# Calculus

> IB Mathematics: Analysis and Approaches SL · IB AA SL
> Source: https://www.owlsprep.com/study/ib-math-aa-sl-u5-overview/
> Weight: 20-23% of overall exam

This unit introduces core calculus concepts including differentiation, integration, and their real-world and analytical applications. It is the highest weighted topic in IB AA SL, forming the basis for many exam problems.

**Prerequisites:** [Mastery of polynomial, rational, trigonometric, exponential and logarithmic functions](https://www.owlsprep.com/study/ib-math-aa-sl-u4-overview/)

## Learning objectives

- Understand the core concepts of limits, differentiation and integration that form the foundation of calculus
- Apply standard differentiation and integration rules to all common function types taught in IB AA SL
- Use calculus to analyze function behavior including monotonicity, stationary points and concavity
- Solve real-world optimization problems and calculate areas/volumes of revolution using integration
- Explain the connection between differentiation and integration via the Fundamental Theorem of Calculus

## 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](https://www.owlsprep.com/study/ib-math-aa-sl-u5-introduction-to-limits-and-differentiation/) — Learn the definition of the derivative via the foundational concept of limits
- [Basic differentiation rules: power, product, quotient, chain rule](https://www.owlsprep.com/study/ib-math-aa-sl-u5-basic-differentiation-rules-power-product/) — Master core rules for differentiating combinations of basic functions
- [Differentiation of trigonometric, exponential and logarithmic functions](https://www.owlsprep.com/study/ib-math-aa-sl-u5-differentiation-of-trigonometric-exponential-and/) — Learn derivative formulas for non-polynomial function types
- [Stationary points, monotonicity and concavity](https://www.owlsprep.com/study/ib-math-aa-sl-u5-stationary-points-monotonicity-and-concavity/) — Use derivatives to analyze the shape and behavior of function graphs
- [Optimization applications](https://www.owlsprep.com/study/ib-math-aa-sl-u5-optimization-applications/) — Apply differentiation to solve practical maximum and minimum problems
- [Indefinite integration and anti-differentiation](https://www.owlsprep.com/study/ib-math-aa-sl-u5-indefinite-integration-and-anti-differentiation/) — Introduce anti-differentiation as the reverse process of differentiation
- [Definite integration and fundamental theorem of calculus](https://www.owlsprep.com/study/ib-math-aa-sl-u5-definite-integration-and-fundamental-theorem/) — Learn the definition of definite integrals and the link to differentiation
- [Integration by substitution, area and volume of revolution](https://www.owlsprep.com/study/ib-math-aa-sl-u5-integration-by-substitution-area-and/) — Master substitution method and apply integration to area and volume problems

## Common pitfalls

- **Wrong:** Forgetting the constant of integration when calculating indefinite integrals
  - Why it fails: This leads to lost easy marks and incorrect results for follow-up problems
  - Correct: Always add $+C$ to the final result of any indefinite integral calculation
- **Wrong:** Misapplying the chain rule for differentiation or forgetting to adjust the differential for substitution
  - Why it fails: This is the most common error in routine calculus questions on IB exams
  - Correct: Always account for the derivative of the inner function for both operations
- **Wrong:** Mixing up volume of revolution formulas for rotation around x vs y axes
  - Why it fails: This leads to incorrect integral setups and lost application marks
  - Correct: Recall $V = \pi \int y^2 dx$ for rotation around x-axis and $V = \pi \int x^2 dy$ for rotation around y-axis

## Cheatsheet

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

## 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.

- [Introduction to limits and differentiation](https://www.owlsprep.com/study/ib-math-aa-sl-u5-introduction-to-limits-and-differentiation/)
- [Basic differentiation rules: power, product, quotient, chain rule](https://www.owlsprep.com/study/ib-math-aa-sl-u5-basic-differentiation-rules-power-product/)
- [Differentiation of trigonometric, exponential and logarithmic functions](https://www.owlsprep.com/study/ib-math-aa-sl-u5-differentiation-of-trigonometric-exponential-and/)

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