Graph near a
Results
Static sample: for f(x)=exp(x) at a=0 and n=6, P6(x)=1 + x + x^2/2! + x^3/3! + x^4/4! + x^5/5! + x^6/6!.
| k | Coefficient c_k | Term |
|---|---|---|
| 0 | 1 | 1 |
| 1 | 1 | x |
| 2 | 1/2 | x^2/2! |
| 3 | 1/6 | x^3/3! |
How it's calculated
- For
exp(x), every derivative at0equals1. - Use
c_k = f^(k)(0)/k!, giving coefficients1/k!. - Stop at degree
6for the displayed Maclaurin polynomial.
Worked example: exp(x) at a = 0
Use this sample as an initial render check before the graph and step log update. Set f(x) = exp(x), a = 0, and n = 6. The Maclaurin polynomial should begin with the known terms of e^x.
| Function | exp(x) |
|---|---|
| Expansion point | a = 0 |
| Degree | n = 6 |
| Expected series | 1 + x + x^2/2! + x^3/3! + x^4/4! + x^5/5! + x^6/6! |
| Check | Coefficients should follow 1/k!, and the plotted polynomial should stay close near x = 0. |
FAQ
How are the coefficients c_k computed?
We evaluate each derivative f^{(k)}(a) symbolically and report c_k = f^{(k)}(a)/k!. If symbolic rules fail, the tool switches to a least-squares fit around a and reports approximate values instead.
When does the numeric fallback activate?
Fallback triggers when derivatives are undefined or unstable (for example abs at 0 or logarithms of negative numbers). The polynomial is then estimated from nearby samples, and the step log highlights the switch.
What should I enter first?
Enter a function and choose an expansion point. Then start with a low order such as 3 or 5 so you can compare the polynomial against the original function before increasing the degree.
How precise are the results?
The calculator keeps internal precision and rounds only for display. Agreement depends on the expansion point, the truncation order, and how far x is from the center of the series.
Why can my result differ from another calculator?
Different tools may expand around a different center, stop at a different order, or switch between symbolic and numeric derivative handling. Match those assumptions before comparing coefficients or plots.