Unit Overview

Calculus

IB Mathematics Applications and Interpretation HL· 5 min read 📊 20-25% of overall IB AI HL exam

1. 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:

Limit definition of the derivative

Introduce the formal limit-based definition of the derivative and its conceptual meaning.

★★★⏱ 10 min

Differentiation rules: power, product, quotient, chain

Learn core rules to quickly compute derivatives of complex combinations of functions.

★★⏱ 12 min

Derivatives of standard functions

Memorize and apply derivative formulas for polynomials, exponentials, logs, and trigonometric functions.

★★⏱ 8 min

Tangents, normals, critical points, inflection points

Use derivatives to find tangent/normal equations and locate key points on function graphs.

★★★⏱ 10 min

Monotonic functions, second derivative test

Use derivatives to classify extrema and identify intervals where functions are increasing or decreasing.

★★★⏱ 10 min

Optimization problems

Apply differentiation to find maximum and minimum values for real-world applied problems.

★★★★⏱ 12 min

Related rates

Solve problems involving connected rates of change of quantities over time.

★★★★⏱ 10 min

Antiderivatives and indefinite integration

Introduce antiderivatives and basic rules for indefinite integration.

★★★⏱ 9 min

Definite integrals and area under curves

Connect definite integrals to area under curves and apply the Fundamental Theorem of Calculus.

★★★⏱ 10 min

Integration techniques: substitution and by parts

Learn two core methods to integrate more complex non-standard functions.

★★★★⏱ 12 min

Area between curves, volumes of revolution

Apply integration to calculate area between curves and volume of solids of revolution.

★★★★⏱ 11 min

First order differential equations: Euler's method

Use numerical Euler's method to approximate solutions to differential equations.

★★★⏱ 10 min

Separation of variables for differential equations

Learn the analytic separation of variables method to solve first-order differential equations.

★★★⏱ 10 min

Differential equations for growth, decay and logistic models

Apply differential equations to model real-world population growth, decay and limited logistic growth.

★★★★⏱ 12 min

2. Common Pitfalls

Wrong move:

Forgetting the constant of integration for indefinite integrals and general differential equation solutions.

Why:

This is a common exam mistake that costs easy method and accuracy marks.

Correct move:

Always add +C to every indefinite integral result, and retain the constant in general DE solutions.

Wrong move:

Mixing up the product rule and chain rule when differentiating composite functions.

Why:

Leads to incorrect derivative calculations for all subsequent steps in a problem.

Correct move:

Confirm if you are multiplying two separate functions (product rule) or composing one function into another (chain rule).

Wrong move:

Forgetting to square the function when setting up the integral for volume of revolution.

Why:

Incorrect integral setup leads to wrong final volume even if integration is calculated correctly.

Correct move:

Recall $V = p i in t_{a}^{b} [f (x)]^{2} d x$ , always square the radius function before integrating.

3. Quick Reference Cheatsheet

Concept	Key Formula/Rule
Limit definition of derivative	$f^{'} (x) = lim_{h \to 0} \frac{f ( x + h ) - f ( x )}{h}$
Chain Rule	$\frac{d y}{d x} = \frac{d y}{d u} \cdot \frac{d u}{d x}$
Fundamental Theorem of Calculus	$\int_{a}^{b} f (x) d x = F (b) - F (a)$ where $F^{'} (x) = f (x)$
Integration by Parts	$\int u \frac{d v}{d x} d x = uv - \int v \frac{d u}{d x} d x$
Volume of revolution (x-axis)	$V = π \int_{a}^{b} [y]^{2} d x$
Second Derivative Test	If $f^{'} (c) = 0$ : $f^{''} (c) > 0$ = minimum, $f^{''} (c) < 0$ = maximum
Separation of Variables	$\int \frac{1}{f ( y )} d y = \int g (x) d x$ for $\frac{d y}{d x} = f (y) g (x)$
Logistic DE Model	$\frac{d P}{d t} = r P (1 - \frac{P}{K})$ 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.