Ontario · Grade 12
Calculus and Vectors, Grade 12 (University)
- MCV4U
Ontario’s Grade 12 math course, from Rate of Change to Geometry and Algebra of Vectors.
- Units
- 3
- Lessons
- 9
- Skills
- 54
- Provincial exam
- None
- Official curriculum
- Ontario Ministry of Education
Unit by unit
Every skill in the course, numbered by unit.
Unit 1 · 21 skills
Rate of Change
Investigating Instantaneous Rate of Change at a Point
Real-world examples of rates of change
I can describe real-world examples of rates of change in different representations.
Skill 1.1
Connecting average rate of change to secants and instantaneous rate to tangents
I can connect average rate of change to the secant's slope, and instantaneous rate of change to the tangent's slope.
Skill 1.2
Approximating instantaneous rate of change using secants approaching the tangent
I can approximate an instantaneous rate of change using secants that approach the tangent.
Skill 1.3
Recognizing limits graphically and numerically
I can recognize graphical and numerical examples of limits and explain the reasoning.
Skill 1.4
Connecting average and instantaneous rate of change to the limit expression
I can connect rate of change to the limit expression (f(a+h) − f(a))/h.
Skill 1.5
Comparing simplified and unsimplified instantaneous rate calculations
I can compare calculating instantaneous rates of change with and without first simplifying the difference quotient.
Skill 1.6
Determining intervals where instantaneous rate of change is positive, negative, or zero
I can determine intervals where the instantaneous rate of change is positive, negative, or zero.
Skill 1.7
Investigating the Concept of the Derivative Function
Graphing the derivative from a table of instantaneous rates of change
I can generate a table of instantaneous rates of change and graph the resulting derivative function.
Skill 1.8
Determining polynomial derivatives using the limit definition
I can determine the derivative of a polynomial function using the limit definition.
Skill 1.9
Investigating the derivative graph of a sinusoidal function
I can investigate the derivative graph of a sinusoidal function.
Skill 1.10
Investigating the derivative graph of an exponential function
I can investigate the derivative graph of an exponential function.
Skill 1.11
Discovering the number e as the base where f'(x) = f(x)
I can discover the number e as the base for which f'(x) = f(x) exactly.
Skill 1.12
The natural logarithm as the inverse of the exponential function e^x
I can recognize the natural logarithm as the inverse of f(x) = e^x.
Skill 1.13
Verifying the derivative of a general exponential function
I can verify that the derivative of f(x) = a^x is f'(x) = a^x ln a.
Skill 1.14
Investigating the Properties of Derivatives
Verifying the power rule for natural-number exponents
I can verify the power rule for functions with natural-number exponents.
Skill 1.15
Verifying the constant, constant multiple, sum, and difference rules
I can verify the constant, constant multiple, sum, and difference rules for derivatives.
Skill 1.16
Determining polynomial derivatives algebraically and using them for rate-of-change problems
I can determine polynomial derivatives algebraically and use them to solve rate-of-change problems.
Skill 1.17
Verifying the power rule for rational exponents
I can verify that the power rule applies to rational exponents.
Skill 1.18
Verifying the chain rule
I can verify the chain rule using monomial functions.
Skill 1.19
Verifying the product rule
I can verify the product rule using polynomial functions.
Skill 1.20
Solving derivative problems using the product and chain rules
I can solve derivative problems using the product and chain rules for a range of function types.
Skill 1.21
The official wording · 21 outcomes
- 1.1
- describe examples of real-world applications of rates of change, represented in a variety of ways (e.g., in words, numerically, graphically, algebraically)
- 1.2
- describe connections between the average rate of change of a function that is smooth (i.e., continuous with no corners) 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
- 1.3
- make 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 contain- ing the point (i.e., by using secants through the given point on a smooth curve to approach the tangent at that point, and determining the slopes of the approaching secants to approxi- mate the slope of the tangent)
- 1.4
- recognize, through investigation with or without technology, graphical and numerical examples of limits, and explain the reasoning involved (e.g., the value of a function approaching an asymptote, the value of the ratio of successive terms in the Fibonacci sequence)
- 1.5
- make 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 , and between the instantaneous rate of change of the function at x = a and the value of the limit
- 1.6
- compare, through investigation, the calcula- tion of instantaneous rates of change at a point (a, f(a)) for polynomial functions [e.g., f(x) = x , f(x) = x ], with and without simplifying the expression before substituting values of h that approach zero
- 1.7
- determine 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 (e.g., by using graphing technology to exam- ine the table of values and the slopes of tan- gents for a function whose equation is given; by examining a given graph), and describe the behaviour of the instantaneous rate of change at and between local maxima and minima
- 1.8
- generate, through investigation using tech- nology, a table of values showing the instan- taneous rate of change of a polynomial function, f(x), for various values of x (e.g., construct a tangent to the function, measure its slope, and create a slider or animation to move the point of tangency), graph the ordered pairs, recognize that the graph represents a function called the derivative, f’(x) or , and make connections between the graphs of f(x) and f’(x) or y and
- 1.9
- determine the derivatives of polynomial func- tions by simplifying the algebraic expression and then taking the limit of the simplified expression as h approaches zero
- 1.10
- determine, through investigation using tech- nology, the graph of the derivative f’(x) or of a given sinusoidal function [i.e., f(x) = sin x, f(x) = cos x] (e.g., by generating a table of values showing the instantaneous rate of change of the function for various values of x and graphing the ordered pairs; by using dynamic geometry software to verify graphi- cally that when f(x) = sin x, f’(x) = cos x, and when f(x) = cos x, f’(x) = – sin x; by using a motion sensor to compare the displacement and velocity of a pendulum)
- 1.11
- determine, through investigation using tech- nology, the graph of the derivative f’(x) or of a given exponential function [i.e., f(x) = a (a > 0, a ≠ 1)] [e.g., by generating a table of values showing the instantaneous rate of change of the function for various values of x and graphing the ordered pairs; by using dynamic geometry software to verify that when f(x) = a , f’(x) = kf(x)], and make con- nections between the graphs of f(x) and f’(x)
- 1.12
- determine, through investigation using tech- nology, the exponential function f(x) = a (a > 0, a ≠ 1) for which f’(x) = f(x) (e.g., by using graphing technology to create a slider that varies the value of a in order to deter- mine the exponential function whose graph is the same as the graph of its derivative), iden- tify the number e to be the value of a for which f’(x) = f(x)
- 1.13
- recognize that the natural logarithmic func- tion f(x) = loge x, also written as f(x) = ln x, is the inverse of the exponential function f(x) = e , and make connections between f(x) = ln x and f(x) = e
- 1.14
- verify, using technology (e.g., calculator, graphing technology), that the derivative of the exponential function f(x) = a is f’(x) = a ln a for various values of a
- 1.15
- verify the power rule for functions of the form f(x) = x , where n is a natural number [e.g., by determining the equations of the derivatives of the functions f(x) = x, f(x) = x , f(x) = x , and f(x) = x algebraically using and graphically using slopes of tangents]
- 1.16
- verify the constant, constant multiple, sum, and difference rules graphically and numeri- cally [e.g., by using the function g(x) = kf(x) and comparing the graphs of g’(x) and kf’(x); by using a table of values to verify that f’(x) + g’(x) = ( f + g)’(x), given f(x) = x and g(x) = 3x], and read and interpret proofs involving of the constant, constant multiple, sum, and difference rules
- 1.17
- determine algebraically the derivatives of polynomial functions, and use these deriva- tives to determine the instantaneous rate of change at a point and to determine point(s) at which a given rate of change occurs
- 1.18
- verify that the power rule applies to functions of the form f(x) = x , where n is a rational number [e.g., by comparing values of the slopes of tangents to the function f(x) = x with values of the derivative function deter- mined using the power rule]
- 1.19
- verify algebraically the chain rule using monomial functions [e.g., by determining the same derivative for f(x) = (5x ) by using the chain rule and by differentiating the simplified form, f(x) = 5 x]
- 1.20
- and the product rule using polynomial functions [e.g., by determining the same derivative for f(x) = (3x + 2)(2x – 1) by using the product rule and by differentiating the expanded form f(x) = 6x + 4x – 3x – 2]
- 1.21
- solve problems, using the product and chain rules, involving the derivatives of polynomial functions, sinusoidal functions, exponential functions, rational functions [e.g., by expressing f(x) = as the product f(x) = (x + 1)(x – 1) ], radical functions [e.g., by expressing f(x) = √x + 5 as the power f(x) = (x + 5) ], and other simple combinations of functions
Key ideas
A rate of change can be described in words, a table, a graph, or an equation — all describing the same underlying idea.
Plotting the instantaneous rate of change at many x-values, then connecting the dots, reveals the derivative function's own shape.
For f(x) = x^n, the power rule gives f'(x) = n·x^(n−1) — verified both algebraically with the limit definition and graphically with tangent slopes.
A secant line crosses a curve at two points — its slope IS the average rate of change between them.
The derivative f'(x) is defined as the limit of (f(x+h) − f(x))/h as h approaches zero.
Watch out for
- Thinking a rate of change is always a positive quantity (it can be negative, e.g., a decreasing tank of fuel).
- Thinking the derivative graph should look similar in shape to the original function.
- Forgetting to reduce the exponent by 1 when applying the power rule.
Every skill’s key ideas, worked examples and checks are in MapleMind.Practise this unit
Unit 2 · 11 skills
Derivatives and Their Applications
Connecting Graphs and Equations of Functions and Their Derivatives
Sketching a derivative graph and recognizing points of inflection
I can sketch a derivative graph and recognize points of inflection.
Skill 2.1
The second derivative as the rate of change of the rate of change
I can recognize the second derivative as the rate of change of the rate of change.
Skill 2.2
Determining the second derivative algebraically
I can determine the second derivative of a function algebraically.
Skill 2.3
Connecting the graphs of a function and its first and second derivatives
I can connect the key features of a function's graph to the graphs of its first and second derivatives.
Skill 2.4
Sketching multiple possible functions from derivative information
I can sketch multiple possible functions consistent with given derivative information.
Skill 2.5
Solving Problems Using Mathematical Models and Derivatives
Sketching a polynomial graph using first/second derivative strategies
I can sketch a polynomial function's graph using first- and second-derivative strategies.
Skill 2.6
Connecting displacement, velocity, and acceleration to the derivative
I can connect displacement, velocity, and acceleration to the derivative.
Skill 2.7
Connecting derivatives to real-world applications
I can connect derivatives to real-world applications like population change and inflation.
Skill 2.8
Solving instantaneous rate-of-change problems using the derivative
I can solve real-world instantaneous rate-of-change problems using the derivative.
Skill 2.9
Solving optimization problems
I can solve optimization problems using derivatives.
Skill 2.10
Solving real-world problems using a mathematical model and the derivative
I can solve real-world problems by building a mathematical model and applying derivative concepts.
Skill 2.11
The official wording · 11 outcomes
- 2.1
- sketch the graph of a derivative function, given the graph of a function that is continu- ous over an interval, and recognize points of inflection of the given function (i.e., points at which the concavity changes)
- 2.2
- recognize the second derivative as the rate of change of the rate of change (i.e., the rate of change of the slope of the tangent), and sketch the graphs of the first and second derivatives, given the graph of a smooth function
- 2.3
- determine algebraically the equation of the second derivative f”(x) of a polynomial or simple rational function f(x)
- 2.4
- and make connections, through investigation using technology, between the key features of the graph of the function (e.g., increasing/ decreasing intervals, local maxima and minima, points of inflection, intervals of con- cavity) and corresponding features of the graphs of its first and second derivatives (e.g., for an increasing interval of the function, the first derivative is positive; for a point of inflection of the function, the slopes of tangents change their behaviour from increasing to decreasing or from decreasing to increasing, the first derivative has a maximum or mini- mum, and the second derivative is zero)
- 2.5
- describe key features of a polynomial function, given information about its first and/or sec- ond derivatives (e.g., the graph of a deriva- tive, the sign of a derivative over specific intervals, the x-intercepts of a derivative), 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
- 2.6
- sketch the graph of a polynomial function, given its equation, by using a variety of strategies (e.g., using the sign of the first derivative; using the sign of the second derivative; identifying even or odd functions) to determine its key features (e.g., increasing/ decreasing intervals, intercepts, local maxima and minima, points of inflection, intervals of concavity), and verify using technology
- 2.7
- make connections between the concept of motion (i.e., displacement, velocity, accelera- tion) and the concept of the derivative in a variety of ways (e.g., verbally, numerically, graphically, algebraically)
- 2.8
- make connections between the graphical or algebraic representations of derivatives and real-world applications (e.g., population and rates of population change, prices and infla- tion rates, volume and rates of flow, height and growth rates)
- 2.9
- solve problems, using the derivative, that involve instantaneous rates of change, includ- ing problems arising from real-world applica- tions (e.g., population growth, radioactive decay, temperature changes, hours of day- light, heights of tides), given the equation of a function
- 2.10
- solve optimization problems involving poly- nomial, simple rational, and exponential func- tions drawn from a variety of applications, including those arising from real-world situations
- 2.11
- solve problems arising from real-world appli- cations by applying a mathematical model and the concepts and procedures associated with the derivative to determine mathemati- cal results, and interpret and communicate the results
Key ideas
A point of inflection is where a function's concavity changes — from curving up to curving down, or vice versa.
Combining the sign of the first derivative, the sign of the second derivative, and even/odd symmetry gives a complete, accurate sketch without plotting many points.
The second derivative measures how quickly the slope of the tangent itself is changing — acceleration is the second derivative of position.
Velocity is the derivative of displacement; acceleration is the derivative of velocity (the second derivative of displacement).
The second derivative is found by differentiating the first derivative — apply the derivative rules a second time.
Watch out for
- Confusing a point of inflection with a local maximum or minimum.
- Plotting only a handful of points and connecting them, instead of using derivative sign analysis for an accurate shape.
- Thinking the second derivative measures the same thing as the first derivative, just 'more.'.
Every skill’s key ideas, worked examples and checks are in MapleMind.Practise this unit
Unit 3 · 22 skills
Geometry and Algebra of Vectors
Representing Vectors Geometrically and Algebraically
A vector as a quantity with magnitude and direction
I can recognize a vector as a quantity with magnitude and direction, and identify real-world applications.
Skill 3.1
Representing a vector in two-space geometrically and algebraically
I can represent a vector in two-space both geometrically and algebraically.
Skill 3.2
Converting between Cartesian form and directed line segment form for two-space vectors
I can convert between the Cartesian form and the directed-line-segment form of a two-space vector.
Skill 3.3
Representing points and vectors in three-space and finding distance and magnitude
I can represent points and vectors in three-space and find distance and magnitude.
Skill 3.4
Operating With Vectors
Adding, subtracting, and scalar-multiplying vectors
I can add, subtract, and scalar-multiply vectors in both directed-segment and Cartesian form.
Skill 3.5
Properties of vector addition, subtraction, and scalar multiplication
I can investigate the properties of vector addition, subtraction, and scalar multiplication.
Skill 3.6
Solving real-world problems with vector addition, subtraction, and scalar multiplication
I can solve real-world problems involving vector addition, subtraction, and scalar multiplication.
Skill 3.7
The dot product and its applications
I can perform the dot product on two vectors and describe its applications.
Skill 3.8
Properties of the dot product
I can investigate the properties of the dot product.
Skill 3.9
The cross product and its applications
I can perform the cross product on two vectors in three-space and describe its applications.
Skill 3.10
Properties of the cross product
I can investigate the properties of the cross product.
Skill 3.11
Solving real-world problems involving the dot and cross product
I can solve real-world problems involving the dot product and the cross product.
Skill 3.12
Describing Lines and Planes Using Linear Equations
Solution points of linear equations in two-space and their intersection
I can recognize that solution points of linear equations in two-space form lines and their intersection point.
Skill 3.13
Solution points of linear equations in three-space as planes and their line of intersection
I can determine that solution points of linear equations in three-space form planes and their line of intersection.
Skill 3.14
Geometric configurations of up to three lines and/or planes in three-space
I can determine and organize different geometric configurations of lines and planes in three-space.
Skill 3.15
Describing Lines and Planes Using Scalar, Vector, and Parametric Equations
Scalar, vector, and parametric equations of a line in two-space
I can represent a line in two-space using scalar, vector, and parametric equations, and connect them.
Skill 3.16
Representing a line in three-space
I can represent a line in three-space using intersecting planes, and vector and parametric equations.
Skill 3.17
The normal to a plane and geometric properties of a plane
I can recognize the normal to a plane and determine geometric properties of the plane.
Skill 3.18
The scalar equation of a plane and finding the intersection of three planes
I can determine the intersection of three planes by solving a system of three equations.
Skill 3.19
Determining the equations of a plane from its properties
I can determine the scalar, vector, and parametric equations of a plane from its properties.
Skill 3.20
Converting between forms of a plane's equation
I can convert a plane's equation between its scalar, vector, and parametric forms.
Skill 3.21
Solving problems involving distances and intersections of lines and planes
I can solve problems involving distances and intersections of lines and planes in three-space.
Skill 3.22
The official wording · 22 outcomes
- 3.1
- recognize a vector as a quantity with both magnitude and direction, and identify, gather, and interpret information about real-world applications of vectors (e.g., displacement, forces involved in structural design, simple animation of computer graphics, velocity determined using GPS)
- 3.2
- represent a vector in two-space geometrically as a directed line segment, with directions ex- pressed in different ways (e.g., 320º; N 40º W), and algebraically (e.g., using Cartesian coordi- nates; using polar coordinates), and recognize vectors with the same magnitude and direc- tion but different positions as equal vectors
- 3.4
- recognize that points and vectors in three-space can both be represented using Cartesian coor- dinates, and determine the distance between two points and the magnitude of a vector using their Cartesian representations
- 3.5
- perform the operations of addition, subtrac- tion, 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
- 3.6
- determine, through investigation with and without technology, some properties (e.g., commutative, associative, and distributive properties) of the operations of addition, subtraction, and scalar multiplication of vectors
- 3.7
- solve problems involving the addition, sub- traction, and scalar multiplication of vectors, including problems arising from real-world applications
- 3.8
- perform the operation of dot product on two vectors represented as directed line segments (i.e., using a•b =|a| |b|cosθ) and in Cartesian form (i.e., using a•b = a 1 b 1 + a 2 b 2 or a•b = a 1 b 1 + a 2 b 2 + a 3 b 3 ) in two-space and three-space, and describe applications of the dot product (e.g., determining the angle between two vectors; determining the projec- tion of one vector onto another)
- 3.9
- determine, through investigation, properties of the dot product (e.g., investigate whether it is commutative, distributive, or associative; investigate the dot product of a vector with itself and the dot product of orthogonal vectors)
- 3.10
- perform the operation of cross product on two vectors represented in Cartesian form in three-space [i.e., using a x b = (a 2 b 3 – a 3 b 2 , a 3 b 1 – a 1 b 3 , a 1 b 2 – a 2 b 1 )], determine the magnitude of the cross product (i.e., using|a x b|=|a| |b|sinθ), and describe applications of the cross product (e.g., deter- mining a vector orthogonal to two given vec- tors; determining the turning effect [or torque] when a force is applied to a wrench at differ- ent angles)
- 3.11
- determine, through investigation, properties of the cross product (e.g., investigate whether it is commutative, distributive, or associative; investigate the cross product of collinear vectors)
- 3.12
- solve problems involving dot product and cross product (e.g., determining projections, the area of a parallelogram, the volume of a parallelepiped), including problems arising from real-world applications (e.g., determin- ing work, torque, ground speed, velocity, force)
- 3.13
- recognize 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
- 3.14
- determine, through investigation with technol- ogy (i.e., 3-D graphing software) 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
- 3.15
- determine, through investigation using a variety of tools and strategies (e.g., modelling with cardboard sheets and drinking straws; sketching on isometric graph paper), different geometric configurations of combinations of up to three lines and/or planes in three-space (e.g., two skew lines, three parallel planes, two intersecting planes, an intersecting line and plane); organize the configurations based on whether they intersect and, if so, how they intersect (i.e., in a point, in a line, in a plane)
- 3.16
- recognize 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 (i.e., r = r0 + tm) and parametric equations, and make connections between a scalar equation, a vector equation, and parametric equations of a line in two-space
- 3.17
- recognize that a line in three-space cannot be represented by a scalar equation, and rep- resent a line in three-space using the scalar equations of two intersecting planes and using vector and parametric equations (e.g., given a direction vector and a point on the line, or given two points on the line)
- 3.18
- recognize a normal to a plane geometrically (i.e., as a vector perpendicular to the plane) and algebraically [e.g., one normal to the plane 3x + 5y – 2z = 6 is (3, 5, –2)], and deter- mine, through investigation, some geometric properties of the plane (e.g., the direction of any normal to a plane is constant; all scalar multiples of a normal to a plane are also nor- mals to that plane; three non-collinear points determine a plane; the resultant, or sum, of any two vectors in a plane also lies in the plane)
- 3.19
- recognize 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 (e.g., by using elimination or substitution), and make connections between the algebraic solution and the geometric configuration of the three planes
- 3.20
- determine, using properties of a plane, the scalar, vector, and parametric equations of a plane
- 3.21
- determine the equation of a plane in its scalar, vector, or parametric form, given another of these forms
- 3.22
- solve problems relating to lines and planes in three-space that are represented in a variety of ways (e.g., scalar, vector, parametric equa- tions) and involving distances (e.g., between a point and a plane; between two skew lines) or intersections (e.g., of two lines, of a line and a plane), and interpret the result geometrically
Key ideas
A vector has both a size (magnitude) and a direction — unlike a scalar, which is just a number.
Vectors add and subtract component-by-component in Cartesian form, or tip-to-tail geometrically.
All (x, y) pairs solving one linear equation in two variables form a line; solving a SYSTEM of two such equations finds where two lines cross.
A line's scalar equation is Ax + By + C = 0; its vector equation is r = r₀ + tm, built from a point and a direction vector.
A vector can be drawn as a directed line segment (an arrow) or written algebraically using Cartesian or polar coordinates.
Watch out for
- Treating a vector's magnitude alone as if it fully described the quantity (ignoring direction).
- Adding vector magnitudes directly without accounting for direction (as if they were scalars).
- Assuming every system of two linear equations must have exactly one solution (parallel or coincident lines are exceptions).
Every skill’s key ideas, worked examples and checks are in MapleMind.Practise this unit
Plan the term
Pick your term and the days you study. Each unit gets its share of the weeks and comes back a week later for a review. The last two weeks go over the whole course.
Which course?
Ontario has 6 math courses for Grade 12. Here they are side by side.
- Advanced Functions, Grade 12 (University)MHF4U4 units · 72 skillsNo provincial examWhat it coversExponential and Logarithmic Functions · Trigonometric Functions · Polynomial and Rational Functions · Characteristics of Functions
- Calculus and Vectors, Grade 12 (University)This courseMCV4U3 units · 54 skillsNo provincial examWhat it coversRate of Change · Derivatives and Their Applications · Geometry and Algebra of Vectors
- Foundations for College Mathematics, Grade 12 (College)MAP4C4 units · 59 skillsNo provincial examWhat it coversMathematical Models · Personal Finance · Geometry and Trigonometry · Data Management
- Mathematics for College Technology, Grade 12 (College)MCT4C4 units · 57 skillsNo provincial examWhat it coversExponential Functions · Polynomial Functions · Trigonometric Functions · Applications of Geometry
- Mathematics for Work and Everyday Life, Grade 12 (Workplace)MEL4E3 units · 52 skillsNo provincial examWhat it coversReasoning With Data · Personal Finance · Applications of Measurement
- Mathematics of Data Management, Grade 12 (University)MDM4U5 units · 57 skillsNo provincial examWhat it coversCounting and Probability · Probability Distributions · Organization of Data for Analysis · Statistical Analysis · Culminating Data Management Investigation
What a university, college or program asks for is theirs to say: check its admission page, and your school’s course guide.
The official curriculum
This page follows Ontario’s own curriculum for Calculus and Vectors, Grade 12 (University).
Textbooks
The textbooks Ontario Ministry of Education lists for Calculus and Vectors, Grade 12 (University):
- Nelson Calculus and Vectors
- McGraw-Hill Ryerson Calculus and Vectors 12
From the Trillium List, read October 27, 2026. ontario.ca
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Questions about Calculus and Vectors, Grade 12 (University)
What is Calculus and Vectors, Grade 12 (University)?
Ontario’s Grade 12 math course: 3 units and 54 skills, from Rate of Change to Geometry and Algebra of Vectors.
What units are in Calculus and Vectors, Grade 12 (University)?
1. Rate of Change, 2. Derivatives and Their Applications and 3. Geometry and Algebra of Vectors.
Is there a provincial exam for Calculus and Vectors, Grade 12 (University)?
No. Ontario gives no provincial exam for Calculus and Vectors, Grade 12 (University).
What textbook does Calculus and Vectors, Grade 12 (University) use?
The textbooks Ontario Ministry of Education lists for Calculus and Vectors, Grade 12 (University): Nelson Calculus and Vectors, Nelson Education Ltd. and McGraw-Hill Ryerson Calculus and Vectors 12, McGraw-Hill Ryerson.
Can MapleMind help with Calculus and Vectors, Grade 12 (University)?
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