Calculus Intermediate

A limit describes the value a function approaches as the input nears a point: lim(x→a) f(x). Limits underpin both derivatives and integrals and make sense of instantaneous behaviour and infinity.

The derivative f'(x) is the instantaneous rate of change — the slope of the tangent. Power rule: d/dx xⁿ = n·xⁿ⁻¹. Product, quotient and chain rules handle combinations. In engineering, derivatives give velocity, acceleration and rates of flow.

The integral accumulates quantity — the area under a curve. The Fundamental Theorem links it to the derivative: integration and differentiation are inverses. ∫xⁿ dx = xⁿ⁺¹/(n+1) + C. Applications: distance from velocity, work from force, charge from current.

Differential equations relate a quantity to its rate of change and model circuits (RC/RL), heat flow, vibrations and control systems. Numerical methods (Euler, Runge–Kutta) approximate solutions when closed forms are unavailable.