatives. Accuracy is usually strongest close to that center and weakens farther away.

2) What is the difference between Taylor and Maclaurin forms?

A Maclaurin polynomial is a Taylor polynomial centered at zero. Any other center value produces a general Taylor expansion.

3) Why does a higher order often improve accuracy?

Higher orders include more derivative information. That usually makes the approximation closer to the original function near the expansion center.

4) Why can the graph differ strongly away from the center?

Taylor polynomials are local models. Outside the nearby region, omitted terms grow in influence, so the approximation can drift from the exact curve.

5) Why is my logarithmic case rejected?

The logarithmic form ln(1 + px) requires 1 + px to stay positive. The center must also lie inside the valid domain for derivatives to exist.

6) What do the coefficient values mean?

Each coefficient equals the corresponding derivative at the center divided by factorial. These coefficients multiply powers of the shifted variable (x − a).

7) Can this tool expand any arbitrary formula?

This version supports selected exact-function families and custom polynomials. That approach keeps derivatives reliable and the calculator fast.

8) When should I use a custom polynomial input?

Use it when your original function is already a polynomial. The tool then computes exact derivatives and the Taylor form around any center.