Overview

Calculus is the mathematical study of continuous change. It has two major branches: Differential Calculus (concerning rates of change and slopes of curves) and Integral Calculus (concerning accumulation of quantities and areas under and between curves).

Core Idea

The core idea is the Limit: analyzing what happens as we get infinitely close to a point. This allows us to handle the concept of “instantaneous” change (speed at a specific moment) and “infinite” sums.

Formal Definition

Calculus relies on the rigorous definition of a limit ($\epsilon-\delta$ definition).

  • Derivative ($f’(x)$): The limit of the difference quotient as the interval approaches zero.
  • Integral ($\int f(x) dx$): The limit of a Riemann sum as the partition size approaches zero.

Intuition

  • Differentiation: Zooming in on a curve until it looks like a straight line. The slope of that line is the derivative.
  • Integration: Slicing a shape into infinitely thin rectangles and adding up their areas to get the total area.

Examples

  • Physics: Velocity is the derivative of position. Acceleration is the derivative of velocity. Distance is the integral of velocity.
  • Optimization: Finding the maximum profit or minimum cost by setting the derivative to zero (finding the peak of the curve).

Common Misconceptions

  • Misconception: Zeno’s paradoxes prove motion is impossible.
    • Correction: Calculus resolves Zeno’s paradoxes by showing that an infinite sum of time intervals can add up to a finite total.
  • Misconception: Derivatives and integrals are unrelated.
    • Correction: The Fundamental Theorem of Calculus connects them: integration and differentiation are inverse operations.
  • Limit: The value that a function (or sequence) “approaches” as the input (or index) approaches some value.
  • Continuity: A function is continuous if you can draw it without lifting your pen.
  • Differential Equations: Equations involving derivatives.

Applications

  • Physics & Engineering: Describing motion, electricity, heat, structures.
  • Economics: Marginal cost/revenue analysis.
  • Biology: Modeling population growth (logistic curves).

Criticism and Limitations

  • Discreteness: Calculus assumes the world is continuous (smooth). At the quantum scale, or in digital computing, the world is discrete, requiring Discrete Math instead.

Further Reading

  • Calculus by Michael Spivak
  • The Calculus Lifesaver by Adrian Banner