# Calculus

> IB Mathematics Applications and Interpretation HL · IB AI HL
> Source: https://www.owlsprep.com/study/ib-math-ai-hl-u4-overview/
> Weight: 20-25% of overall IB AI HL exam

This unit covers core calculus from differentiation through integration to differential equations, the highest-weighted topic in IB AI HL, used for modeling real-world phenomena across applied contexts.

**Prerequisites:** [Unit 3: Functions and Equations](https://www.owlsprep.com/study/ib-math-ai-hl-u3-overview/)

## Learning objectives

- Explain the conceptual foundations of differentiation and integration using the idea of limits
- Apply differentiation and integration rules to compute derivatives and integrals of common functions
- Use calculus to analyze function behavior and solve applied problems including optimization and related rates
- Set up and solve first-order differential equations for modeling real-world growth, decay and logistic processes
- Use integration to calculate areas and volumes of revolution for applied and graphical problems

## Unit at a Glance

We progress from conceptual foundations of derivatives to applied problem-solving and modeling, building each new skill on the previous one. We start with differentiation, then use it to analyze functions and solve applied problems. Next, we cover integration, its techniques and applications for area and volume. Finally, we end with first-order differential equations for real-world modeling.

This unit includes the following sub-topics:
- [Differentiation rules: power, product, quotient, chain](https://www.owlsprep.com/study/ib-math-ai-hl-u4-differentiation-rules-power-product-quotient/) — Learn core rules to quickly compute derivatives of complex combinations of functions.
- [Derivatives of standard functions](https://www.owlsprep.com/study/ib-math-ai-hl-u4-derivatives-of-standard-functions/) — Memorize and apply derivative formulas for polynomials, exponentials, logs, and trigonometric functions.
- [Tangents, normals, critical points, inflection points](https://www.owlsprep.com/study/ib-math-ai-hl-u4-tangents-normals-critical-points-inflection/) — Use derivatives to find tangent/normal equations and locate key points on function graphs.
- [Monotonic functions, second derivative test](https://www.owlsprep.com/study/ib-math-ai-hl-u4-monotonic-functions-second-derivative-test/) — Use derivatives to classify extrema and identify intervals where functions are increasing or decreasing.
- [Optimization problems](https://www.owlsprep.com/study/ib-math-ai-hl-u4-optimization-problems/) — Apply differentiation to find maximum and minimum values for real-world applied problems.
- [Related rates](https://www.owlsprep.com/study/ib-math-ai-hl-u4-related-rates/) — Solve problems involving connected rates of change of quantities over time.
- [Antiderivatives and indefinite integration](https://www.owlsprep.com/study/ib-math-ai-hl-u4-antiderivatives-and-indefinite-integration/) — Introduce antiderivatives and basic rules for indefinite integration.
- [Definite integrals and area under curves](https://www.owlsprep.com/study/ib-math-ai-hl-u4-definite-integrals-and-area-under/) — Connect definite integrals to area under curves and apply the Fundamental Theorem of Calculus.
- [Integration by substitution](https://www.owlsprep.com/study/ib-math-ai-hl-u4-integration-techniques-substitution-and-by/) — Learn the substitution method to integrate more complex non-standard functions.
- [Area between curves, volumes of revolution](https://www.owlsprep.com/study/ib-math-ai-hl-u4-area-between-curves-volumes-of/) — Apply integration to calculate area between curves and volume of solids of revolution.
- [First order differential equations: Euler's method](https://www.owlsprep.com/study/ib-math-ai-hl-u4-first-order-differential-equations-euler/) — Use numerical Euler's method to approximate solutions to differential equations.
- [Separation of variables for differential equations](https://www.owlsprep.com/study/ib-math-ai-hl-u4-separation-of-variables-for-differential/) — Learn the analytic separation of variables method to solve first-order differential equations.
- [Differential equations for growth, decay and logistic models](https://www.owlsprep.com/study/ib-math-ai-hl-u4-differential-equations-for-growth-decay/) — Apply differential equations to model real-world population growth, decay and limited logistic growth.

## Common pitfalls

- **Wrong:** Forgetting the constant of integration for indefinite integrals and general differential equation solutions.
  - Why it fails: This is a common exam mistake that costs easy method and accuracy marks.
  - Correct: Always add +C to every indefinite integral result, and retain the constant in general DE solutions.
- **Wrong:** Mixing up the product rule and chain rule when differentiating composite functions.
  - Why it fails: Leads to incorrect derivative calculations for all subsequent steps in a problem.
  - Correct: Confirm if you are multiplying two separate functions (product rule) or composing one function into another (chain rule).
- **Wrong:** Forgetting to square the function when setting up the integral for volume of revolution.
  - Why it fails: Incorrect integral setup leads to wrong final volume even if integration is calculated correctly.
  - Correct: Recall $V = \pi \int_a^b [f(x)]^2 dx$, always square the radius function before integrating.

## Cheatsheet

| Concept | Key Formula/Rule |
| --- | --- |
| Chain Rule | $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$ |
| Fundamental Theorem of Calculus | $\int_a^b f(x) dx = F(b) - F(a)$ where $F'(x)=f(x)$ |
| Volume of revolution (x-axis) | $V = \pi \int_a^b [y]^2 dx$ |
| Second Derivative Test | If $f'(c)=0$: $f''(c) > 0$ = minimum, $f''(c) < 0$ = maximum |
| Separation of Variables | $\int \frac{1}{f(y)} dy = \int g(x) dx$ for $\frac{dy}{dx} = f(y)g(x)$ |
| Logistic DE Model | $\frac{dP}{dt} = rP\left(1-\frac{P}{K}\right)$ where $r$ = growth rate, $K$ = carrying capacity |

## What's next

Begin your study of this unit with the first sub-topic, which builds the conceptual foundation of derivatives from limits that all subsequent calculus topics rely on. Once you complete all sub-topics in this unit, you can move on to the next core unit of IB AI HL focused on statistics and probability.

- [Unit 5: Statistics and Probability](https://www.owlsprep.com/study/ib-math-ai-hl-u5-overview/)
- [Differentiation rules: power, product, quotient, chain](https://www.owlsprep.com/study/ib-math-ai-hl-u4-differentiation-rules-power-product-quotient/)

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