IB AA: Calculus

Calculus is the mathematical study of continuous change. This chapter introduces its two main branches: differentiation and integration. Differentiation is concerned with finding the instantaneous rate of change, or the gradient of a function at a specific point. You will start with the formal definition of a derivative from first principles before learning rules to differentiate functions quickly. These include the power, product, quotient, and chain rules, which are applied to polynomial, trigonometric, exponential, and logarithmic functions.

The second part of the chapter, integration, can be understood as the reverse process of differentiation, also known as anti-differentiation. It is used to find the area under a curve between two points. You will learn to compute both indefinite and definite integrals. Key applications include calculating the area enclosed by curves and finding the volume of solids formed by revolving a curve around an axis.

This chapter relies heavily on your skills from Algebra and Functions. A strong ability to manipulate expressions and understand the behaviour of graphs is essential for success. Students often find the application problems, such as optimisation (finding maximum or minimum values), to be the most challenging part, as they require translating a real-world problem into a mathematical function before applying calculus techniques.

Formulas for IB AA: Calculus

Derivative from first principles: $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$ The formal definition of the derivative of a function f(x).
Power rule for differentiation: $\frac{d}{dx}(ax^n) = anx^{n-1}$ For differentiating terms of the form ax^n.
Product rule: $\frac{d}{dx}(uv) = u\frac{dv}{dx} + v\frac{du}{dx}$ For differentiating the product of two functions, u and v.
Quotient rule: $\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}$ For differentiating the quotient of two functions, u and v.
Chain rule: $\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$ For differentiating a composite function y = f(g(x)).
Derivatives of trigonometric functions: $\frac{d}{dx}(\sin x) = \cos x, \quad \frac{d}{dx}(\cos x) = -\sin x$ The variable x must be in radians.
Derivatives of exponential and log functions: $\frac{d}{dx}(e^x) = e^x, \quad \frac{d}{dx}(\ln x) = \frac{1}{x}$ Core derivatives for the natural exponential and logarithm.
Power rule for integration: $\int x^n \,dx = \frac{x^{n+1}}{n+1} + C$ For n ≠ -1. C is the constant of integration.
The definite integral: $\int_a^b f(x) \,dx = [F(x)]_a^b = F(b) - F(a)$ Where F(x) is the antiderivative of f(x).
Area under a curve: $A = \int_a^b y \,dx$ Area between the curve y=f(x), the x-axis, and lines x=a, x=b.
Volume of revolution (around x-axis): $V = \pi \int_a^b y^2 \,dx$ Volume formed by rotating the curve y=f(x) around the x-axis.