Simple Calculus

Understanding Calculus

The mathematics of change and motion

What is Calculus?

Calculus is a branch of mathematics that studies continuous change. While algebra focuses on static equations, calculus provides the framework to analyze moving objects, changing rates, and complex curves. It is divided into two primary fields linked by the Fundamental Theorem of Calculus.

Core Concepts

1. Limits: The Foundation +

A limit evaluates what value a function approaches as the input gets infinitely close to a specific point. It allows mathematicians to safely explore values like "zero divided by zero" without breaking algebraic rules.

$$\lim_{x \to c} f(x) = L$$

2. Differential Calculus (Derivatives) +

Differential calculus cuts shapes into tiny pieces to find instantaneous rates of change. Geometrically, the derivative is the exact slope of a curve's tangent line at any single point.

$$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$

3. Integral Calculus (Integrals) +

Integral calculus accumulation values together to find total quantities, such as the exact area under an irregular curve. It acts as the inverse operation to differentiation.

$$\int_{a}^{b} f(x) \, dx = F(b) - F(a)$$

Interactive Derivative Visualizer

Move the slider to see how the tangent line changes along the curve $f(x) = x^2$

X Position: 1.0