Introductory Calculus (often called Calculus I or AP Calculus) is the mathematics of change. It formalizes the concepts of instantaneous rate of change and accumulation.

Key Topics

• Limits — Understanding what happens to a function as its input approaches a value: \(\lim_{x \to a} f(x)\).
• Derivatives — The slope of a curve at any point, representing instantaneous rate of change: \(f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\).
• Differentiation Rules — Power rule, product rule, quotient rule, and chain rule.
• Applications of Derivatives — Optimization, related rates, curve sketching, and motion.
• Integrals — Accumulation of quantities and area under curves: \(\int_a^b f(x)\,dx\).
• Antiderivatives — Reversing differentiation.
• Fundamental Theorem of Calculus — The deep connection between differentiation and integration.

Why It Matters

Calculus is the language of science and engineering. It describes how planets orbit, how bridges bear loads, how populations grow, and how machines learn. Nearly every branch of physics and engineering is built on calculus.