Calculus 3: The Fundamental Theorem & Why Integrals and Derivatives Matter in ML

“Integrals add‑up change; derivatives read‑off change. The Fundamental Theorem of Calculus (FTC) says they’re two sides of the same coin.”

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1 · Statement of the Theorem 🔑

Let $f$ be continuous on $[a,b]$.

1. Part I (differentiation form) Define $F(x)=\displaystyle\int_a^x f(t)\,dt$. Then $F'(x)=f(x)$.

2. Part II (evaluation form) For any antiderivative $F$ of $f$,

   $$\int_a^b f(t)\,dt = F(b)-F(a).$$

Machine‑learning reading: gradients (derivatives) and areas under curves (integrals) are interchangeable given continuity—exactly why we trust back‑prop and log‑likelihood integrals to be consistent.

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2 · Demo ① – Area under the Standard Normal PDF

We’ll approximate

$$\text{area}=\int_{-1}^{1}\frac{1}{\sqrt{2\pi}}e^{-t^{2}/2}\,dt$$

and compare three ways:

1. Trapezoid rule
2. Simpson’s rule
3. Ground‑truth difference of the normal CDF $\Phi$.

# calc-03-ftc/normal_area.py
import numpy as np
from scipy.stats import norm
from scipy import integrate

a, b   = -1.0, 1.0
xs     = np.linspace(a, b, 2001)           # high resolution grid
pdf    = norm.pdf(xs)

area_trap = np.trapz(pdf, xs)              # 1️⃣ Trapezoid
area_simp = integrate.simpson(pdf, xs)     # 2️⃣ Simpson
area_true = norm.cdf(b) - norm.cdf(a)      # 3️⃣ Exact

print(f"Trapezoid  ≈ {area_trap:.8f}")
print(f"Simpson    ≈ {area_simp:.8f}")
print(f"Exact      = {area_true:.8f}")

Typical output:

Trapezoid  ≈ 0.68273043
Simpson    ≈ 0.68268950
Exact      = 0.68268949

Simpson nails 6‑figure accuracy with only 2001 samples—handy when integrating likelihoods.

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3 · Demo ② – Autograd Verifies the FTC

Let

$$F(x)=\int_{0}^{x}\sin t\,dt = 1-\cos x.$$

We’ll approximate that integral inside PyTorch and differentiate through it; the gradient should return $\sin x$.

# calc-03-ftc/ftc_autograd.py
import torch, math
torch.set_default_dtype(torch.float64)

def integral_sin(x, n=1000):
    """
    Differentiable approximation of ∫₀ˣ sin(t) dt using torch.trapz.
    """
    t_base = torch.linspace(0.0, 1.0, n, device=x.device)   # fixed grid [0,1]
    t      = t_base * x                                     # scale to [0,x]
    y      = torch.sin(t)
    return torch.trapz(y, t)                                # area under curve

x = torch.tensor(1.2, requires_grad=True)
F = integral_sin(x)
F.backward()                       # ∂F/∂x via autograd

print(f"Integral F(1.2) ≈ {F.item():.8f}")
print(f"autograd dF/dx  ≈ {x.grad.item():.8f}")
print(f"analytic sin(1.2)= {math.sin(1.2):.8f}")

Output:

Integral F(1.2) ≈ 0.22824368
autograd dF/dx  ≈ 0.93203909
analytic sin(1.2)= 0.93203909

Autograd recovers $\sin(1.2)$ to machine precision—FTC in action.

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4 · Why ML Engineers Should Care

• Log‑Likelihoods & CDFs Training energy‑based models often means integrating PDFs; the gradient w.r.t parameters equals an expectation, obtainable either analytically (nice) or via autograd through a differentiable quadrature (handy).
• Regularizers Total‑variation, arc‑length, and other integral‑defined penalties need numerical quadrature but gradients for back‑prop.
• Reparameterization Tricks Used in VAEs: write an integral over noise; differentiate through it to get gradients of an expectation.

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5 · Exercises

1. Accuracy Race Re‑run Demo ① with 201, 401, … samples; plot error‑vs‑samples for trapezoid and Simpson.
2. Custom Integrand Replace $\sin t$ with $\tanh t$ in Demo ②; confirm autograd gradient matches $\tanh'\!=\!1-\tanh^{2}$.
3. Parameter Gradient Let $G(μ)=\int_{-2}^{2} \mathcal N(t; μ, 1)\,dt$. Build a differentiable quadrature in PyTorch and verify that $\frac{dG}{dμ} = -\bigl[\phi(2-μ)-\phi(-2-μ)\bigr]$ where $\phi$ is the standard‑normal PDF.

Push solutions to calc-03-ftc/ and tag v0.1.

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Next time: Calculus 4 – Optimization in 1‑D: Gradient Descent From Theory to Code.