Unit Overview

Calculus

IB Mathematics: Analysis and Approaches Higher Level· 6 min read 📊 24-26% of total exam

1. Unit at a Glance

This unit follows a clear, logical learning progression: we start with the foundational concept of limits to build a rigorous definition of the derivative. Next, we master basic and advanced differentiation techniques before covering common applications of differentiation.

After differentiation, we move to integration, learning core integration techniques, applications of integration for area and volume, and the fundamental theorem that connects the two big ideas of calculus. The unit concludes with HL-only advanced topics including improper integrals, differential equations, and Taylor/Maclaurin series.

This unit includes the following sub-topics:

Limits and continuity

Introduces the formal definition of limits and tests for continuity of functions.

★★⏱ 10 min

L'Hôpital's rule (HL only)

Evaluate indeterminate limits using this efficient derivative-based rule.

★★★⏱ 8 min

Definition of the derivative

Derive derivatives from first principles using the limit definition.

★★⏱ 7 min

Differentiation rules

Master core rules for differentiating all common core functions.

★★⏱ 12 min

Implicit, parametric and logarithmic differentiation

Advanced differentiation techniques for non-explicit functions.

★★★⏱ 15 min

Higher order derivatives

Calculate first, second and higher-order derivatives for any function.

★★⏱ 6 min

Applications of differentiation

Solve tangents/normals, optimization, and related rates problems.

★★★⏱ 14 min

Indefinite and definite integration

Introduce integration as the reverse of differentiation and definite integrals.

★★⏱ 10 min

Integration techniques: substitution, parts, partial fractions

Master the three core integration methods required for all exam questions.

★★★★⏱ 18 min

Improper integrals (HL only)

Evaluate integrals with infinite bounds or discontinuous integrands.

★★★★⏱ 9 min

Applications of integration: area and volume

Calculate areas between curves and volumes of revolution.

★★★⏱ 12 min

First order differential equations (HL only)

Solve and interpret separable and homogeneous first order differential equations.

★★★★⏱ 15 min

Second order linear differential equations (HL only)

Solve homogeneous and non-homogeneous second order linear DEs.

★★★★★⏱ 16 min

Maclaurin and Taylor series (HL only)

Expand functions as power series and approximate functions with series.

★★★★★⏱ 18 min

Fundamental theorem of calculus

Formal connection between differentiation and integration.

★★★⏱ 7 min

2. Common Pitfalls

Wrong move:

Forgetting to adjust integral bounds when using u-substitution.

Why:

This is one of the most common exam errors, leading to wrong answers even if your antiderivative is correct.

Correct move:

Always convert the bounds of integration to the new variable when substituting.

Wrong move:

Mixing up product rule and chain rule for composite functions.

Why:

Many students misapply rules when differentiating products of composite functions.

Correct move:

Always apply chain rule first for inner functions, then product rule for products.

Wrong move:

Incorrectly setting up volume of revolution integrals.

Why:

Students often mix formulas for rotation around x vs y axis, or forget to subtract inner curves.

Correct move:

Always sketch the region to confirm the integral setup before solving.

3. Quick Reference Cheatsheet

Concept / Key Formula	Description
Derivative from first principles	$f^{'} (x) = lim_{h \to 0} \frac{f ( x + h ) - f ( x )}{h}$
Chain Rule	$\frac{d y}{d x} = \frac{d y}{d u} \cdot \frac{d u}{d x}$
Integration by Parts	$\int u \frac{d v}{d x} d x = uv - \int v \frac{d u}{d x} d x$
Fundamental Theorem of Calculus	$\int_{a}^{b} f^{'} (x) d x = f (b) - f (a)$
L'Hôpital's Rule	$lim_{x \to a} \frac{f ( x )}{g ( x )} = lim_{x \to a} \frac{f ^{'} ( x )}{g ^{'} ( x )}$
Volume of revolution (x-axis)	$V = π \int_{a}^{b} y^{2} d x$
Separable First Order DE	$\frac{d y}{d x} = f (x) g (y) ⟹ \int \frac{1}{g ( y )} d y = \int f (x) d x$
Taylor Series (centered at $a$ )	$f (x) = \sum_{n = 0}^{\infty} \frac{f ^{(n)} ( a )}{n !} (x - a)^{n}$

What's Next

Begin your study of this unit with the foundational topic of limits and continuity, the building block for all further calculus concepts. Once you complete all sub-topics in this unit, you can progress to the next unit on Probability and Statistics. The links below will take you to the first sub-topic of this unit and the opening of the next unit.