OntarioGrade 12 Math Courses - Ontario Curriculum
Master Grade 12 Calculus and Vectors (MCV4U) with comprehensive support covering derivatives, rates of change, and vector operations. Navigate university preparation mathematics with confidence through structured learning pathways.
Ontario Grade 12 Math Curriculum - MCV4U Calculus & VectorsHelp
OE_ID | Expectations | StudyPug Topic |
|---|---|---|
ON.OE.12CV.A1.1 | 12CV.A1.1: Describe examples of real-world applications of rates of change, represented in a variety of ways |
ON.OE.12CV.A1.2 | 12CV.A1.2: Describe 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 |
ON.OE.12CV.A1.3 | 12CV.A1.3: 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 containing the point |
ON.OE.12CV.A1.4 | 12CV.A1.4: Recognize, through investigation with and without technology, graphical and numerical examples of limits, and explain the reasoning involved |
ON.OE.12CV.A1.5 | 12CV.A1.5: 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 [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 |
ON.OE.12CV.A1.6 | 12CV.A1.6: Compare, 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 |
ON.OE.12CV.A2.1 | 12CV.A2.1: 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 |
ON.OE.12CV.A2.2 | 12CV.A2.2: Generate, 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) |
ON.OE.12CV.A2.3 | 12CV.A2.3: Determine 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 |
ON.OE.12CV.A2.4 | 12CV.A2.4: Determine, through investigation using technology, the graph of the derivative f'(x) of a given sinusoidal function |
ON.OE.12CV.A2.5 | 12CV.A2.5: Determine, through investigation using technology, the graph of the derivative f'(x) of a given exponential function |
ON.OE.12CV.A2.6 | 12CV.A2.6: Determine, 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 |
ON.OE.12CV.A2.7 | 12CV.A2.7: Recognize 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 |
ON.OE.12CV.A2.8 | 12CV.A2.8: Verify, 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 |
ON.OE.12CV.A3.1 | 12CV.A3.1: Verify the power rule for functions of the form f(x) = x^n, where n is a natural number |
ON.OE.12CV.A3.2 | 12CV.A3.2: Verify the constant, constant multiple, sum, and difference rules graphically and numerically |
ON.OE.12CV.A3.3 | 12CV.A3.3: Determine 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 |
ON.OE.12CV.A3.4 | 12CV.A3.4: Verify 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 |
ON.OE.12CV.A3.5 | 12CV.A3.5: Solve 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 |
ON.OE.12CV.B1.1 | 12CV.B1.1: Sketch 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 |
ON.OE.12CV.B1.2 | 12CV.B1.2: Recognize 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 |
ON.OE.12CV.B1.3 | 12CV.B1.3: Determine 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 |
ON.OE.12CV.B1.4 | 12CV.B1.4: Describe 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 |
ON.OE.12CV.B1.5 | 12CV.B1.5: Sketch the graph of a polynomial function, given its equation, by using a variety of strategies to determine its key features |
ON.OE.12CV.B2.1 | 12CV.B2.1: Make connections between the concept of motion and the concept of the derivative in a variety of ways |
ON.OE.12CV.B2.2 | 12CV.B2.2: Make connections between the graphical or algebraic representations of derivatives and real-world applications |
ON.OE.12CV.B2.3 | 12CV.B2.3: Solve problems, using the derivative, that involve instantaneous rates of change, including problems arising from real-world applications, given the equation of a function |
ON.OE.12CV.B2.4 | 12CV.B2.4: Solve optimization problems involving polynomial, simple rational, and exponential functions drawn from a variety of applications, including those arising from real-world situations |
ON.OE.12CV.B2.5 | 12CV.B2.5: Solve 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 |
ON.OE.12CV.C1.1 | 12CV.C1.1: Recognize a vector as a quantity with both magnitude and direction, and identify, gather, and interpret information about real-world applications of vectors |
ON.OE.12CV.C1.2 | 12CV.C1.2: Represent 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 |
ON.OE.12CV.C1.3 | 12CV.C1.3: Determine, 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 |
ON.OE.12CV.C1.4 | 12CV.C1.4: Recognize 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 |
ON.OE.12CV.C2.1 | 12CV.C2.1: Perform 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 |
ON.OE.12CV.C2.2 | 12CV.C2.2: Determine, through investigation with and without technology, some properties of the operations of addition, subtraction, and scalar multiplication of vectors |
ON.OE.12CV.C2.3 | 12CV.C2.3: Solve problems involving the addition, subtraction, and scalar multiplication of vectors, including problems arising from real-world applications |
ON.OE.12CV.C2.4 | 12CV.C2.4: Perform 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 |
ON.OE.12CV.C2.5 | 12CV.C2.5: Determine, through investigation, properties of the dot product |
ON.OE.12CV.C2.6 | 12CV.C2.6: Perform 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 |
ON.OE.12CV.C2.7 | 12CV.C2.7: Determine, through investigation, properties of the cross product |
ON.OE.12CV.C2.8 | 12CV.C2.8: Solve problems involving dot product and cross product, including problems arising from real-world applications |
ON.OE.12CV.C3.1 | 12CV.C3.1: 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 |
ON.OE.12CV.C3.2 | 12CV.C3.2: Determine, 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 |
ON.OE.12CV.C3.3 | 12CV.C3.3: Determine, 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 |
ON.OE.12CV.C4.1 | 12CV.C4.1: 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 and parametric equations, and make connections between a scalar equation, a vector equation, and parametric equations of a line in two-space |
ON.OE.12CV.C4.2 | 12CV.C4.2: Recognize 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 |
ON.OE.12CV.C4.3 | 12CV.C4.3: Recognize a normal to a plane geometrically and algebraically, and determine, through investigation, some geometric properties of the plane |
ON.OE.12CV.C4.4 | 12CV.C4.4: 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, and make connections between the algebraic solution and the geometric configuration of the three planes |
ON.OE.12CV.C4.5 | 12CV.C4.5: Determine, using properties of a plane, the scalar, vector, and parametric equations of a plane |
ON.OE.12CV.C4.6 | 12CV.C4.6: Determine the equation of a plane in its scalar, vector, or parametric form, given another of these forms |
ON.OE.12CV.C4.7 | 12CV.C4.7: Solve 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 |