MCV4U – Calculus and Vectors Grade 12 builds on students’ understanding of functions and rates of change. Students solve problems involving vectors, as well as lines and planes in three-dimensional space, while expanding their knowledge of derivatives across various function types. They apply these concepts to model real-world relationships and strengthen key mathematical processes. This course is designed for students pursuing post-secondary studies in fields such as science, engineering, economics, and business.
Course curriculum
Inside %course-code%: from limits to three-space
The engineering and science prerequisite
Intended for students pursuing science, engineering, economics and some areas of business — including anyone who will take a university-level calculus, linear algebra or physics course.
The derivative, built up step by step
From average and instantaneous rates of change to limits, then derivative rules you verify graphically and algebraically — applied to polynomial, rational, radical, sinusoidal and exponential functions.
Optimization and modelling
Curve sketching from first and second derivatives, then optimization problems drawn from real-world applications, where you build the mathematical model yourself.
Vectors, lines and planes in 3D
Vectors in two- and three-space, algebraically and geometrically, then scalar, vector and parametric equations of lines and planes — and the distance and intersection problems they make solvable.
MCV4U Course outline
Full unit-by-unit breakdown.
Connect average rate of change over an interval to instantaneous rate of change at a point, using the slopes of secants and tangents and the concept of the limit. Learn limit properties and the difference between continuous and discontinuous functions.
Key topics
Secant and tangent slopes · the limit · limit properties · continuity
Assessed by
Quiz · end-of-unit meeting · proctored unit test
Graph the derivatives of polynomial functions and connect their numeric, graphical and algebraic representations. Verify the derivative rules graphically and algebraically, then apply them to polynomial, rational and radical functions and simple combinations of functions.
Key topics
Derivative of a polynomial · power, product, quotient and chain rules · rational and radical functions
Assessed by
Quiz · culminating assignment · end-of-unit meeting · proctored unit test
Make connections, graphically and algebraically, between the key features of a function and its first and second derivatives, and use them to sketch curves. Solve optimization problems, including those from real-world applications that involve building a mathematical model.
Key topics
Increasing and decreasing intervals · concavity and points of inflection · curve sketching · optimization problems
Assessed by
Quiz · culminating assignment · end-of-unit meeting · proctored unit test
Graph the derivatives of sinusoidal and exponential functions, connect their representations, and verify and apply the derivative rules to these functions and simple combinations of them.
Key topics
Derivatives of sine and cosine · derivatives of exponential functions · combinations of functions
Assessed by
Quiz · culminating assignment · end-of-unit meeting · proctored unit test
Represent vectors in two-space and three-space algebraically and geometrically, recognize their real-world applications, and perform and apply vector operations to solve problems.
Key topics
Geometric and algebraic vectors · vector addition and scalar multiplication · dot and cross products · applications
Assessed by
Culminating assignment · end-of-unit meeting · proctored unit test
Distinguish the geometric representations of a single linear equation or a system of two in two-space and three-space, determine configurations of lines and planes in three-space, and represent them with scalar, vector and parametric equations to solve distance and intersection problems.
Key topics
Vector, parametric and scalar equations · lines and planes in three-space · intersections and distances · systems of equations
Assessed by
Research-based culminating project (applications of derivatives, vectors and rates of change) · end-of-unit meeting · proctored unit test
