Taylor Expansion¶

Formula¶

\[ f(x)\approx f(a)+f'(a)(x-a)+\frac{f''(a)}{2}(x-a)^2+\cdots \]

Parameters¶

\(a\): expansion point
\(f^{(k)}(a)\): \(k\)-th derivative at \(a\)

What it means¶

Approximates a smooth function locally by a polynomial around a point.

What it's used for¶

Local approximations and error analysis.
Deriving Newton and second-order optimization methods.

Key properties¶

Accuracy improves near the expansion point (under smoothness).
First-order term gives linearization; second-order adds curvature.

Common gotchas¶

Approximation quality can be poor far from \(a\).
Not every smooth function equals its Taylor series globally.

Example¶

\(e^x \approx 1+x\) near \(x=0\) (first-order Taylor approximation).

How to Compute (Pseudocode)¶

Input: function f, expansion point a, evaluation point x, order K
Output: Taylor approximation T_K(x)

T <- 0
for k from 0 to K:
  deriv_k <- k-th derivative of f evaluated at a
  T <- T + deriv_k * (x - a)^k / k!
return T

Complexity¶

Time: \(O(K)\) term accumulation once derivatives \(f^{(k)}(a)\) are available; derivative computation cost is additional
Space: \(O(1)\) extra space (or \(O(K)\) if storing coefficients)
Assumptions: Scalar Taylor expansion shown; symbolic or automatic derivative computation can dominate runtime