Calculator
Formula Used
The Taylor polynomial of degree n about center a is:
Pn(x) = Σ [f(k)(a) / k!] (x - a)k, for k = 0 to n.
The actual approximation error is:
Error = |f(x) - Pn(x)|.
The Lagrange bound is estimated as:
|Rn(x)| ≤ M |x - a|n+1 / (n + 1)!.
Here, M is the largest absolute value of the next derivative on the interval between a and x.
How to Use This Calculator
- Select the function you want to approximate.
- Enter the center value where derivatives are taken.
- Enter the target x value for approximation.
- Choose the Taylor polynomial degree.
- Select decimal places and graph point density.
- Press Calculate to show the result above the form.
- Use CSV for term data or PDF for a compact report.
Example Data Table
| Function | Center a | Target x | Degree | Use Case |
|---|---|---|---|---|
| e^x | 0 | 1 | 5 | Estimate Euler growth near zero |
| sin(x) | 0 | 0.5 | 7 | Check trigonometric approximation |
| ln(x) | 1 | 1.2 | 4 | Estimate natural log near one |
| sqrt(1 + x) | 0 | 0.3 | 5 | Approximate root expansion |
Understanding Taylor Error
A Taylor polynomial replaces a smooth function with a finite sum of powers. The result is useful because powers are easy to evaluate. The approximation improves when the degree rises or the input stays close to the center. Error measures the gap between the real function value and the polynomial value. This calculator shows that gap directly, so the estimate is not only symbolic. It also gives a Lagrange style bound, when the selected function allows a safe derivative check.
Why the Center Matters
The center is the point where derivatives are measured. A good center is near the target value. That choice keeps the power term small. It usually makes each added term more effective. For example, estimating sin x near zero works well with a Maclaurin polynomial. Estimating the same function far away may need a higher degree. This tool lets you change the center, degree, and target value. You can compare several setups and see which one controls error best.
Reading the Results
The result panel lists the exact value, polynomial estimate, absolute error, relative error, and a bound. The term table shows every derivative, coefficient, power, and term contribution. Large final terms are a warning sign. They may mean the degree is too low, the input is too far from the center, or the chosen function has a nearby restriction. The curve chart gives another view. It compares the exact curve with the polynomial curve across the interval.
Practical Study Tips
Use small degrees first. Then raise the degree and watch the error fall. Change only one setting at a time. This makes the effect easier to understand. For logarithms, reciprocal functions, and roots, respect the stated domain. Do not cross points where the function is undefined. Download the CSV when you need term details for notes. Download the PDF when you need a quick report for assignments, checks, or teaching. Taylor error is not just a final number. It explains how reliable an approximation is.
The best workflow is simple. Pick a function. Enter the center. Select the degree. Review both errors. If the bound stays high, raise the degree or move the center closer to the target point before trusting it.
FAQs
What is Taylor polynomial approximation error?
It is the difference between the exact function value and the Taylor polynomial value. This calculator reports absolute error, relative error, percent error, and an estimated Lagrange bound.
What does the center value mean?
The center is the point where the derivatives are evaluated. A center close to the target x value usually improves the approximation and lowers the error.
What degree should I choose?
Start with a low degree, such as 3 or 5. Increase it until the error or bound is small enough for your required accuracy.
Why is the Lagrange bound different from actual error?
The bound is a safe estimate based on the next derivative. It may be larger than the actual error because it protects against the worst case on the interval.
Can this calculator handle Maclaurin polynomials?
Yes. A Maclaurin polynomial is a Taylor polynomial centered at zero. Set the center value to 0 and choose your function and degree.
Why do some inputs show domain errors?
Some functions are undefined at certain points. For example, ln(x) needs positive values, and 1 / (1 - x) cannot use x equal to one.
What is included in the CSV file?
The CSV includes the main result summary and the full term table. It lists derivatives, coefficients, powers, term values, and partial sums.
What is the PDF download for?
The PDF gives a compact report of the chosen inputs and final outputs. It is useful for homework notes, teaching records, and quick comparisons.