Calculus
The mathematics of change and accumulation. From a single idea — taking a limit — calculus builds derivatives (instantaneous rates of change) and integrals (continuous totals), and the Fundamental Theorem reveals they are two sides of the same coin.
Limits
"Approaches but never reaches." The single foundational idea — what value a function tends to as the input approaches some point.
Continuity
When the graph has no gaps, jumps, or holes. The four discontinuity types, plus the Intermediate and Extreme Value Theorems.
Derivatives
The instantaneous rate of change. The slope of the tangent line. The first big payoff of calculus.
Rules of Differentiation
Power rule, product rule, quotient rule, chain rule. The toolkit that turns calculus into algebra.
Applications of Derivatives
Optimization, related rates, and the meaning of the second derivative. Where calculus earns its keep.
Integrals
Continuous accumulation. The area under a curve, and the conceptual leap from "rate" to "total."
The Fundamental Theorem of Calculus
The shock that ties the whole subject together: differentiation and integration are inverse operations.
Techniques of Integration
Substitution, integration by parts, partial fractions. The standard moves for evaluating integrals by hand.
Applications of Integration
Area between curves, volumes of revolution, arc length, average value, and work. Every "total" quantity is a Riemann sum in disguise.
Calculus of Parametric & Polar Curves
When a curve isn't y = f(x): parametric paths (x(t), y(t)) and polar shapes (r, θ). Derivatives, arc length, and area generalized to both.
Infinite Series & Convergence
When an infinite sum has a finite value. Geometric and harmonic series, the seven standard convergence tests, absolute vs conditional.
Power Series
Series in a variable. Radius of convergence, Taylor and Maclaurin series, and the moment Euler's identity falls out for free.
Partial Derivatives
Calculus on f(x, y). Partials as cross-section slopes, the gradient as direction of steepest ascent, and tangent planes.
Vector-Valued Functions
Curves in space as r(t). Velocity and acceleration, the TNB frame, arc-length parameterization, and curvature κ.
Multiple Integrals
Double and triple integrals as volumes under surfaces. Fubini, non-rectangular regions, polar/cylindrical/spherical change of variables.
Vector Calculus
Gradient, divergence, curl, line and surface integrals, and the three "Stokes" theorems unified as boundary = bulk of a derivative.