Mathematical Approximations

Explore different methods for approximating functions and calculations

Taylor Series

Approximate functions using infinite sums of polynomial terms centered around a point.

f(x) ≈ ∞ Σ n=0 f(n)(a)(x−a)n n!

Arc Length

Calculate the length of a curve using linear segment approximations.

L = b ∫ a √(1 + [f'(x)]²) dx

Riemann Sums

Approximate definite integrals using rectangular area approximations.

b ∫ a f(x) dx ≈ n Σ i=1 f(xi)Δx

Newton's Method

Find roots of equations using iterative tangent line approximations.

xn+1 = xn − f(xn) f'(xn)

Fourier Series

Decompose periodic functions into sums of sine and cosine waves.

f(x) = a₀ + ∞ Σ n=1 [ancos(nx) + bnsin(nx)]

Euler's Method

Numerically solve differential equations using tangent line steps.

yn+1 = yn + hf(xn, yn)