# Calculus

> IB Mathematics AI SL · IB Math AI SL
> Source: https://www.owlsprep.com/study/ib-math-ai-sl-u5-overview/
> Weight: 16-18% of overall IB Mathematics AI SL examination

This unit introduces core calculus concepts for IB AI SL, covering differentiation and integration of polynomials, and their applications to optimisation, area calculation, and kinematics. It is one of the most heavily weighted topics on the exam.

**Prerequisites:** [Foundational knowledge of polynomial functions and coordinate geometry](https://www.owlsprep.com/study/ib-math-ai-sl-u3-functions-overview/)

## Learning objectives

- Understand the derivative as the gradient of a tangent and rate of change for real-world functions
- Apply the power rule to differentiate and integrate polynomial functions correctly
- Use differentiation to find stationary points and solve real-world optimisation problems
- Calculate the area under a curve using definite integration
- Apply both differentiation and integration to solve kinematic motion problems

## 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](https://www.owlsprep.com/study/ib-math-ai-sl-u5-derivative-as-gradient-of-tangent/) — Introduces the geometric interpretation of the derivative and its meaning as a rate of change.
- [Differentiation of polynomial functions (power rule)](https://www.owlsprep.com/study/ib-math-ai-sl-u5-differentiation-of-polynomial-functions/) — Teaches the power rule for differentiating any polynomial function.
- [Tangents, gradients, and stationary points](https://www.owlsprep.com/study/ib-math-ai-sl-u5-tangents-gradients-and-stationary-points/) — Shows how to use derivatives to find tangent equations and classify stationary points.
- [Optimisation problems with differentiation](https://www.owlsprep.com/study/ib-math-ai-sl-u5-optimisation-problems-with-differentiation/) — Applies stationary point analysis to solve real-world maximum/minimum problems.
- [Indefinite integration of polynomials](https://www.owlsprep.com/study/ib-math-ai-sl-u5-indefinite-integration-of-polynomials/) — Introduces integration as the reverse of differentiation and the power rule for integration.
- [Definite integrals and area under a curve](https://www.owlsprep.com/study/ib-math-ai-sl-u5-definite-integrals-and-area-under/) — Explains how to use definite integration to calculate the area between a curve and the x-axis.
- [Kinematic applications of calculus](https://www.owlsprep.com/study/ib-math-ai-sl-u5-kinematic-applications-of-calculus/) — Connects displacement, velocity and acceleration using differentiation and integration.

## Common pitfalls

- **Wrong:** Forgetting the constant of integration $+c$ when calculating indefinite integrals
  - Why it fails: This leads to automatic mark loss even if your integration calculation is correct
  - Correct: Always add $+c$ to the end of any indefinite integral result
- **Wrong:** Mixing up differentiation vs integration for kinematic problems
  - Why it fails: Many students reverse the relationship between displacement, velocity and acceleration
  - Correct: Differentiate to go from displacement → velocity → acceleration; integrate to go the opposite direction
- **Wrong:** Not adjusting for curves that dip below the x-axis when calculating area
  - Why it fails: Definite integrals return negative values for area below the x-axis, leading to incorrect net results
  - Correct: Split the integral into intervals, take the absolute value of each interval's result, then add them together

## Cheatsheet

| Concept | Key Formula/Rule |
| --- | --- |
| Power rule for differentiation | $\frac{d}{dx}(ax^n) = nax^{n-1}$ |
| Gradient of tangent at $x=a$ | $m = f'(a)$ |
| Stationary point condition | $f'(x) = 0$ |
| Power rule for integration | $\int ax^n dx = \frac{ax^{n+1}}{n+1} + c \quad (n \neq -1)$ |
| Area under a positive curve | $\text{Area} = \int_{a}^{b} f(x) dx$ |
| Acceleration from displacement | $a = \frac{d^2s}{dt^2} = \frac{dv}{dt}$ |
| Displacement from velocity | $s = \int v \, dt + c$ |

## 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.

- [Derivative as gradient of tangent and rate of change](https://www.owlsprep.com/study/ib-math-ai-sl-u5-derivative-as-gradient-of-tangent/)
- [Differentiation of polynomial functions (power rule)](https://www.owlsprep.com/study/ib-math-ai-sl-u5-differentiation-of-polynomial-functions/)
- [Tangents, gradients, and stationary points](https://www.owlsprep.com/study/ib-math-ai-sl-u5-tangents-gradients-and-stationary-points/)

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From [OwlsPrep](https://www.owlsprep.com) — free study guides for A-Level, IB, AP and IGCSE, written against the official syllabus. Canonical page: https://www.owlsprep.com/study/ib-math-ai-sl-u5-overview/