Fundamental Theorem of Calculus

Fundamental Theorem of Calculus

The FTC connects differentiation and integration -- the two central operations of calculus.

Part 1

If (f) is continuous on ([a, b]) and (F(x) = \int_a^x f(t),dt), then (F) is differentiable and (F'(x) = f(x)).

Part 2

If (f) is continuous on ([a, b]) and (F) is any antiderivative of (f), then:

$\int_a^b f(x)\,dx = F(b) - F(a)$

Numerical Verification

We can verify the FTC numerically:

1. Compute (F(x) = \int_a^x f(t),dt) using numerical integration
2. Compute (F'(x)) using numerical differentiation
3. Check that (F'(x) \approx f(x))

Simpson's Rule

For more accurate numerical integration, use Simpson's rule:

$\int_a^b f(x)\,dx \approx \frac{\Delta x}{3} \left[f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + \cdots + f(x_n)\right]$

where (n) must be even.

Your Task

Implement:

1. integrate(f, a, b, n) -- Simpson's rule with (n) subintervals ((n) must be even)
2. antiderivative_at(f, a, x, n) -- computes (F(x) = \int_a^x f(t),dt) using Simpson's rule
3. verify_ftc(f, a, x, h, n) -- checks that the numerical derivative of (F(x)) approximately equals (f(x)) (within 0.001). Uses (F'(x) \approx (F(x+h) - F(x-h)) / (2h)).