Will this help with my child's Grade 12 Calculus and Vectors homework?

Yes. StudyPug covers every MCV4U topic, from limits and differentiation rules to dot products and equations of planes, with worked examples and practice problems that mirror exactly what Ontario teachers assign.

How long are the video lessons for MCV4U?

Most video lessons run between 5 and 15 minutes, so your child can watch a focused explanation for one concept, pause, and immediately try the matching practice problems without losing momentum.

Is there a provincial test for Grade 12 math in Ontario?

Grade 12 is not an EQAO assessment year. EQAO assesses math in Grades 3, 6, and 9. MCV4U students are typically assessed through school-based tests, culminating tasks, and university admission requirements.

Can my child use StudyPug on a tablet or phone?

Yes. StudyPug works on tablets, phones, and computers, so your child can review a calculus lesson on a tablet at home or check a vectors worked example on a phone before class.

Can I try StudyPug before paying?

Yes. StudyPug offers free sample lessons so you can explore the content before committing. There is also a 30-day money-back guarantee, giving your family time to make sure it is the right fit.

ON Grade 12 Math Curriculum

Give your Grade 12 student the support they need to succeed in Calculus and Vectors. Every MCV4U topic is covered with clear lessons and practice problems matched to what Ontario schools teach.

ON Grade 12 Math Curriculum

OE_IDExpectationsStudyPug Topic

ON.OE.12CV.A1.1Describe examples of real-world applications of rates of change, represented in a variety of ways
- Rate of change
- Difference quotient: applications of functions
ON.OE.12CV.A1.2Describe connections between the average rate of change of a function that is smooth over an interval and the slope of the corresponding secant, and between the instantaneous rate of change of a smooth function at a point and the slope of the tangent at that point
- Slope and equation of tangent line
- Linear approximation
ON.OE.12CV.A1.3Make connections, with or without graphing technology, between an approximate value of the instantaneous rate of change at a given point on the graph of a smooth function and average rates of change over intervals containing the point
- Critical number & maximum and minimum values
- Estimating Derivatives from a table
ON.OE.12CV.A1.4Recognize, through investigation with and without technology, graphical and numerical examples of limits, and explain the reasoning involved
- Finding limits from graphs
- Introduction to Calculus - Limits
ON.OE.12CV.A1.5Make connections, for a function that is smooth over the interval a ? x ? a + h, between the average rate of change of the function over this interval and the value of the expression [f(a + h) - f(a)]/h, and between the instantaneous rate of change of the function at x = a and the value of the limit of this expression as h approaches 0
- Definition of derivative
- Limit laws
ON.OE.12CV.A1.6Compare, through investigation, the calculation of instantaneous rates of change at a point for polynomial functions, with and without simplifying the expression [f(a + h) - f(a)]/h before substituting values of h that approach zero
- Finding limits algebraically - direct substitution
- Finding limits algebraically - when direct substitution is not possible
ON.OE.12CV.A2.1Determine numerically and graphically the intervals over which the instantaneous rate of change is positive, negative, or zero for a function that is smooth over these intervals
- Intermediate value theorem
- Curve sketching
ON.OE.12CV.A2.2Generate, through investigation using technology, a table of values showing the instantaneous rate of change of a polynomial function for various values of x, graph the ordered pairs, recognize that the graph represents a function called the derivative, and make connections between the graphs of f(x) and f'(x)
- Power rule
- Derivative of trigonometric functions
ON.OE.12CV.A2.3Determine the derivatives of polynomial functions by simplifying the algebraic expression [f(x + h) - f(x)]/h and then taking the limit of the simplified expression as h approaches zero
- Derivative of exponential functions
- Derivative of logarithmic functions
ON.OE.12CV.A2.4Determine, through investigation using technology, the graph of the derivative f'(x) of a given sinusoidal function
- Implicit differentiation
- Higher order derivatives
ON.OE.12CV.A2.5Determine, through investigation using technology, the graph of the derivative f'(x) of a given exponential function
- Product rule
- Quotient rule
ON.OE.12CV.A2.6Determine, through investigation using technology, the exponential function f(x) = a^x (a > 0, a ? 1) for which f'(x) = f(x), identify the number e to be the value of a for which f'(x) = f(x), and recognize that for the exponential function f(x) = e^x the slope of the tangent at any point on the function is equal to the value of the function at that point
- Chain rule
- l'Hospital's rule
ON.OE.12CV.A2.7Recognize that the natural logarithmic function f(x) = log_e x, also written as f(x) = ln x, is the inverse of the exponential function f(x) = e^x, and make connections between f(x) = ln x and f(x) = e^x
- Natural log: ln
- Evaluating logarithms using change-of-base formula
- Evaluating logarithms using logarithm rules
ON.OE.12CV.A2.8Verify, using technology, that the derivative of the exponential function f(x) = a^x is f'(x) = a^x ln a for various values of a
- Derivative of inverse trigonometric functions
- Mean value theorem
ON.OE.12CV.A3.1Verify the power rule for functions of the form f(x) = x^n, where n is a natural number
- Position velocity acceleration
- Related rates
ON.OE.12CV.A3.2Verify the constant, constant multiple, sum, and difference rules graphically and numerically
- Optimization
- Demand, revenue, cost & profit
ON.OE.12CV.A3.3Determine algebraically the derivatives of polynomial functions, and use these derivatives to determine the instantaneous rate of change at a point and to determine point(s) at which a given rate of change occurs
- Marginal revenue, and maximizing revenue & average revenue
- Marginal cost, and minimizing cost & average cost
ON.OE.12CV.A3.4Verify that the power rule applies to functions of the form f(x) = x^n, where n is a rational number, and verify algebraically the chain rule using monomial functions and the product rule using polynomial functions
- Marginal profit, and maximizing profit & average profit
- Elasticity of demand
ON.OE.12CV.A3.5Solve problems, using the product and chain rules, involving the derivatives of polynomial functions, sinusoidal functions, exponential functions, rational functions, radical functions, and other simple combinations of functions
- Consumer and producer surplus
- Continuous money flow
ON.OE.12CV.B1.1Sketch the graph of a derivative function, given the graph of a function that is continuous over an interval, and recognize points of inflection of the given function
- Antiderivatives
- Riemann sum
ON.OE.12CV.B1.2Recognize the second derivative as the rate of change of the rate of change, and sketch the graphs of the first and second derivatives, given the graph of a smooth function
- Fundamental theorem of calculus
- Definite integral
ON.OE.12CV.B1.3Determine algebraically the equation of the second derivative f"(x) of a polynomial or simple rational function f(x), and make connections between the key features of the graph of the function and corresponding features of the graphs of its first and second derivatives
- U-Substitution
- Integration using trigonometric identities
ON.OE.12CV.B1.4Describe key features of a polynomial function, given information about its first and/or second derivatives, sketch two or more possible graphs of the function that are consistent with the given information, and explain why an infinite number of graphs is possible
- Trigonometric substitution
- Integration by parts
ON.OE.12CV.B1.5Sketch the graph of a polynomial function, given its equation, by using a variety of strategies to determine its key features
- Integration of rational functions by partial fractions
- Numerical integration
ON.OE.12CV.B2.1Make connections between the concept of motion and the concept of the derivative in a variety of ways
- Areas between curves
- Average value of a function
ON.OE.12CV.B2.2Make connections between the graphical or algebraic representations of derivatives and real-world applications
- Volumes of solids of revolution - Disc method
- Volumes of solid with known cross-sections
ON.OE.12CV.B2.3Solve problems, using the derivative, that involve instantaneous rates of change, including problems arising from real-world applications, given the equation of a function
- Volumes of solid of revolution - Shell method
- Arc length
ON.OE.12CV.B2.4Solve optimization problems involving polynomial, simple rational, and exponential functions drawn from a variety of applications, including those arising from real-world situations
- Order and solutions to differential equations
- Separable equations
ON.OE.12CV.B2.5Solve problems arising from real-world applications by applying a mathematical model and the concepts and procedures associated with the derivative to determine mathematical results, and interpret and communicate the results
- Modeling with differential equations
- Slope fields
ON.OE.12CV.C1.1Recognize a vector as a quantity with both magnitude and direction, and identify, gather, and interpret information about real-world applications of vectors
- Introduction to vectors
- Magnitude of a vector
ON.OE.12CV.C1.2Represent a vector in two-space geometrically as a directed line segment, with directions expressed in different ways, and algebraically, and recognize vectors with the same magnitude and direction but different positions as equal vectors
- Direction angle of a vector
- Scalar multiplication of vectors
ON.OE.12CV.C1.3Determine, using trigonometric relationships, the Cartesian representation of a vector in two-space given as a directed line segment, or the representation as a directed line segment of a vector in two-space given in Cartesian form
- Equivalent vectors
- Adding and subtracting vectors in component form
ON.OE.12CV.C1.4Recognize that points and vectors in three-space can both be represented using Cartesian coordinates, and determine the distance between two points and the magnitude of a vector using their Cartesian representations
- Operations on vectors in magnitude and direction form
- Unit vector
ON.OE.12CV.C2.1Perform the operations of addition, subtraction, and scalar multiplication on vectors represented as directed line segments in two-space, and on vectors represented in Cartesian form in two-space and three-space
- Notation of matrices
- Word problems on vectors
ON.OE.12CV.C2.2Determine, through investigation with and without technology, some properties of the operations of addition, subtraction, and scalar multiplication of vectors
- Adding and subtracting matrices
- Scalar multiplication of matrices
ON.OE.12CV.C2.3Solve problems involving the addition, subtraction, and scalar multiplication of vectors, including problems arising from real-world applications
- Matrix multiplication
- The three types of matrix row operations
ON.OE.12CV.C2.4Perform the operation of dot product on two vectors represented as directed line segments and in Cartesian form in two-space and three-space, and describe applications of the dot product
- Representing a linear system as a matrix
- Solving a linear system with matrices using Gaussian elimination
ON.OE.12CV.C2.5Determine, through investigation, properties of the dot product
- Zero matrix
- Identity matrix
ON.OE.12CV.C2.6Perform the operation of cross product on two vectors represented in Cartesian form in three-space, determine the magnitude of the cross product, and describe applications of the cross product
- Properties of matrix addition
- Properties of scalar multiplication
ON.OE.12CV.C2.7Determine, through investigation, properties of the cross product
- Properties of matrix multiplication
- The determinant of a 2 x 2 matrix
ON.OE.12CV.C2.8Solve problems involving dot product and cross product, including problems arising from real-world applications
- The determinant of a 3 x 3 matrix (General & Shortcut Method)
- The Inverse of a 2 x 2 matrix
ON.OE.12CV.C3.1Recognize that the solution points (x, y) in two-space of a single linear equation in two variables form a line and that the solution points (x, y) in two-space of a system of two linear equations in two variables determine the point of intersection of two lines, if the lines are not coincident or parallel
- Graphing linear inequalities in two variables
- The inverse of 3 x 3 matrices with matrix row operations
- The inverse of 3 x 3 matrix with determinants and adjugate
ON.OE.12CV.C3.2Determine, through investigation with technology and without technology, that the solution points (x, y, z) in three-space of a single linear equation in three variables form a plane and that the solution points (x, y, z) in three-space of a system of two linear equations in three variables form the line of intersection of two planes, if the planes are not coincident or parallel
- 2 x 2 invertible matrix
- Solving linear systems using Cramer's Rule
ON.OE.12CV.C3.3Determine, through investigation using a variety of tools and strategies, different geometric configurations of combinations of up to three lines and/or planes in three-space
- Solving linear systems using 2 x 2 inverse matrices
- Transforming vectors with matrices
ON.OE.12CV.C4.1Recognize a scalar equation for a line in two-space to be an equation of the form Ax + By + C = 0, represent a line in two-space using a vector equation and parametric equations, and make connections between a scalar equation, a vector equation, and parametric equations of a line in two-space
- General form: Ax + By + C = 0
- Point-slope form: y - y_1 = m(x - x_1)
- Transforming shapes with matrices
- Finding the transformation matrix
ON.OE.12CV.C4.2Recognize that a line in three-space cannot be represented by a scalar equation, and represent a line in three-space using the scalar equations of two intersecting planes and using vector and parametric equations
- 3-D coordinate system
- 3-Dimensional vectors
ON.OE.12CV.C4.3Recognize a normal to a plane geometrically and algebraically, and determine, through investigation, some geometric properties of the plane
- Dot product
- Cross product
ON.OE.12CV.C4.4Recognize a scalar equation for a plane in three-space to be an equation of the form Ax + By + Cz + D = 0 whose solution points make up the plane, determine the intersection of three planes represented using scalar equations by solving a system of three linear equations in three unknowns algebraically, and make connections between the algebraic solution and the geometric configuration of the three planes
- 3-Dimensional lines
- 3-Dimensional planes
ON.OE.12CV.C4.5Determine, using properties of a plane, the scalar, vector, and parametric equations of a plane
- Cylindrical coordinates
- Spherical-coordinates
ON.OE.12CV.C4.6Determine the equation of a plane in its scalar, vector, or parametric form, given another of these forms
- Calculus for vector functions
- Tangent, normal, and binormal vectors
ON.OE.12CV.C4.7Solve problems relating to lines and planes in three-space that are represented in a variety of ways and involving distances or intersections, and interpret the result geometrically
- Arc length with vector functions
- Curvature with vector functions

Grade 12 Calculus and Vectors (MCV4U)

About Grade 12 Ontario math

Ontario follows the Ontario Curriculum, organized around Big Ideas and expectations that develop both conceptual understanding and procedural fluency. MCV4U covers rates of change and derivatives, including limits, differentiation rules, and applications of derivatives to curve sketching and optimization, as well as a full strand on vectors, lines, and planes in two-space and three-space. StudyPug provides step-by-step lessons and practice problems aligned to every MCV4U expectation so students can work through any topic at their own pace.

Strands: Rates of Change, Derivatives and Their Applications, Vectors in Two-Space and Three-Space, and Lines and Planes
Step-by-step lessons and worked examples for every MCV4U topic
Practice problems with full solutions aligned to the Ontario Curriculum
Accessible on any device so your child can study at home or on the go

Curriculum topics by strand

Find lessons and practice for every Grade 12 MCV4U topic, aligned to the Ontario Curriculum.

Topic includes:

Practice

Video

Quiz

OE_ID

Expectations

StudyPug Topic

Ontario Grade 12 Math Topics

Rates of Change: Real-world applications of rates of change, average and instantaneous rates of change, secant and tangent slopes, limits, and the difference quotient.
Derivatives and Their Applications: Derivatives of polynomial, sinusoidal, exponential, rational, and radical functions; the power, product, and chain rules; first and second derivatives; curve sketching; and optimization problems.
Vectors in Two-Space and Three-Space: Geometric and algebraic representation of vectors, vector addition, subtraction, scalar multiplication, dot product, cross product, and real-world applications.
Lines and Planes: Scalar, vector, and parametric equations of lines in two-space and three-space; equations of planes; intersections and geometric configurations of lines and planes; and distance problems.

How StudyPug Helps Ontario Grade 12 Math Students

When a student gets stuck on a calculus problem late at night, StudyPug is there with a clear, step-by-step explanation of exactly that concept. Instead of waiting until the next class, they can watch a focused lesson, work through practice problems, and confirm they understand the idea before moving on.

For the vectors and lines-and-planes strands, where three-dimensional reasoning can be challenging, StudyPug breaks each expectation into manageable steps. Students can revisit any lesson as many times as they need, building the confidence required for university-level mathematics.

Aligned to Ontario Grade 12 Math Curriculum

Every lesson and practice problem on StudyPug is built to match the Ontario Curriculum expectations for MCV4U, covering all strands from rates of change and differentiation through to lines and planes in three-space, so what your child studies here is exactly what their Ontario teacher is teaching in class.

Ready to put your Grade 12 Math skills to the test?

Jump into practice questions for every ON Grade 12 topic - with instant feedback and step-by-step video solutions when you get stuck.

Start Practicing

Written by the StudyPug Math Team

Last updated June 5, 2026

3 M+

students

★★★★★

4.8 average

K-12

all grades