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Skills available for Ontario grade 12 math curriculum

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12.A Rate of Change

  • 12.A.1 demonstrate an understanding of rate of change by making connections between average rate of change over an interval and instantaneous rate of change at a point, using the slopes of secants and tangents and the concept of the limit;

    • 12.A.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)

    • 12.A.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

    • 12.A.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 containing 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 approximate the slope of the tangent)

    • 12.A.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)

    • 12.A.1.5 make connections, for a function that is smooth over the interval a less than or equal to x less than or equal to 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 (f(a + h) - f(a))/h as h approaches 0

    • 12.A.1.6 compare, through investigation, the calculation of instantaneous rates of change at a point (a, f(a)) for polynomial functions [e.g., f(x) = x², f(x) = x³], with and without simplifying the expression (f(a + h) - f(a))/h before substituting values of h that approach zero [e.g., for f(x) = x² at x = 3, by determining (f(3 + 1) - f(3))/1 = 7, (f(3 + 0.1) - f(3))/0.1 = 6.1, (f(3 + 0.01) - f(3))/0.001 = 6.01, and (f(3 + 0.001 - f(3))/0.001 = 6.001, and by first simplifying (f(3 + h) - f(3))/h as ((3 + h)² - 3²)/h = 6 + h and then substituting the same values of h to give the same results]

  • 12.A.2 graph the derivatives of polynomial, sinusoidal, and exponential functions, and make connections between the numeric, graphical, and algebraic representations of a function and its derivative;

    • 12.A.2.1 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 (e.g., by using graphing technology to examine the table of values and the slopes of tangents 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

    • 12.A.2.2 generate, through investigation using technology, a table of values showing the instantaneous 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 (dy)/(dx), and make connections between the graphs of f(x) and f'(x) or y and (dy)/(dx) [e.g., when f(x) is linear, f'(x) is constant; when f(x) is quadratic, f'(x) is linear; when f(x) is cubic, f'(x) is quadratic]

    • 12.A.2.3 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 [i.e., determining the limit of (f(x + h) - f(x))/h as h approaches 0]

    • 12.A.2.4 determine, through investigation using technology, the graph of the derivative f'(x) or (dy)/(dx) 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 graphically 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)

    • 12.A.2.5 determine, through investigation using technology, the graph of the derivative f,(x) or (dy)/(dx) of a given exponential function [i.e., f(x) = a to the x power (a greater than 0, a does not equal 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 to the x power, f'(x) = kf(x)], and make connections between the graphs of f(x) and f'(x) or y and (dy)/(dx) [e.g., f(x) and f'(x) are both exponential; the ratio (f'(x))/(f(x)) is constant, or f'(x) = kf(x); f'(x) is a vertical stretch from the x-axis of f(x)]

    • 12.A.2.6 determine, through investigation using technology, the exponential function f(x) = a to the x power (a is greater than 0, a does not equal 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 determine the exponential function whose graph is the same as the graph of its derivative), identify the number e to be the value of a for which f'(x) = f(x) [i.e., given f(x) = e to the x power, f'(x) = e to the x power], and recognize that for the exponential function f(x) = e to the x power the slope of the tangent at any point on the function is equal to the value of the function at that point

    • 12.A.2.7 recognize that the natural logarithmic function f(x) = log base e of x, also written as f(x) = ln x, is the inverse of the exponential function f(x) = e to the x power, and make connections between f(x) = ln x and f(x) = e to the x power [e.g., f(x) = ln x reverses what f(x) = e to the x power does; their graphs are reflections of each other in the line y = x; the composition of the two functions, e to the lnx power or ln e to the x power, maps x onto itself, that is, e to the lnx power = x and ln e to the x power = x]

    • 12.A.2.8 verify, using technology (e.g., calculator, graphing technology), that the derivative of the exponential function f(x) = a to the x power is f'(x) = a to the x power ln a for various values of a [e.g., verifying numerically for f(x) = 2 to the x power that f'(x) = 2 to the x power ln 2 by using a calculator to show that the limit of (2 to the h power - 1)/h as h approaches 0 is ln 2 or by graphing f(x) = 2 to the x power, determining the value of the slope and the value of the function for specific x-values, and comparing the ratio (f'(x))/(f(x)) with ln 2]

  • 12.A.3 verify graphically and algebraically the rules for determining derivatives; apply these rules to determine the derivatives of polynomial, sinusoidal, exponential, rational, and radical functions, and simple combinations of functions; and solve related problems.

    • 12.A.3.1 verify the power rule for functions of the form f(x) = x to the n power, where n is a natural number [e.g., by determining the equations of the derivatives of the functions f(x) = x, f(x) = x², f(x) = x³, and f(x) = x to the 4th power algebraically using the limit of (f(x + h) - f(x))/h, as h approaches 0, and graphically using slopes of tangents]

    • 12.A.3.2 verify the constant, constant multiple, sum, and difference rules graphically and numerically [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 the limit of (f(x + h) - f(x))/h, as h approaches 0, of the constant, constant multiple, sum, and difference rules (student reproduction of the development of the general case is not required)

    • 12.A.3.3 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

    • 12.A.3.4 verify that the power rule applies to functions of the form f(x) = x to the n power, where n is a rational number [e.g., by comparing values of the slopes of tangents to the function f(x) = x to the ½ power with values of the derivative function determined using the power rule], and verify algebraically the chain rule using monomial functions [e.g., by determining the same derivative for f(x) = (5x³) to the 1/3 power by using the chain rule and by differentiating the simplified form, f(x) = (5 to the 1/3 power) times x] and the product rule using polynomial functions [e.g., by determining the same derivative for f(x) = (3x + 2)(2x² – 1) by using the product rule and by differentiating the expanded form f(x) = 6x³ + 4x² – 3x – 2]

    • 12.A.3.5 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) = x² + 1/x - 1 as the product f(x) = (x² + 1)(x – 1) to the -1 power], radical functions [e.g., by expressing f(x) = the square root of x² + 5 as the power f(x) = (x² + 5) to the ½ power], and other simple combinations of functions [e.g., f(x) = x sin x, f(x) = sin x/cos x]

12.B Derivatives and Their Applications

  • 12.B.1 make connections, graphically and algebraically, between the key features of a function and its first and second derivatives, and use the connections in curve sketching;

    • 12.B.1.1 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 (i.e., points at which the concavity changes)

    • 12.B.1.2 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

    • 12.B.1.3 determine algebraically the equation of the second derivative f"(x) of a polynomial or simple rational function f(x), 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 concavity) 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 minimum, and the second derivative is zero)

    • 12.B.1.4 describe key features of a polynomial function, given information about its first and/or second derivatives (e.g., the graph of a derivative, 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

    • 12.B.1.5 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

  • 12.B.2 solve problems, including optimization problems, that require the use of the concepts and procedures associated with the derivative, including problems arising from real-world applications and involving the development of mathematical models.

    • 12.B.2.1 make connections between the concept of motion (i.e., displacement, velocity, acceleration) and the concept of the derivative in a variety of ways (e.g., verbally, numerically, graphically, algebraically)

    • 12.B.2.2 make connections between the graphical or algebraic representations of derivatives and real-world applications (e.g., population and rates of population change, prices and inflation rates, volume and rates of flow, height and growth rates)

    • 12.B.2.3 solve problems, using the derivative, that involve instantaneous rates of change, including problems arising from real-world applications (e.g., population growth, radioactive decay, temperature changes, hours of daylight, heights of tides), given the equation of a function.

    • 12.B.2.4 solve optimization problems involving polynomial, simple rational, and exponential functions drawn from a variety of applications, including those arising from real-world situations

    • 12.B.2.5 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

12.C Geometry and Algebra of Vectors

  • 12.C.1 demonstrate an understanding of vectors in two-space and three-space by representing them algebraically and geometrically and by recognizing their applications;

  • 12.C.2 perform operations on vectors in two-space and three-space, and use the properties of these operations to solve problems, including those arising from real-world applications;

    • 12.C.2.1 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

    • 12.C.2.2 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

    • 12.C.2.3 solve problems involving the addition, subtraction, and scalar multiplication of vectors, including problems arising from real-world applications

    • 12.C.2.4 perform the operation of dot product on two vectors represented as directed line segments (i.e., using vector a times vector b = (absolute value of vector a)(absolute value of vector b)(cos Theta)) and in Cartesian form (i.e., using vector a times vector b = (a base 1 of b base 1) + (a base 2 of b base 2) or (vector a times vector b) = (a base 1 of b base 1) + (a base 2 of b base 2) + (a base 3 of b base 3)) in two-space and three-space, and describe applications of the dot product (e.g., determining the angle between two vectors; determining the projection of one vector onto another)

    • 12.C.2.5 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)

    • 12.C.2.6 perform the operation of cross product on two vectors represented in Cartesian form in three-space [i.e., using vector a times vector b = ((a base 2 of b base 3) – (a base 3 of b base 2), (a base 3 of b base 1) – (a base 1 of b base 3), (a base 1 of b base 2) – (a base 2 of b base 1))], determine the magnitude of the cross product (i.e., using absolute value of (vector a times vector b) = (absolute value of vector a) (absolute value of vector b)(sin Theta)), and describe applications of the cross product (e.g., determining a vector orthogonal to two given vectors; determining the turning effect [or torque] when a force is applied to a wrench at different angles)

    • 12.C.2.7 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)

    • 12.C.2.8 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., determining work, torque, ground speed, velocity, force)

  • 12.C.3 distinguish between the geometric representations of a single linear equation or a system of two linear equations in two-space and three-space, and determine different geometric configurations of lines and planes in three-space;

  • 12.C.4 represent lines and planes using scalar, vector, and parametric equations, and solve problems involving distances and intersections.

    • 12.C.4.1 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 (i.e., vector r = (vector r base 0) + (t times vector m)) and parametric equations, and make connections between a scalar equation, a vector equation, and parametric equations of a line in two-space

    • 12.C.4.2 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 (e.g., given a direction vector and a point on the line, or given two points on the line)

    • 12.C.4.3 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 determine, 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 normals 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)

    • 12.C.4.4 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 (e.g., by using elimination or substitution), and make connections between the algebraic solution and the geometric configuration of the three planes

    • 12.C.4.5 determine, using properties of a plane, the scalar, vector, and parametric equations of a plane

    • 12.C.4.6 determine the equation of a plane in its scalar, vector, or parametric form, given another of these forms

    • 12.C.4.7 solve problems relating to lines and planes in three-space that are represented in a variety of ways (e.g., scalar, vector, parametric equations) 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