Taylor Polynomial T3(x) Calculator

Analyze cubic Taylor models for many functions. Inspect derivatives, errors, tables, and plotted trends easily. Build intuition around local curves with structured numerical results.

Calculator

Choose a supported model, enter the expansion point a, then evaluate the cubic Taylor approximation at x.

Function model

Use A for scale, B for growth rate, and D for vertical shift.

Expansion point a

Derivatives are evaluated at this point.

Evaluation point x

This is where T₃(x) and f(x) are compared.

Graph half range

Graph spans from a − range to a + range.

Polynomial Coefficients

Use these only when Polynomial is selected.

Example Data Table

This sample uses f(x) = e^x, expansion point a = 0, and the cubic Taylor model T₃(x) = 1 + x + x²/2 + x³/6.

Model	x	T₃(x)	Actual f(x)	Absolute Error
e^x	0.0	1.000000	1.000000	0.000000
e^x	0.2	1.221333	1.221403	0.000069
e^x	0.4	1.490667	1.491825	0.001158

Formula Used

T₃(x) = f(a) + f′(a)(x − a) + [f″(a) / 2!](x − a)² + [f‴(a) / 3!](x − a)³

f(a) is the original function value at the expansion point.
f′(a), f″(a), and f‴(a) are the first three derivatives at that same point.
x − a measures how far the target point is from the expansion point.
The calculator reports the approximation, exact model value, and approximation error.
Accuracy usually improves when x stays near a.

How to Use This Calculator

Select a supported function model from the dropdown list.
Enter the expansion point a and the evaluation point x.
Provide the model parameters A, B, C, and D when required.
If Polynomial is selected, enter coefficients from c0 through c6.
Set the graph half range to control the visible x interval.
Click Calculate T₃(x) to generate the result, graph, and comparison table.
Use the CSV and PDF buttons to export the displayed output.

Frequently Asked Questions

1. What does T₃(x) mean?

T₃(x) is the third order Taylor polynomial. It uses function values and derivatives up to the third derivative. It approximates a function near the chosen expansion point.

2. Why does the approximation work best near a?

Taylor polynomials are local models. Their coefficients come from derivatives at one point. When x stays close to a, the truncated series usually matches the function more closely.

3. Which functions can this calculator handle?

The calculator supports exponential, sine, cosine, natural log, reciprocal, and polynomial models. These cover many common classroom examples and let you compare exact values with cubic approximations.

4. What do A, B, C, and D represent?

They control scale, rate, phase, and vertical shift. Their exact meaning depends on the selected model. The note below the dropdown explains how each parameter is used.

5. Why can log or reciprocal inputs fail?

Those models have domain restrictions. Natural log needs 1 + Bx greater than zero. Reciprocal needs 1 − Bx not equal to zero. Invalid points stop derivative evaluation.

6. What is the absolute error?

Absolute error is the distance between the exact model value and the Taylor approximation. It shows approximation quality in the same units as the function output.

7. Can I use this for a polynomial function?

Yes. Choose Polynomial and enter coefficients through degree six. The calculator then builds derivatives directly from the polynomial and evaluates its cubic Taylor model.

8. What do the CSV and PDF buttons export?

They export the current results, including summary values and the comparison table. This makes it easier to save calculations, print reports, or share work with others.