Calculus is the central topic in Grade 12 mathematics, focusing on limits, derivatives, and integration to study change, motion, and accumulation in mathematical and real-world situations.

Real-life Applications

• Calculus: Used to study motion, growth, and change, such as calculating speed or optimizing production in engineering.
• Derivatives: Help determine rates of change, such as how quickly a car’s speed is increasing or decreasing.
• Integration: Used to calculate quantities like total distance traveled, area under curves, or accumulated growth.
• Differential Equations: Help model real-world systems such as population growth, electrical circuits, and heat transfer.

AP Calculus AB

Equivalent to one semester of college Calculus I. Covers limits, derivatives, and integrals. Most common calculus path for 12th graders.

1. Limits & Continuity

Limits

• Limits in Calculus
• Estimating Limits from Graphs
• Estimating Limits from Tables
• One-Sided Limits
• When Does a Limit Not Exist
• Limit of a Function
• Real-life Applications of Limits

Calculate Limits

• Strategy in Finding Limits
• Limit Formula
• Calculating Limit of a Function
• Limits by Rationalization
• Limits using Algebraic Manipulation
• Limits at Infinity

Asymptotes

• Vertical Asymptote
• Horizontal Asymptote

Continuity

• Continuity of Functions
• Continuity at a Point
• Continuity and Discontinuity
• Intermediate Value Theorem
• Real-life Applications of Continuity

2. Differentiation — Definition & Basics

Derivatives

• Introduction to Derivatives
• Average and Instantaneous Rate of Change
• Derivatives as Rate of Change
• Limit Definition of Derivative

Derivative Rules

• Algebra of Derivative
• Derivative Rules
• Product Rule
• Quotient Rule
• Power Rule
• Chain Rule

Derivatives of Functions

• Derivative of a Function
• Derivatives of Polynomial Functions
• Derivatives of Trigonometric Functions
• Derivatives of Inverse Functions
• Derivative of Exponential Functions
• Derivative of Logarithmic Functions
• Derivative of Inverse Trigonometric Functions
• Derivative of Root x

3. Differentiation — Composite & Implicit

Derivative Strategies

• Implicit Differentiation
• Logarithmic Differentiation
• Finding Derivative with Fundamental Theorem of Calculus

Higher Order Derivatives

• Higher Order Derivatives
• Second-Order Derivative
• Advanced Differentiation

4. Contextual Applications of Differentiation

L'Hôpital's Rule

• L'Hôpital's Rule in Calculus
• When Does L'Hôpital's Rule Fail?

Applied Differentiation

• Linear Approximation Formula
• Derivatives as Rate of Change
• Related Rates

5. Analytical Applications of Differentiation

Examining Functions

• First Derivative Test
• Maxima and Minima
• Mean Value Theorem
• Absolute Minima and Maxima
• Second Derivative Test
• Extreme Value Theorem
• Curve Sketching

6. Integration & Accumulation of Change

Introduction to Integration

• Introduction to Integration
• Antiderivatives — Indefinite Integrals
• Power Rule of Integration

Techniques of Integration

• Integration by Substitution (U-sub)
• Properties of Definite Integrals

Definite Integrals

• Definite Integrals
• Area as Definite Integral
• Calculating Definite Integral
• Applications of Definite Integrals
• Fundamental Theorem of Calculus
• Finding Derivative with Fundamental Theorem of Calculus

Applications of Integration

• Area Under the Curve
• Riemann Sum
• Trapezoidal Rule
• Improper Integrals
• Methods of Integration
• Applications of Integration
• Average Value of a Function
• Volumes of Revolution: Disk & Washer Method
• Arc Length of a Curve

7. Differential Equations

Differential Equations

• Ordinary Differential Equations (ODEs)
• Linear Differential Equations
• Linear vs Nonlinear Differential Equations
• Homogeneous Differential Equations
• Second-Order Differential Equation
• Slope Fields
• Euler's Method
• Separable Differential Equations

AP Calculus BC

Includes everything in AP Calculus AB plus the following BC-exclusive topics. Equivalent to a full year of college Calculus I + II.

Note: Only the BC section is covered.

8. Advanced Integration

BC Integration Techniques

• Integration by Parts
• Improper Integrals
• Integration by Partial Fractions
• Tabular Integration

9. Parametric Equations

Parametric Curves

• Parametric Equations & Curves
• Derivatives of Parametric Equations
• Second Derivatives of Parametric Curves
• Arc Length of Parametric Curves

10. Polar Coordinates

Polar Functions

• Polar Coordinate System
• Derivatives of Polar Functions
• Areas Bounded by Polar Curves

11. Vector-Valued Functions

Vectors & Motion

• Vector-Valued Functions
• Vector Differentiation
• Integrating Vector-Valued Functions

12. Infinite Sequences & Series

Sequences & Convergence Tests

• Convergence & Divergence of Sequences
• Geometric Progression (Geometric Series)
• Important Conversions Tests
• Integral Test & p-Series Test
• Absolute and Conditional Convergence
• Ratio Test & Root Test

Power & Taylor Series

• Power Series & Radius of Convergence
• Taylor Series
• Maclaurin Series

AP Statistics

Equivalent to a one-semester, non-calculus college statistics course. Covers data analysis, probability, experimental design, and inference.

13. Exploring One-Variable Data

Data Types & Displays

• Types of Data in Statistics
• Descriptive Statistics
• Graphical Representation of Data
• Frequency Distribution Table
• Measures of Central Tendency
• Measures of Dispersion
• Boxplots & Five-Number Summary
• Normal Distribution & Z-Scores
• Empirical Rule

14. Exploring Two-Variable Data

Regression & Correlation

• Scatter Plot
• Correlation Coefficient Formula
• Regression Analysis
• Regression Line
• Residual Analysis
• Coefficient of Determination (r²)
• Outliers

15. Collecting Data (Study Design)

Sampling & Experimental Design

• Data Organization
• Sampling Methods
• Sampling Bias & Errors
• Principles of Experimental Design

16. Probability & Distributions

Probability Rules

• Basic Concepts of Probability
• Conditional Probability
• Bayes' Theorem
• Probability Distribution
• Random Variable
• Binomial Distribution
• Geometric Distribution
• Normal Distribution

17. Sampling Distributions

CLT & Sampling

• Inferential Statistics
• Central Limit Theorem
• Sampling Distribution
• Bias & Variability

18. Inference — Confidence Intervals

• Introduction to Confidence Intervals
• Introduction to Hypothesis Testing
• One-Proportion z-test
• Two-Proportion z-test

Pre-Calculus (Non-AP)

For students who completed Algebra 2 in Grade 11. Strong foundation for college-level math.

19. Functions & Graphs

Function Analysis

• Functions in Maths
• Domain and Range of a Function
• Graphing Function
• Relations in Maths
• Derivatives of Inverse Functions
• Even/Odd Functions
• Composition of Functions

20. Exponential & Logarithmic Functions

Exponential & Log

• Exponential Functions
• Exponential Graph
• Introduction to Logarithms
• Properties of Logarithms
• Logarithmic Function
• Solving Exponential Equations
• Change-of-Base Formula

21. Trigonometry

Trig Functions & Identities

• Unit Circle
• Radian Measure
• Trigonometric Functions
• Trigonometric Identities
• Trigonometric Graph
• Inverse Trigonometric Functions
• Inverse Trigonometric Identities
• Trigonometric Equations
• Sum & Difference Angle Formulas
• Double-Angle & Half-Angle Formulas

22. Conic Sections

Conics

• Circles
• Equation of a Circle
• Parabola
• Standard Equation of a Parabola
• Focus and Directrix of a Parabola
• Ellipse
• Ellipse Formula
• Hyperbola
• Hyperbola Formula
• Completing the Square to Identify Conics

23. Matrices & Linear Systems

Matrices

• Matrices
• Partial Differential Equations
• Determinant of a Matrix
• Inverse of a Matrix
• Cramer's Rule
• Row Reduction / Gaussian Elimination

24. Sequences, Series & Probability

Sequences & Probability

• Arithmetic Progression
• Geometric Progression
• Permutations and Combinations
• Principle of Counting
• Infinite Geometric Series
• Binomial Theorem
• Mathematical Induction

25. Introduction to Calculus

Limits & Derivatives Preview

• Limits in Calculus
• One-Sided Limits
• Continuity of Functions
• Average and Instantaneous Rate of Change
• Introduction to Derivatives
• Power Rule
• Introduction to Integration
• Area Under the Curve