Calculus

Elements

The fundamental element of calculus is the function — a mapping from inputs to outputs that describes how quantities vary in relation to each other.

The Three Core Forms

• Limit — the value a function approaches as its input approaches a point; the conceptual foundation on which both derivative and integral rest
• Derivative — the instantaneous rate of change: how fast the output changes per unit change in input
• Integral — the accumulated quantity: the area under a curve, the total change over an interval

Concept Forms

• Convergence / Divergence — whether a sequence or series approaches a finite value or escapes to infinity
• Stationary points — where $f'(x) = 0$ — the equilibrium states of a function (local minima, maxima, saddle points)

Axiomatic Foundations

Calculus rests on the epsilon-delta definition of the limit — a precise logical statement replacing the intuitive notion of “approaching”:

$\lim_{x \to a} f(x) = L \iff \forall \varepsilon > 0,\; \exists \delta > 0 : |x - a| < \delta \implies |f(x) - L| < \varepsilon$

From this, continuity and differentiability are derived notions, not primitive ones.

Key Theorems

• If $f$ is continuous on $[a,b]$ and $A$ is compact, then $f$ is bounded and attains its maximum and minimum.
• The image of a connected set under a continuous function is connected.
• Fundamental Theorem of Calculus: Differentiation and integration are inverse operations:

$\frac{d}{dx}\int_a^x f(t)\,dt = f(x), \qquad \int_a^b f(x)\,dx = F(b) - F(a)$

This theorem unifies the two halves of calculus — change and accumulation — as a single structure.

Procedures

The algorithmic lens reduces calculus to systematic rule application — finite procedures that transform one function form into another.

Differentiation Rules

Finding Extrema

1. Compute $f'(x)$.
2. Solve $f'(x) = 0$ for critical points.
3. Evaluate $f''(x)$ at each critical point: $f'' > 0$ is a minimum, $f'' < 0$ is a maximum.

Integration Techniques

• Substitution: $u = g(x)$, transform the integrand
• Integration by parts: $\int u\,dv = uv - \int v\,du$
• Partial fractions: decompose rational functions before integrating

Calculus as a System

Calculus provides the natural language for dynamic systems — systems where quantities change over time.

• Functions are the stocks: the current state of a quantity
• Derivatives are the flows: the rate at which the stock changes
• Integrals accumulate flows back into stocks

This is the mathematical substrate beneath every systems model: a differential equation $\frac{dy}{dt} = f(y, t)$ specifies exactly how the stock $y$ changes through time.

Equilibrium

Stationary points ($f'(x) = 0$) are the system’s equilibria — where the rate of change is zero. The second derivative determines whether equilibrium is stable (a minimum: $f'' > 0$) or unstable (a maximum: $f'' < 0$).

Convergence and divergence are the system’s fates: does it settle, or does it escape?

Calculus as a Tool

The engineering lens treats calculus as the instrument for optimizing and modeling:

• Optimization: find the input that minimizes cost or maximizes output by solving $f'(x) = 0$
• Modeling change: express physical laws as differential equations — $F = ma$ becomes $m\ddot{x} = F(x, \dot{x}, t)$
• Control systems: differential equations govern how systems respond to inputs and disturbances

Practical Constraints

• Not every function has a closed-form antiderivative — numerical integration (e.g., Simpson’s rule, Gaussian quadrature) introduces bounded discretization error.
• Differentiability requires continuity, but continuity does not guarantee differentiability — the absolute value function $|x|$ is continuous but not differentiable at $x = 0$.