This course builds on students’ previous experience with functions and their developing
understanding of rates of change. Students will solve problems involving geometric and
algebraic representations of vectors and representations of lines and planes in three dimensional space; broaden their understanding of rates of change to include the
derivatives of polynomial, sinusoidal, exponential, rational, and radical functions; and
apply these concepts and skills to the modelling of real-world relationships. Students
will also refine their use of the mathematical processes necessary for success in senior
mathematics. This course is intended for students who choose to pursue careers in fields
such as science, engineering, economics, and some areas of business, including those
students who will be required to take a university-level calculus, linear algebra, or
physics course.

Prerequisite: Advanced Functions, Grade 12 (University Preparation)

Unit 1: Limits and Derivatives

In this unit, students will review important pre-calculus algebra skills, including rationalizing radical expressions and manipulating a variety of functions algebraically.

Unit 2: First Principles and Derivative Rules

In this unit, students will use limits and continuity to generalize rates of change for different types of functions, deriving the derivative of functions from first principles. Students will then generalize derivative rules for polynomial functions and a variety of functions, including the sum, difference, product, quotient, and composition of two or more functions.

Unit 3: Optimization and Curve Sketching

In this unit, students will explore the connection between algebraic and graphical features of functions. Students will investigate local and global extrema, intervals of increase/decrease, asymptotes, concavity, inflection points, and critical points to accurately sketch a variety of functions. They will learn to extend and apply these skills to solve real-world optimization and curve sketching problems.

Unit 4: Derivatives of Exponential and Trigonometric Functions

In this unit, students will learn to derive general derivative rules for trigonometric, exponential, and logarithmic functions and will use these rules to solve real-world optimization and geometry problems.

Unit 5: Vectors I

In this unit, students will learn to distinguish between vector and scalar values and explore the geometric, and algebraic definitions, and applications of vectors. Students will also perform basic algebraic and graphical vector manipulations in 2D and 3D to solve real-world problems.

Unit 6: Vectors II

In this unit, students will learn to represent real-world problems using vectors and explore different ways vectors can be manipulated in 2D and 3D. Students will evaluate the dot product and cross product of two vectors algebraically and geometrically, applying these skills to solve real-world problems. They will also explore vector projection and its relation to the dot and cross products.

Unit 7: Lines and Planes I

In this unit, students will explore the Cartesian coordinate system, using this system to describe points, lines, and planes. Students will explore different forms of equations that describe these geometric objects, including parametric, vector, and standard-form equations. They will synthesize these skills to describe and sketch planes in 3D.

Unit 8: Lines and Planes II

In this unit, students will explore the intersection of lines and planes in 3 dimensions and distinguish between different types of intersections. Students will also find the solution to a system of equations in 3 dimensions and subsequently find the intersection of 2D and 3D planes. Further, they will explore the distance between a point and a plane or line using systems of equations.