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Taylor's Formula Calculator

Taylor formula calculator. Enter f(x), center a, order n and point to compute the Taylor polynomial and remainder—for approximation.

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f(x) = Σ f^(k)(a)/k!·(x−a)^k + R_n
Supports + - * / ^ and sin cos tan sqrt exp log pi e
Expand about here
Integer 1~6
Compare approximation here

📖 Tutorial｜Taylor's Formula

1. Definition
Taylor's formula: f(x)=Σ f^(k)(a)/k!·(x−a)^k + R_n, a polynomial approximation near a with remainder R_n.
2. Symbols
SymbolMeaning
aCenter
nOrder
f^(k)(a)k-th derivative at a
R_nRemainder (error)
3. How it works
  • We numerically compute derivatives of all orders;
  • Compute coefficients f^(k)(a)/k!;
  • Compare approximation and true value at x0.
4. Steps
  1. Enter f(x);
  2. Enter center a, order n, point x0;
  3. Click Calculate.
5. Example
Example: eˣ at a=0 to order 3: 1+x+x²/2+x³/6. At x0=0.1, P≈1.10517.
6. Pitfalls
Higher order improves accuracy but numerical error accumulates;
The closer a is to x0, the better;
The remainder is an estimate of the error.

❓ FAQ｜Taylor's Formula

What is the essence of Taylor's formula?
Locally approximate any smooth function near a center by a polynomial; higher order is more accurate.
What is the remainder R_n?
The difference between the true value and the Taylor polynomial—the approximation error.
How to choose the center?
Near the point you want to approximate; at x0=a the error is 0.
Relation to Maclaurin?
Maclaurin is Taylor with center a=0.
Is higher order always better?
In theory yes, but numerical error grows at high order; keep it within about 6.