Unit Overview

Calculus

IB Mathematics: Analysis and Approaches SL· 5 min read 📊 20-23% of overall exam

1. Unit at a Glance

This unit follows a natural learning arc that builds from foundational concepts to applied problems. We start with limits, the core idea that makes calculus work, to introduce the definition of the derivative. You will then build a complete toolkit of differentiation rules for all common function types, before moving to applications of differentiation to analyze function graphs and solve practical problems.

After mastering differentiation, we turn to integration, starting with anti-differentiation as the reverse process of differentiation. We then introduce definite integrals, the Fundamental Theorem of Calculus that links the two core concepts of calculus, and end with applications of integration to calculate areas and volumes of revolution.

This unit includes 8 ordered sub-topics, progressing from foundation to application:

Introduction to limits and differentiation

Learn the definition of the derivative via the foundational concept of limits

★★⏱ 15 min

Basic differentiation rules: power, product, quotient, chain rule

Master core rules for differentiating combinations of basic functions

★★⏱ 18 min

Differentiation of trigonometric, exponential and logarithmic functions

Learn derivative formulas for non-polynomial function types

★★★⏱ 15 min

Stationary points, monotonicity and concavity

Use derivatives to analyze the shape and behavior of function graphs

★★★⏱ 20 min

Optimization applications

Apply differentiation to solve practical maximum and minimum problems

★★★★⏱ 15 min

Indefinite integration and anti-differentiation

Introduce anti-differentiation as the reverse process of differentiation

★★⏱ 15 min

Definite integration and fundamental theorem of calculus

Learn the definition of definite integrals and the link to differentiation

★★★⏱ 18 min

Integration by substitution, area and volume of revolution

Master substitution method and apply integration to area and volume problems

★★★★⏱ 20 min

2. Common Pitfalls

Wrong move:

Forgetting the constant of integration when calculating indefinite integrals

Why:

This leads to lost easy marks and incorrect results for follow-up problems

Correct move:

Always add $+ C$ to the final result of any indefinite integral calculation

Wrong move:

Misapplying the chain rule for differentiation or forgetting to adjust the differential for substitution

Why:

This is the most common error in routine calculus questions on IB exams

Correct move:

Always account for the derivative of the inner function for both operations

Wrong move:

Mixing up volume of revolution formulas for rotation around x vs y axes

Why:

This leads to incorrect integral setups and lost application marks

Correct move:

Recall $V = π \int y^{2} d x$ for rotation around x-axis and $V = π \int x^{2} d y$ for rotation around y-axis

3. Quick Reference Cheatsheet

Concept	Key Formula
Derivative definition	$\frac{d y}{d x} = lim_{h \to 0} \frac{f ( x + h ) - f ( x )}{h}$
Chain rule	$\frac{d}{d x} [f (g (x))] = f^{'} (g (x)) g^{'} (x)$
Derivative of $sin x$	$\frac{d}{d x} (sin x) = cos x$
Derivative of $ln x$	$\frac{d}{d x} (ln x) = \frac{1}{x}$
Fundamental Theorem of Calculus	$\int_{a}^{b} f^{'} (x) d x = f (b) - f (a)$
Indefinite power rule	$\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + C, n \neq = - 1$
Area between two curves	$Area = \int_{a}^{b} ∣ f (x) - g (x) ∣ d x$
Volume of revolution (x-axis)	$V = π \int_{a}^{b} [y (x)]^{2} d x$

What's Next

Begin your study of this calculus unit with the first sub-topic, which introduces the concept of limits that forms the foundation for all differentiation and integration work. After completing all 8 sub-topics in this unit, you will progress to the next unit in the IB AA SL syllabus, focused on statistics and probability.