Calculus

Introduction to Calculus

Calculus is the mathematics of change. Where algebra describes static relationships, calculus describes how quantities grow, shrink, accelerate, accumulate, and approach limits. Invented — or discovered — independently by Newton and Leibniz in the 17th century, it remains the single most powerful mathematical tool in physics, engineering, economics, and biology. The AA course treats calculus with analytical depth: you will not only apply differentiation and integration rules, but derive them from first principles, prove that a function is continuous or differentiable at a point, and construct series approximations that turn transcendental functions like $e^x$ and $\\sin x$ into infinite polynomials. The section begins with the concept of a limit — the idea that makes calculus logically sound — and builds outward from there: differentiation (rates of change, optimisation, kinematics), integration (accumulation, areas, volumes), and at HL, differential equations and Maclaurin series that close the loop between functions and power series.

Section Map — What Is Covered

• Limits and Differentiation from First Principles (5.1): the limit $\\lim_{x\\to a}f(x)$ as the foundation, and the derivative defined as $f'(x)=\\lim_{h\\to0}\\frac{f(x+h)-f(x)}{h}$
• The Derivative and Rates of Change (5.2): the power rule, derivatives of sin/cos/tan, $e^x$, and $\\ln x$; and using the sign of $f'(x)$ to find where a function increases or decreases
• Tangents, Normals and Optimisation (5.3): equations of tangent and normal lines, and the systematic strategy for maximising or minimising quantities within given constraints
• The Chain, Product and Quotient Rules (5.4): differentiating composite functions, products, and quotients — the complete differentiation toolkit
• Curve Sketching (5.5): concavity, points of inflexion, and the full checklist for producing a labelled sketch from intercepts through stationary points to asymptotes
• Kinematics via Calculus (5.6): displacement → velocity → acceleration via differentiation, and the reverse via integration; the crucial distinction between total distance and net displacement when direction changes
• Integration as Antidifferentiation (5.7): reversing differentiation — the power rule for integration, standard integrals, and the definite integral as signed area between a curve and the x-axis
• Definite Integrals and Further Techniques (5.8): integration by substitution — the reverse chain rule — and the trapezoidal rule for numerical approximation when an exact antiderivative is unavailable
• Limits, Continuity and Differentiability (5.9, HL): why differentiability implies continuity but not the reverse; one-sided limits; and classifying discontinuities as removable, jump, or infinite
• Further Differentiation (5.10, HL): derivatives of arcsin, arctan, sec, csc, cot; implicit differentiation for curves not in the form $y=f(x)$; and related rates problems
• Further Integration Techniques (5.11, HL): integration by parts — the reverse product rule — and volumes of revolution via the disc method $V=\\pi\\int[f(x)]^2dx$
• Differential Equations (5.12, HL): separation of variables for first-order equations, particular solutions from initial conditions, slope fields as a visual tool, and Euler's method for numerical approximation
• Maclaurin Series (5.13, HL): expressing functions as infinite polynomials built from their derivatives at x = 0; standard series for $e^x,\\ \\sin x,\\ \\cos x$; and using truncation for approximation with error bounds