Unit Overview

Calculus

IB Mathematics AI SL· 5 min read 📊 16-18% of overall IB Mathematics AI SL examination

1. Unit at a Glance

We build this unit from the ground up, starting with the intuitive geometric meaning of the derivative before moving to procedural skills and real-world applications. First you will learn to differentiate polynomials, then use derivatives to solve optimisation problems. We then reverse differentiation to introduce integration, learn to calculate area under curves, and finally apply both tools to kinematic motion problems. Each sub-topic builds directly on the previous one, so work through them in order.

Sub-topics in this unit:

Derivative as gradient of tangent and rate of change

Introduces the geometric interpretation of the derivative and its meaning as a rate of change.

★★⏱ 8 min

Differentiation of polynomial functions (power rule)

Teaches the power rule for differentiating any polynomial function.

★★⏱ 7 min

Tangents, gradients, and stationary points

Shows how to use derivatives to find tangent equations and classify stationary points.

★★★⏱ 9 min

Optimisation problems with differentiation

Applies stationary point analysis to solve real-world maximum/minimum problems.

★★★★⏱ 12 min

Indefinite integration of polynomials

Introduces integration as the reverse of differentiation and the power rule for integration.

★★★⏱ 8 min

Definite integrals and area under a curve

Explains how to use definite integration to calculate the area between a curve and the x-axis.

★★★⏱ 10 min

Kinematic applications of calculus

Connects displacement, velocity and acceleration using differentiation and integration.

★★★★⏱ 11 min

2. Common Pitfalls

Wrong move:

Forgetting the constant of integration $+ c$ when calculating indefinite integrals

Why:

This leads to automatic mark loss even if your integration calculation is correct

Correct move:

Always add $+ c$ to the end of any indefinite integral result

Wrong move:

Mixing up differentiation vs integration for kinematic problems

Why:

Many students reverse the relationship between displacement, velocity and acceleration

Correct move:

Differentiate to go from displacement → velocity → acceleration; integrate to go the opposite direction

Wrong move:

Not adjusting for curves that dip below the x-axis when calculating area

Why:

Definite integrals return negative values for area below the x-axis, leading to incorrect net results

Correct move:

Split the integral into intervals, take the absolute value of each interval's result, then add them together

3. Quick Reference Cheatsheet

Concept	Key Formula/Rule
Power rule for differentiation	$\frac{d}{d x} (a x^{n}) = na x^{n - 1}$
Gradient of tangent at $x = a$	$m = f^{'} (a)$
Stationary point condition	$f^{'} (x) = 0$
Power rule for integration	$\int a x^{n} d x = \frac{a x ^{n + 1}}{n + 1} + c (n \neq = - 1)$
Area under a positive curve	$Area = \int_{a}^{b} f (x) d x$
Acceleration from displacement	$a = \frac{d ^{2} s}{d t ^{2}} = \frac{d v}{d t}$
Displacement from velocity	$s = \int v d t + c$

What's Next

Start with the first sub-topic of this unit to build your foundational intuitive understanding of derivatives as gradients of tangents and rates of change. Work through each sub-topic in order, as each concept builds directly on the previous one. Memorize the key formulas listed in the cheatsheet above, as they appear in nearly every calculus exam question. Once you complete all Calculus sub-topics, you can move on to the next unit in the IB Math AI SL syllabus.