Calculator Inputs
This page uses a single-column structure for the content sections, while the calculator fields switch to three columns on large screens, two on medium screens, and one on mobile screens.
Formula Used
The calculator uses the Lagrange form of the Taylor remainder:
|Rn(x)| ≤ M × |x - a|n+1 / (n+1)!
Here, a is the expansion center, x is the target point, n is the polynomial order, and M is an upper bound for |f(n+1)(t)| on every point between a and x.
- Step size:
h = x - a - Factorial growth: larger
(n+1)!often shrinks the error bound. - Interval choice: a tighter interval can produce a smaller and more realistic M.
- Guarantee: the result is a proven ceiling when M really bounds the next derivative.
How to Use This Calculator
- Choose a supported function family or switch to a custom derivative bound.
- Enter the Taylor center a, target point x, and order n.
- Provide interval endpoints if you want to control where the derivative maximum is measured.
- Keep the automatic mode for built-in functions, or enter your own verified M value.
- Press Calculate Bound to see the remainder ceiling above the form.
- Use the CSV or PDF buttons to save the result summary.
Example Data Table
| Function | a | x | n | M | Bound | Actual error |
|---|---|---|---|---|---|---|
| e^x | 0 | 0.5 | 3 | 1.648721 | 0.00429354 | 0.00288794 |
| sin(x) | 0 | 0.4 | 5 | 1 | 0.00000568889 | 0.000000324358 |
| ln(1+x) | 0 | 0.5 | 2 | 2 | 0.0416667 | 0.0304651 |
| 1/(1-x) | 0 | 0.2 | 3 | 73.2422 | 0.00488281 | 0.002 |
Frequently Asked Questions
1. What does the remainder bound represent?
It gives a guaranteed upper limit for the truncation error after stopping a Taylor series at degree n. The actual error can be smaller, but not larger, when the chosen M is valid on the full interval.
2. Why is M so important?
M controls the size of the next derivative. If M is too large, the bound becomes conservative. If M is too small, the guarantee fails. Good interval selection usually makes M more realistic.
3. Why can the actual error be much smaller?
The Lagrange bound protects against the worst admissible case over the interval. Real functions often behave more gently than that maximum, so the true approximation error may be far below the bound.
4. Should I always use the interval from a to x?
Usually yes, because the Lagrange theorem only needs the points between the center and evaluation point. A wider interval may still be valid, but it often increases M and makes the bound looser.
5. Why does a higher order often reduce the bound?
The exponent on |x−a| grows, and the factorial in the denominator grows quickly. Those effects commonly dominate, making the remainder bound smaller as the Taylor polynomial order increases.
6. Can I use the custom M mode for other functions?
Yes. If you know a trustworthy bound for the next derivative on the chosen interval, you can enter it manually. The calculator then applies the generic Lagrange remainder formula directly.
7. Why does 1/(1-x) produce warnings near x = 1?
That function has a singularity at x = 1. Derivatives grow rapidly as the interval approaches that point, and the Taylor series around a may fail to converge once the target lies beyond the radius.
8. Are the CSV and PDF files the same as the screen output?
They export the current result summary shown above the form. This makes it easy to save inputs, derivative bounds, and the computed ceiling for reports, homework, or technical notes.