Calculus

Single-variable calculus: derivatives, integrals, the classic theorems, and their applications.

Derivatives

Derivatives Warmup & Power Rule — Differentiate c x^n one piece at a time.
Quotient Rule — (f/g)' = (f'g - fg') / g^2 — low d-high minus high d-low, over the square of the low.
Special Derivatives — (e^x)' = e^x, ( x)' = 1/x, ( )' = , ( )' = - , ( )' = ^2.
Product Rule — (fg)' = f'g + fg' — differentiate each factor, swap, add.
Chain Rule — d/dx[f(g(x))] = f'(g(x)) · g'(x) — outer, then inner.

Continuity — _x → a f(x) = f(a) — no holes, no jumps, no blow-ups.
Intermediate Value Theorem — Continuous, with values straddling N, so some c has f(c) = N. Existence of roots, made easy.
Extreme Value Theorem — Continuous on a closed, bounded interval [a, b] means the max and min are always attained.
Rolle's Theorem — Continuous, differentiable, and f(a) = f(b) force some c ∈ (a, b) with f'(c) = 0.
Mean Value Theorem — Some tangent matches the secant slope: f'(c) = (f(b) - f(a))/(b - a).
L'Hôpital's Rule — For 0/0 or ∞/∞: f/g = f'/g'.
Convex Functions — f'' ≥ 0 means every chord sits on or above the graph. The unifying notion of optimization.

Integration — Adding up infinitely many infinitesimally small pieces.
Definite Integral — Cashing in the Fundamental Theorem: find an antiderivative, subtract.
Fundamental Theorem of Calculus — Why a limit of infinitely many sums collapses to one subtraction.
Taylor Series — f(x) = ∑ f^(n)(a)n!(x-a)^n — polynomials become any smooth function.

Increasing & Decreasing — f' > 0 rises, f' < 0 falls — read monotonicity off the derivative.
Tangents & Normals — Tangent slope f'(a); normal slope -1/f'(a).
Second Derivative — f'' sets concavity and tests maxima vs minima.
Local Maxima & Minima — Set f'(x) = 0, classify, optimize.
Kinematics — v = s'(t), a = v'(t), displacement = ∫ v dt.
Implicit Differentiation — Differentiate both sides; chain-rule every y; solve for dy/dx.

Related Rates — Differentiate a shared equation in time to link two rates.
Linear Approximation & Differentials — Estimate values and errors with the tangent line and differentials.
Newton's Method — Tangent-line iteration that converges fast to a root.
Improper Integrals — Infinite intervals and unbounded integrands, handled by limits.
Average Value of a Function — The mean of f over an interval, and where it is attained.
Surface Area of Revolution — Revolve a curve and sum the bands: S = ∫ 2π r ds.
Parametric Curves & Calculus — Slope dy/dx, tangents, and arc length of a curve traced by x(t), y(t).
Polar Coordinates & Area — Graph r = f(θ) and find enclosed area with 12∫ r^2 dθ.
Series Convergence Tests & Power Series — Geometric, p-series, ratio test, and the radius of convergence.