Calculus

Limits

Aspect	Limits	Continuity
Definition	Describes the value a function approaches as input nears a point or infinity	A function is continuous at a point if it has no breaks, jumps, or holes there
Formal Expression	$\lim_{x \to c} f(x) = L$, meaning as $x$ approaches $c$, $f(x)$ approaches $L$	$f(x)$ is continuous at $c$ if: $f(c)$ is defined, $\lim_{x \to c} f(x)$ exists, and both are equal
Purpose	Helps analyze function behavior at undefined points or extremes	Ensures smoothness of function behavior across its domain
Examples	Linear: $\lim_{x \to 3} (2x+1)=7$ Hole: $\lim_{x \to 1} \frac{x^2-1}{x-1} = 2$ Asymptote: $\lim_{x \to \infty} \frac{1}{x} = 0$	Removable: hole in function though limit exists Jump: left- and right-hand limits differ Infinite: vertical asymptote, e.g., $f(x)=1/x$ at $x=0$
Key Role in Calculus	Foundation for defining derivatives and integrals	Precondition for differentiability and smooth curve behavior
Relevance to Data Analysis	Models at undefined/extreme input values Asymptotic properties of estimators (e.g., CLT) Anticipating numerical instabilities	Smoothness needed for optimization (e.g., gradient descent) Assumption in regression, neural nets, interpolation Understanding thresholds in piecewise models like decision trees

Derivatives

Concept	Definition / Formula	Key Example	Relevance to Data Analysis
Derivative (Rate of Change)	$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$	For $f(x) = x^3$, $f'(x) = 3x^2$	Measures sensitivity of outputs to inputs, marginal effects, rates of change
Average vs Instantaneous Rate	Avg: $\frac{f(x_2)-f(x_1)}{x_2-x_1}$; Inst: limit form above	Slope of secant vs slope of tangent	Distinguishes between overall vs. pointwise change
Notation	$f'(x)$, $\frac{dy}{dx}$, $\frac{d}{dx} f(x)$	$y'$	Different notations useful in various contexts (Leibniz, Lagrange)
Power Rule	$\frac{d}{dx}(x^n) = nx^{n-1}$	$(x^3)' = 3x^2$, $(\sqrt{x})' = \frac{1}{2\sqrt{x}}$	Core tool for polynomial/exponential changes
Constant Multiple Rule	$(cf(x))' = c f'(x)$	$(5x^3)' = 15x^2$	Simplifies scaling derivatives in models
Sum/Difference Rule	$(f(x)\pm g(x))' = f'(x)\pm g'(x)$	$(4x^2+7x-2)'= 8x+7$	Enables decomposition of model functions
Product Rule	$(fg)' = f'g + fg'$	$(x^2 e^x)' = e^x(2x+x^2)$	Needed for features interacting multiplicatively
Quotient Rule	$\Big(\frac{f}{g}\Big)' = \frac{f'g - fg'}{g^2}$	$\Big(\frac{x^2}{x+1}\Big)' = \frac{x^2+2x}{(x+1)^2}$	Used when variables appear in ratios
Chain Rule	$(f(g(x)))' = f'(g(x))g'(x)$	$((x^2+3x)^5)' = 5(x^2+3x)^4(2x+3)$	Central to backpropagation in neural networks
Common Functions	$(e^x)'=e^x$, $(\ln x)'=\frac{1}{x}$, $(\sin x)'=\cos x$, $(\cos x)'=-\sin x$	Trigonometric and exponential forms	Key in interpreting exponential growth/periodicity
Partial Derivatives	$\frac{\partial f}{\partial x}$, hold others constant	For $f(x,y)=x^2y+3xy^3$, $\frac{\partial f}{\partial x}=2xy+3y^3$	Measures effect of single feature
Gradient	$\nabla f = \big(\frac{\partial f}{\partial x_1}, \ldots, \frac{\partial f}{\partial x_n}\big)^T$	For earlier $f(x,y)$: $(2xy+3y^3,\; x^2+9xy^2)$	Fundamental to gradient descent optimization
Second-order Derivatives	Pure: $\frac{\partial^2 f}{\partial x^2}$; Mixed: $\frac{\partial^2 f}{\partial x \partial y}$	Concavity/curvature along axes	Determines shape and inflection, used in convexity analysis
Hessian Matrix	Matrix of all 2nd-order partials	For multivariable $f(x_1,\ldots,x_n)$ → Hessian $H$	Used in Newton's method, uncertainty estimation
Applications	Optimization, sensitivity, feature importance, backpropagation, marginal analysis	Loss minimization, elasticity, customer growth rates	Core to machine learning training and interpretability

Integrals

Concept	Formula	Geometric Interpretation	Use Cases
Integral (general)	Summation of infinitesimal changes; inverse of derivative. Notation: $\int f(x)dx$	Area under a curve of $f(x)$	Fundamental for accumulation, total quantities, probability, and advanced methods
Indefinite Integral	$\int f(x) dx = F(x) + C$ where $F'(x) = f(x)$	Family of anti-derivatives, no fixed interval	Useful for theoretical derivations, recovering functions from their rates
Basic Rules (examples)	Power Rule: $\int x^n dx = \frac{x^{n+1}}{n+1} + C$, $\int e^x dx = e^x + C$, etc	Reverse of differentiation rules	Needed for solving analytical integration problems in modeling
Definite Integral	$\int_a^b f(x) dx$. Results in a number	Net area under curve between $a$ and $b$; negative if below x-axis	Widely used (e.g., probability, computing totals from rates)
Fundamental Theorem of Calculus	$\int_a^b f(x) dx = F(b) - F(a)$	Connects indefinite and definite integrals	Basis for practical computation—greatly simplifies area and probability calculations
Probability via Integrals	$P(a \leq X \leq b) = \int_a^b f(x) dx$. CDF: $F(x) = \int_{-\infty}^x f(t) dt$. Expectation: $E[X] = \int_{-\infty}^\infty x f(x) dx$	Probability is area under the PDF	Core use in continuous probability, distributions, inference, and statistics
Total Change from a Rate	$\int_{t_1}^{t_2} R(t) dt$	Accumulated value from rate function	Total sales, leads, or growth from a rate function in business/analytics
Geometric Applications	Integrals compute area, volume, surface, etc	Shapes and 3D distributions	Less common but used in capacity, resource estimation, or spatial data
Machine Learning Applications	Used in kernel density estimation, Fourier transforms, and some loss functions	Transformations of functions (time ↔ frequency)	Important in advanced analytics, signal processing, regularization, density models

Partial Derivatives

Concept	Definition	Key Properties	Applications
Partial Derivatives	Rate of change of a multivariable function with respect to one variable while holding others constant	Notation: $\frac{\partial f}{\partial x_i}$. Chain rule: $\frac{\partial f}{\partial x} = \frac{\partial f}{\partial u} \frac{\partial u}{\partial x}$	Feature importance analysis, sensitivity analysis in models
Gradient Vector	Vector of all partial derivatives: $\nabla f = \begin{pmatrix} \frac{\partial f}{\partial x_1} \\ \vdots \\ \frac{\partial f}{\partial x_n} \end{pmatrix}$	Points in direction of steepest ascent. Magnitude gives rate of change	Gradient descent optimization, backpropagation in neural networks
Jacobian Matrix	Matrix of all first-order partial derivatives of a vector-valued function	For $\mathbf{f}: \mathbb{R}^m \to \mathbb{R}^n$, $\mathbf{J} \in \mathbb{R}^{n \times m}$. Determinant shows volume scaling	Coordinate transformations, stability analysis, robotics
Hessian Matrix	Square matrix of second-order partial derivatives	$\mathbf{H}_{ij} = \frac{\partial^2 f}{\partial x_i \partial x_j}$. Eigenvalues determine local convexity	Newton's optimization method, curvature analysis, saddle point detection
Multiple Integrals	Integration over regions in multiple dimensions	Double integrals for area/volume, triple integrals for 3D volumes. Fubini's theorem for iteration	Probability density functions, expected values in multiple dimensions

Mathematical Analysis

Concept	Definition	Key Components	Applications
Real Analysis Fundamentals	Rigorous study of real numbers, sequences, and functions	Set theory, limits of sequences, continuity, differentiation, integration in abstract terms	Theoretical foundation for advanced ML topics, measure theory, functional analysis
Vector Calculus	Differentiation and integration of vector fields	Vector fields, line integrals, surface integrals, volume integrals. Green's, Stokes', Divergence theorems	Physics-based simulations, advanced geometric ML, fluid dynamics, electromagnetism