Calculator input
Use the responsive grid below. Large screens show three columns, medium screens show two, and phones stack fields in one column.
Formula used
The calculator applies the truncated Taylor expansion around a center a:
Each coefficient is cₙ = f(n)(a) / n!. The calculator sums the first chosen terms, compares the result with the exact value, and reports the next omitted term as a quick truncation clue.
- ex: every derivative remains ea.
- sin(x) and cos(x): derivatives cycle every four orders.
- sinh(x) and cosh(x): derivatives alternate every two orders.
- ln(1+x): higher derivatives follow alternating signs and powers of
1 + a. - 1 / (1 - x): coefficients depend on the distance from the singular point at x = 1.
How to use this calculator
- Choose one of the supported functions.
- Select Taylor mode for any center or Maclaurin mode for a = 0.
- Enter the evaluation point, term count, decimals, and graph range.
- Press the calculate button to show the result above the form.
- Review the polynomial, coefficient table, graph, and error metrics.
- Download the coefficient table as CSV or save the results as PDF.
Example data table
| Function | Center a | x | Terms | Approximation | Exact value | Absolute error |
|---|---|---|---|---|---|---|
| e^x | 0 | 1 | 6 | 2.716667 | 2.718282 | 0.001615 |
| sin(x) | 0 | 0.5 | 5 | 0.479427 | 0.479426 | 0.000002 |
| ln(1 + x) | 0 | 0.3 | 5 | 0.262461 | 0.262364 | 0.000097 |
FAQs
1. What does this calculator return?
It returns a truncated Taylor polynomial, an approximate function value, the exact value, absolute and percent error, a coefficient table, and a comparison graph.
2. What is the difference between Taylor and Maclaurin series?
A Taylor series expands around any center a. A Maclaurin series is the special case where the center equals zero.
3. Why can the approximation still be poor?
Accuracy depends on the number of terms, the chosen center, and whether the evaluation point sits inside the convergence radius for that function.
4. Why are only certain functions supported?
This version uses exact derivative rules for common functions. That keeps the coefficients stable, transparent, and fast without adding a symbolic algebra engine.
5. What does the next omitted term mean?
It is a quick estimate of the first neglected contribution. Small omitted terms often suggest better accuracy, though they do not replace a full remainder bound.
6. Can I use negative centers?
Yes, as long as the selected center remains inside the function domain. For example, ln(1+x) requires a greater than negative one.
7. Why does convergence radius matter?
The radius shows how far the series can reliably extend from the center before singularities or divergence limit the approximation.
8. What downloads are included?
You can export the coefficient table and result summary as CSV, and you can save the visible result section, including the graph, as a PDF.