Mathematics for AI 2

Module 1: Continuous Functions and Limits (Chapter 4)

Continuous Functions

The precise mathematical definition of a continuous function relies on limits. A function $f: D \to \mathbb{R}$ is continuous at $x_0 \in D$ if for all sequences $(x_n)_{n \in \mathbb{N}}$ with $x_n \to x_0$, the limit $\lim_{n \to \infty} f(x_n)$ exists and satisfies:

An equivalent definition is the $\epsilon-\delta$ criterion. A function is continuous at $x_0$ if and only if:

Standard calculation rules apply: if $f$ and $g$ are continuous, then $f+g$, $f \cdot g$, and $c \cdot f$ are continuous. Moreover, the composition $g \circ f$ is continuous. The inverse function of a continuous, bijective function on an interval is also continuous.

Limits and Accumulation Points

An accumulation point $x_0$ of a set $D \subset \mathbb{R}$ is a point where there exists a sequence $(x_n)_{n \in \mathbb{N}}$ in $D$ such that $x_0 = \lim_{n \to \infty} x_n$ and $x_n \neq x_0$ for all $n$. If $y$ is the limit of $f(x)$ as $x$ approaches an accumulation point $x_0$, we write:

The Sandwich Rule states that if $f(x) \leq g(x) \leq h(x)$ for all $x \in D$ and $\lim_{x \to x_0} f(x) = \lim_{x \to x_0} h(x) = L$, then the limit of $g(x)$ exists and $\lim_{x \to x_0} g(x) = L$.

Module 2: Differential Calculus (Chapter 5)

Derivatives and Differentiability

A function $f: (a, b) \to \mathbb{R}$ is differentiable at $x_0$ if the difference quotient limit exists:

This limit is denoted as $f'(x_0)$ or $\frac{df}{dx}(x_0)$. Geometrically, $f'(x_0)$ represents the slope of the tangent line to the graph of $f$ at $x_0$. Any function that is differentiable at a point is necessarily continuous at that point.

Key Calculation Rules:

• Product Rule:
  $$(fg)'(x_0) = f'(x_0)g(x_0) + g'(x_0)f(x_0)$$
• Chain Rule:
  $$(g \circ f)'(x_0) = g'(f(x_0))f'(x_0)$$
• Quotient Rule:
  $$\left(\frac{f}{g}\right)'(x_0) = \frac{f'(x_0)g(x_0) - f(x_0)g'(x_0)}{g(x_0)^2}$$
• Inverse Functions:
  $$(f^{-1})'(y_0) = \frac{1}{f'(x_0)}$$

Extrema and Mean Value Theorems

A necessary condition for a local extreme value at $x_0$ for a differentiable function is $f'(x_0) = 0$ (a stationary point). The second derivative test provides a sufficient condition: if $f'(x_0) = 0$ and $f''(x_0) > 0$, $x_0$ is a strict local minimum; if $f''(x_0) < 0$, it is a strict local maximum.

The Mean Value Theorem asserts that for a continuous function on $[a, b]$ that is differentiable on $(a, b)$, there exists $\xi \in (a, b)$ such that:

L'Hospital's Rule and Taylor's Theorem

For limits producing indeterminate forms $\frac{0}{0}$ or $\frac{\pm \infty}{\pm \infty}$, L'Hospital's rule can be applied:

Taylor's Theorem allows approximating a function using polynomials. The Taylor polynomial of order $n$ at $x_0$ is:

The remainder term $R_n(x) = f(x) - T_n(x)$ takes the Lagrange form:

Newton's Method

Newton's method iteratively finds roots for $f(x) = 0$ using the sequence:

Module 3: Basic Integration Theory (Chapter 6)

Antiderivatives and the Riemann Integral

An antiderivative of $f$ on an open set $\Omega$ is a differentiable function $F$ such that $F'(x) = f(x)$.

The definite integral computes the area under the curve of $f$ over $[a, b]$ by taking the limit of Riemann sums (quadrature rule):

This value is bounded by the limits of the lower sums $L_n(f)$ and upper sums $U_n(f)$.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus connects differentiation and integration:

• Part I: $F(x) = \int_a^x f(y) dy$ is an antiderivative of $f$.
• Part II: For any antiderivative $F$, the definite integral is calculated as:
  $$\int_a^b f(x) dx = F(b) - F(a)$$

Integration Rules:

• Integration by Parts: derived from the product rule.
  $$\int f'(x)g(x) dx = f(x)g(x) - \int f(x)g'(x) dx$$
• Substitution Rule: derived from the chain rule.
  $$\int_a^b f(g(x))g'(x) dx = \int_{g(a)}^{g(b)} f(y) dy$$

Module 4: Fourier Series (Chapter 7)

Trigonometric Polynomials and Fourier Coefficients

Fourier series approximate periodic functions (e.g., $f(x+1) = f(x)$ ) via superposition of waves. A trigonometric polynomial is defined as:

By Euler's formula $e^{it} = \cos(t) + i\sin(t)$, this can be written using sines and cosines.
For a given periodic function $f$, its $k$ -th Fourier coefficient is:

Convergence

The $n$ -th partial sum is:

• Point-wise convergence: $\lim_{n \to \infty} f_n(x) = f(x)$ for each $x$.
• Uniform convergence: $\lim_{n \to \infty} \sup_{x \in D} |f_n(x) - f(x)| = 0$.
  Dirichlet's theorem dictates that if $f$ is periodic, piecewise continuous, and differentiable at $x_0$, its Fourier series converges point-wise to $f(x_0)$. If $f$ is twice continuously differentiable, the Fourier series converges uniformly.

Module 5: Multivariate Calculus (Chapter 8)

Scalar Fields and Vector Fields

Functions mapping $D \subset \mathbb{R}^n \to \mathbb{R}$ are scalar fields (e.g., population density). Functions mapping $D \subset \mathbb{R}^n \to \mathbb{R}^m$ are vector fields (e.g., wind velocity).

Norms and Distances

A norm assigns a non-negative length to vectors. Common $p$ -norms include:

• 1-norm: $\|v\|_1 = \sum_{i=1}^d |v_i|$
• Euclidean norm: $\|v\|_2 = \sqrt{\sum_{i=1}^d v_i^2}$
• Maximum norm: $\|v\|_\infty = \max \{|v_1|, \dots, |v_d|\}$

Norms satisfy the Triangle Inequality ($\|u + v\| \leq \|u\| + \|v\|$) and the Cauchy-Schwarz Inequality:

Partial Derivatives and Gradients

Multivariate continuity means $x_k \to x_0 \implies f(x_k) \to f(x_0)$.
The partial derivative of $f$ with respect to the $i$-th coordinate is:

The Gradient collects all partial derivatives into a vector: