#### AF.A: Exponential and Logarithmic Functions

AF.A.1: demonstrate an understanding of the relationship between exponential expressions and logarithmic expressions, evaluate logarithms, and apply the laws of logarithms to simplify numeric expressions;

AF.A.1.1: recognize the logarithm of a number to a given base as the exponent to which the base must be raised to get the number, recognize the operation of finding the logarithm to be the inverse operation (i.e., the undoing or reversing) of exponentiation, and evaluate simple logarithmic expressions

AF.A.2: identify and describe some key features of the graphs of logarithmic functions, make connections among the numeric, graphical, and algebraic representations of logarithmic functions, and solve related problems graphically;

AF.A.2.1: determine, through investigation with technology (e.g., graphing calculator, spreadsheet) and without technology, key features (i.e., vertical and horizontal asymptotes, domain and range, intercepts, increasing/decreasing behaviour) of the graphs of logarithmic functions of the form f(x) = log base b of x, and make connections between the algebraic and graphical representations of these logarithmic functions

AF.A.2.2: recognize the relationship between an exponential function and the corresponding logarithmic function to be that of a function and its inverse, deduce that the graph of a logarithmic function is the reflection of the graph of the corresponding exponential function in the line y = x, and verify the deduction using technology

AF.A.2.3: determine, through investigation using technology, the roles of the parameters d and c in functions of the form y = log base 10 of (x - d) + c and the roles of the parameters a and k in functions of the form y = alog base 10 of (kx), and describe these roles in terms of transformations on the graph of f(x) = log base 10 of x (i.e., vertical and horizontal translations; reflections in the axes; vertical and horizontal stretches and compressions to and from the x- and y-axes)

AF.A.2.4: pose problems based on real-world applications of exponential and logarithmic functions (e.g., exponential growth and decay, the Richter scale, the pH scale, the decibel scale), and solve these and other such problems by using a given graph or a graph generated with technology from a table of values or from its equation

#### AF.B: Trigonometric Functions

AF.B.1: demonstrate an understanding of the meaning and application of radian measure;

AF.B.1.1: recognize the radian as an alternative unit to the degree for angle measurement, define the radian measure of an angle as the length of the arc that subtends this angle at the centre of a unit circle, and develop and apply the relationship between radian and degree measure

AF.B.1.3: determine, with technology, the primary trigonometric ratios (i.e., sine, cosine, tangent) and the reciprocal trigonometric ratios (i.e., cosecant, secant, cotangent) of angles expressed in radian measure

AF.B.1.4: determine, without technology, the exact values of the primary trigonometric ratios and the reciprocal trigonometric ratios for the special angles 0, pi/6, pi/4, pi/3, pi/2, and their multiples less than or equal to 2pi

AF.B.2: make connections between trigonometric ratios and the graphical and algebraic representations of the corresponding trigonometric functions and between trigonometric functions and their reciprocals, and use these connections to solve problems;

AF.B.2.1: sketch the graphs of f(x) = sin x and f(x) = cos x for angle measures expressed in radians, and determine and describe some key properties (e.g., period of 2pi, amplitude of 1) in terms of radians

AF.B.2.2: make connections between the tangent ratio and the tangent function by using technology to graph the relationship between angles in radians and their tangent ratios and defining this relationship as the function f(x) = tan x, and describe key properties of the tangent function

AF.B.2.3: graph, with technology and using the primary trigonometric functions, the reciprocal trigonometric functions (i.e., cosecant, secant, cotangent) for angle measures expressed in radians, determine and describe key properties of the reciprocal functions (e.g., state the domain, range, and period, and identify and explain the occurrence of asymptotes), and recognize notations used to represent the reciprocal functions [e.g., the reciprocal of f(x) = sin x can be represented using csc x, 1/f(x), or 1/sin x, but not using f to the -1 power (x) or sin to the -1 one power times x, which represent the inverse function]

AF.B.2.4: determine the amplitude, period, and phase shift of sinusoidal functions whose equations are given in the form f(x) = a sin (k(x - d)) + c or f(x) = a cos(k(x - d)) + c, with angles expressed in radians

AF.B.2.5: sketch graphs of y = a sin (k(x - d)) + c and y = a cos(k(x - d)) + c by applying transformations to the graphs of f(x) = sin x and f(x) = cos x with angles expressed in radians, and state the period, amplitude, and phase shift of the transformed functions

AF.B.2.6: represent a sinusoidal function with an equation, given its graph or its properties, with angles expressed in radians

AF.B.2.7: pose problems based on applications involving a trigonometric function with domain expressed in radians (e.g., seasonal changes in temperature, heights of tides, hours of daylight, displacements for oscillating springs), and solve these and other such problems by using a given graph or a graph generated with or without technology from a table of values or from its equation

AF.B.3: solve problems involving trigonometric equations and prove trigonometric identities.

AF.B.3.1: recognize equivalent trigonometric expressions [e.g., by using the angles in a right triangle to recognize that sin x and cos (pi/2 - x) are equivalent; by using transformations to recognize that cos (x + pi/2) and -sin x are equivalent], and verify equivalence using graphing technology

AF.B.3.2: explore the algebraic development of the compound angle formulas (e.g., verify the formulas in numerical examples, using technology; follow a demonstration of the algebraic development [student reproduction of the development of the general case is not required]), and use the formulas to determine exact values of trigonometric ratios [e.g., determining the exact value of sin (pi/12) by first rewriting it in terms of special angles as sin (pi/4 - pi/6)]

AF.B.3.3: recognize that trigonometric identities are equations that are true for every value in the domain (i.e., a counter-example can be used to show that an equation is not an identity), prove trigonometric identities through the application of reasoning skills, using a variety of relationships (e.g., tan x = sin x / cos x; sin²x + cos²x = 1; the reciprocal identities; the compound angle formulas), and verify identities using technology

AF.B.3.4: solve linear and quadratic trigonometric equations, with and without graphing technology, for the domain of real values from 0 to 2pi, and solve related problems

#### AF.C: Polynomial and Rational Functions

AF.C.1: identify and describe some key features of polynomial functions, and make connections between the numeric, graphical, and algebraic representations of polynomial functions;

AF.C.1.1: recognize a polynomial expression (i.e., a series of terms where each term is the product of a constant and a power of x with a nonnegative integral exponent, such as x³ - 5x² + 2x - 1); recognize the equation of a polynomial function, give reasons why it is a function, and identify linear and quadratic functions as examples of polynomial functions

AF.C.1.2: compare, through investigation using graphing technology, the numeric, graphical, and algebraic representations of polynomial (i.e., linear, quadratic, cubic, quartic) functions (e.g., compare finite differences in tables of values; investigate the effect of the degree of a polynomial function on the shape of its graph and the maximum number of x-intercepts; investigate the effect of varying the sign of the leading coefficient on the end behaviour of the function for very large positive or negative x-values)

AF.C.1.3: describe key features of the graphs of polynomial functions (e.g., the domain and range, the shape of the graphs, the end behaviour of the functions for very large positive or negative x-values)

AF.C.1.4: distinguish polynomial functions from sinusoidal and exponential functions [e.g., f(x) = sin x, g(x) = 2 to the x power], and compare and contrast the graphs of various polynomial functions with the graphs of other types of functions

AF.C.1.5: make connections, through investigation using graphing technology (e.g., dynamic geometry software), between a polynomial function given in factored form [e.g., f(x) = 2(x - 3)(x + 2)(x - 1)] and the x-intercepts of its graph, and sketch the graph of a polynomial function given in factored form using its key features (e.g., by determining intercepts and end behaviour; by locating positive and negative regions using test values between and on either side of the x-intercepts)

AF.C.1.6: determine, through investigation using technology, the roles of the parameters a, k, d, and c in functions of the form y = af (k(x - d)) + c, and describe these roles in terms of transformations on the graphs of f(x) = x³ and f(x) = x to the 4th power (i.e., vertical and horizontal translations; reflections in the axes; vertical and horizontal stretches and compressions to and from the x- and y-axes)

AF.C.1.7: determine an equation of a polynomial function that satisfies a given set of conditions (e.g., degree of the polynomial, intercepts, points on the function), using methods appropriate to the situation (e.g., using the x-intercepts of the function; using a trial-and-error process with a graphing calculator or graphing software; using finite differences), and recognize that there may be more than one polynomial function that can satisfy a given set of conditions (e.g., an infinite number of polynomial functions satisfy the condition that they have three given x-intercepts)

AF.C.1.8: determine the equation of the family of polynomial functions with a given set of zeros and of the member of the family that passes through another given point [e.g., a family of polynomial functions of degree 3 with zeros 5, -3, and -2 is defined by the equation f(x) = k(x - 5)(x + 3)(x + 2), where k is a real number, k does not equal 0; the member of the family that passes through (-1, 24) is f(x) = -2(x - 5)(x + 3)(x + 2)]

AF.C.1.9: determine, through investigation, and compare the properties of even and odd polynomial functions [e.g., symmetry about the y-axis or the origin; the power of each term; the number of x-intercepts; f(x) = f(- x) or f(- x) = - f(x)], and determine whether a given polynomial function is even, odd, or neither

AF.C.2: identify and describe some key features of the graphs of rational functions, and represent rational functions graphically;

AF.C.2.1: determine, through investigation with and without technology, key features (i.e., vertical and horizontal asymptotes, domain and range, intercepts, positive/negative intervals, increasing/decreasing intervals) of the graphs of rational functions that are the reciprocals of linear and quadratic functions, and make connections between the algebraic and graphical representations of these rational functions [e.g., make connections between f(x) = 1/(x² - 4) and its graph by using graphing technology and by reasoning that there are vertical asymptotes at x = 2 and x = -2 and a horizontal asymptote at y = 0 and that the function maintains the same sign as f(x) = x² - 4]

AF.C.2.2: determine, through investigation with and without technology, key features (i.e., vertical and horizontal asymptotes, domain and range, intercepts, positive/negative intervals, increasing/decreasing intervals) of the graphs of rational functions that have linear expressions in the numerator and denominator [e.g., f(x) = 2x/(x - 3), h(x)= (x - 2)/(3x + 4)], and make connections between the algebraic and graphical representations of these rational functions

AF.C.2.3: sketch the graph of a simple rational function using its key features, given the algebraic representation of the function

AF.C.3: solve problems involving polynomial and simple rational equations graphically and algebraically;

AF.C.3.1: make connections, through investigation using technology (e.g., computer algebra systems), between the polynomial function f(x), the divisor x - a, the remainder from the division f(x)/(x - a), and f(a) to verify the remainder theorem and the factor theorem

AF.C.3.2: factor polynomial expressions in one variable, of degree no higher than four, by selecting and applying strategies (i.e., common factoring, difference of squares, trinomial factoring, factoring by grouping, remainder theorem, factor theorem)

AF.C.3.3: determine, through investigation using technology (e.g., graphing calculator, computer algebra systems), the connection between the real roots of a polynomial equation and the x-intercepts of the graph of the corresponding polynomial function, and describe this connection [e.g., the real roots of the equation x to the 4th power - 13x² + 36 = 0 are the x-intercepts of the graph of f(x) = x to the 4th power - 13x² + 36]

AF.C.3.4: solve polynomial equations in one variable, of degree no higher than four (e.g., 2x³ - 3x² + 8x - 12 = 0), by selecting and applying strategies (i.e., common factoring, difference of squares, trinomial factoring, factoring by grouping, remainder theorem, factor theorem), and verify solutions using technology (e.g., using computer algebra systems to determine the roots; using graphing technology to determine the x-intercepts of the graph of the corresponding polynomial function)

AF.C.3.5: determine, through investigation using technology (e.g., graphing calculator, computer algebra systems), the connection between the real roots of a rational equation and the x-intercepts of the graph of the corresponding rational function, and describe this connection [e.g., the real root of the equation (x - 2)/(x - 3) = 0 is 2, which is the x-intercept of the function f(x) = (x - 2)/(x - 3); the equation 1/(x - 3) = 0 has no real roots, and the function f(x) = 1/(x - 3) does not intersect the x-axis]

AF.C.3.6: solve simple rational equations in one variable algebraically, and verify solutions using technology (e.g., using computer algebra systems to determine the roots; using graphing technology to determine the x-intercepts of the graph of the corresponding rational function)

AF.C.3.7: solve problems involving applications of polynomial and simple rational functions and equations [e.g., problems involving the factor theorem or remainder theorem, such as determining the values of k for which the function f(x) = x³ + 6x² + kx - 4 gives the same remainder when divided by x - 1 and x + 2]

AF.C.4: demonstrate an understanding of solving polynomial and simple rational inequalities.

AF.C.4.1: explain, for polynomial and simple rational functions, the difference between the solution to an equation in one variable and the solution to an inequality in one variable, and demonstrate that given solutions satisfy an inequality (e.g., demonstrate numerically and graphically that the solution to (1/(x + 1)) less than (5 is x) less than -1 or x greater than -4/5);

AF.C.4.2: determine solutions to polynomial inequalities in one variable [e.g., solve f(x) ≥ 0, where f(x) = x³ - x² + 3x - 9] and to simple rational inequalities in one variable by graphing the corresponding functions, using graphing technology, and identifying intervals for which x satisfies the inequalities

AF.C.4.3: solve linear inequalities and factorable polynomial inequalities in one variable (e.g., (x³ + x²) greater than 0) in a variety of ways (e.g., by determining intervals using x-intercepts and evaluating the corresponding function for a single x-value within each interval; by factoring the polynomial and identifying the conditions for which the product satisfies the inequality), and represent the solutions on a number line or algebraically (e.g., for the inequality (x to the 4th power - 5x² + 4) less than 0, the solution represented algebraically is -2 less than x less than -1 or 1 less than x less than 2)

#### AF.D: Characteristics of Functions

AF.D.1: demonstrate an understanding of average and instantaneous rate of change, and determine, numerically and graphically, and interpret the average rate of change of a function over a given interval and the instantaneous rate of change of a function at a given point;

AF.D.1.1: gather, interpret, and describe information about real-world applications of rates of change, and recognize different ways of representing rates of change (e.g., in words, numerically, graphically, algebraically)

AF.D.1.2: recognize that the rate of change for a function is a comparison of changes in the dependent variable to changes in the independent variable, and distinguish situations in which the rate of change is zero, constant, or changing by examining applications, including those arising from real-world situations (e.g., rate of change of the area of a circle as the radius increases, inflation rates, the rising trend in graduation rates among Aboriginal youth, speed of a cruising aircraft, speed of a cyclist climbing a hill, infection rates)

AF.D.1.3: sketch a graph that represents a relationship involving rate of change, as described in words, and verify with technology (e.g., motion sensor) when possible

AF.D.1.4: calculate and interpret average rates of change of functions (e.g., linear, quadratic, exponential, sinusoidal) arising from real-world applications (e.g., in the natural, physical, and social sciences), given various representations of the functions (e.g., tables of values, graphs, equations)

AF.D.1.5: recognize examples of instantaneous rates of change arising from real-world situations, and make connections between instantaneous rates of change and average rates of change (e.g., an average rate of change can be used to approximate an instantaneous rate of change)

AF.D.1.6: determine, through investigation using various representations of relationships (e.g., tables of values, graphs, equations), approximate instantaneous rates of change arising from real-world applications (e.g., in the natural, physical, and social sciences) by using average rates of change and reducing the interval over which the average rate of change is determined

AF.D.1.7: make connections, through investigation, between the slope of a secant on the graph of a function (e.g., quadratic, exponential, sinusoidal) and the average rate of change of the function over an interval, and between the slope of the tangent to a point on the graph of a function and the instantaneous rate of change of the function at that point

AF.D.1.8: determine, through investigation using a variety of tools and strategies (e.g., using a table of values to calculate slopes of secants or graphing secants and measuring their slopes with technology), the approximate slope of the tangent to a given point on the graph of a function (e.g., quadratic, exponential, sinusoidal) by using the slopes of secants through the given point (e.g., investigating the slopes of secants that approach the tangent at that point more and more closely), and make connections to average and instantaneous rates of change

AF.D.1.9: solve problems involving average and instantaneous rates of change, including problems arising from real-world applications, by using numerical and graphical methods (e.g., by using graphing technology to graph a tangent and measure its slope)

AF.D.2: determine functions that result from the addition, subtraction, multiplication, and division of two functions and from the composition of two functions, describe some properties of the resulting functions, and solve related problems;

AF.D.2.1: determine, through investigation using graphing technology, key features (e.g., domain, range, maximum/minimum points, number of zeros) of the graphs of functions created by adding, subtracting, multiplying, or dividing functions [e.g., f(x) = 2 to the -x power sin 4x, g(x) = x² + 2 to the x power, h(x) = (sin x)/(cos x)], and describe factors that affect these properties

AF.D.2.2: recognize real-world applications of combinations of functions (e.g., the motion of a damped pendulum can be represented by a function that is the product of a trigonometric function and an exponential function; the frequencies of tones associated with the numbers on a telephone involve the addition of two trigonometric functions), and solve related problems graphically

AF.D.2.3: determine, through investigation, and explain some properties (i.e., odd, even, or neither; increasing/decreasing behaviours) of functions formed by adding, subtracting, multiplying, and dividing general functions [e.g., f(x) + g(x), f(x)g(x)]

AF.D.2.4: determine the composition of two functions [i.e., f(g(x))] numerically (i.e., by using a table of values) and graphically, with technology, for functions represented in a variety of ways (e.g., function machines, graphs, equations), and interpret the composition of two functions in real-world applications

AF.D.2.7: demonstrate, by giving examples for functions represented in a variety of ways (e.g., function machines, graphs, equations), the property that the composition of a function and its inverse function maps a number onto itself [i.e., f to the -1 power (f(x)) = x and f(f to the -1 power (x)) = x demonstrate that the inverse function is the reverse process of the original function and that it undoes what the function does]

AF.D.2.8: make connections, through investigation using technology, between transformations (i.e., vertical and horizontal translations; reflections in the axes; vertical and horizontal stretches and compressions to and from the x- and y-axes) of simple functions f(x) [e.g., f(x) = x to the 3rd power + 20, f(x) = sin x, f(x) = log x] and the composition of these functions with a linear function of the form g(x) = A(x + B)

AF.D.3: compare the characteristics of functions, and solve problems by modelling and reasoning with functions, including problems with solutions that are not accessible by standard algebraic techniques.

AF.D.3.1: compare, through investigation using a variety of tools and strategies (e.g., graphing with technology; comparing algebraic representations; comparing finite differences in tables of values) the characteristics (e.g., key features of the graphs, forms of the equations) of various functions (i.e., polynomial, rational, trigonometric, exponential, logarithmic)

AF.D.3.2: solve graphically and numerically equations and inequalities whose solutions are not accessible by standard algebraic techniques

AF.D.3.3: solve problems, using a variety of tools and strategies, including problems arising from real-world applications, by reasoning with functions and by applying concepts and procedures involving functions (e.g., by constructing a function model from data, using the model to determine mathematical results, and interpreting and communicating the results within the context of the problem)

### CV: Calculus and Vectors

#### CV.A: Rate of Change

CV.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;

CV.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

CV.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)

CV.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

CV.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]

CV.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;

CV.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

CV.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]

CV.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]

CV.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)

CV.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)]

CV.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

CV.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]

CV.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]

CV.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.

CV.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)

CV.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

CV.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]

CV.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]

#### CV.B: Derivatives and Their Applications

CV.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;

CV.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)

CV.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

CV.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

CV.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.

CV.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

#### CV.C: Geometry and Algebra of Vectors

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

CV.C.1.1: 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)

CV.C.1.2: represent a vector in two-space geometrically as a directed line segment, with directions expressed in different ways (e.g., 320º; N 40º W), and algebraically (e.g., using Cartesian coordinates; using polar coordinates), and recognize vectors with the same magnitude and direction but different positions as equal vectors

CV.C.1.3: determine, using trigonometric relationships [e.g., x = rcos Theta, y = rsin Theta, Theta = tan to the -1 power (y / x) or tan to the -1 power (y / x) + 180º, r = square root of (x² + y²)], 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 [e.g., representing the vector (8, 6) as a directed line segment]

CV.C.1.4: 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

CV.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;

CV.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

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

CV.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)

CV.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)

CV.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)

CV.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;

CV.C.3.1: 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

CV.C.3.2: determine, through investigation with technology (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

CV.C.3.3: 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)

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

CV.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

CV.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)

CV.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)

CV.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

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

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

CV.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

### MDM: Mathematics and Data Management

#### MDM.A: Counting and Probability

MDM.A.1: solve problems involving the probability of an event or a combination of events for discrete sample spaces;

MDM.A.1.1: recognize and describe how probabilities are used to represent the likelihood of a result of an experiment (e.g., spinning spinners; drawing blocks from a bag that contains different-coloured blocks; playing a game with number cubes; playing Aboriginal stick-and-stone games) and the likelihood of a real-world event (e.g., that it will rain tomorrow, that an accident will occur, that a product will be defective)

MDM.A.1.2: describe a sample space as a set that contains all possible outcomes of an experiment, and distinguish between a discrete sample space as one whose outcomes can be counted (e.g., all possible outcomes of drawing a card or tossing a coin) and a continuous sample space as one whose outcomes can be measured (e.g., all possible outcomes of the time it takes to complete a task or the maximum distance a ball can be thrown)

MDM.A.1.3: determine the theoretical probability, P (i.e., a value from 0 to 1), of each outcome of a discrete sample space (e.g., in situations in which all outcomes are equally likely), recognize that the sum of the probabilities of the outcomes is 1 (i.e., for n outcomes, (P base 1) + (P base 2) + (P base 3) +... + (P base n) = 1), recognize that the probabilities P form the probability distribution associated with the sample space, and solve related problems

MDM.A.1.4: determine, through investigation using class-generated data and technology-based simulation models (e.g., using a random-number generator on a spreadsheet or on a graphing calculator; using dynamic statistical software to simulate repeated trials in an experiment), the tendency of experimental probability to approach theoretical probability as the number of trials in an experiment increases (e.g., "If I simulate tossing two coins 1000 times using technology, the experimental probability that I calculate for getting two tails on the two tosses is likely to be closer to the theoretical probability of ¼ than if I simulate tossing the coins only 10 times")

MDM.A.1.5: recognize and describe an event as a set of outcomes and as a subset of a sample space, determine the complement of an event, determine whether two or more events are mutually exclusive or non-mutually exclusive (e.g., the events of getting an even number or getting an odd number of heads from tossing a coin 5 times are mutually exclusive), and solve related probability problems [e.g., calculate P(~A), P(A and B), P(A or B)] using a variety of strategies (e.g., Venn diagrams, lists, formulas)

MDM.A.1.6: determine whether two events are independent or dependent and whether one event is conditional on another event, and solve related probability problems [e.g., calculate P(A and B), P(A or B), P(A given B)] using a variety of strategies (e.g., tree diagrams, lists, formulas)

MDM.A.2: solve problems involving the application of permutations and combinations to determine the probability of an event.

MDM.A.2.1: recognize the use of permutations and combinations as counting techniques with advantages over other counting techniques (e.g., making a list; using a tree diagram; making a chart; drawing a Venn diagram), distinguish between situations that involve the use of permutations and those that involve the use of combinations (e.g., by considering whether or not order matters), and make connections between, and calculate, permutations and combinations

MDM.A.2.2: solve simple problems using techniques for counting permutations and combinations, where all objects are distinct, and express the solutions using standard combinatorial notation [e.g., n!, P(n, r),(n factorial r factorial)]

MDM.A.2.4: make connections, through investigation, between combinations (i.e., n choose r) and Pascal's triangle [e.g., between (2 factorial r factorial) and row 3 of Pascal's triangle, between (n factorial 2 factorial) and diagonal 3 of Pascal's triangle]

MDM.A.2.5: solve probability problems using counting principles for situations involving equally likely outcomes

#### MDM.B: Probability Distributions

MDM.B.1: demonstrate an understanding of discrete probability distributions, represent them numerically, graphically, and algebraically, determine expected values, and solve related problems from a variety of applications;

MDM.B.1.1: recognize and identify a discrete random variable X (i.e., a variable that assumes a unique value for each outcome of a discrete sample space, such as the value x for the outcome of getting x heads in 10 tosses of a coin), generate a probability distribution [i.e., a function that maps each value x of a random variable X to a corresponding probability, P(X= x)] by calculating the probabilities associated with all values of a random variable, with and without technology, and represent a probability distribution numerically using a table

MDM.B.1.2: calculate the expected value for a given probability distribution [i.e., using E(X)= Sigma xP(X= x)], interpret the expected value in applications, and make connections between the expected value and the weighted mean of the values of the discrete random variable

MDM.B.1.3: represent a probability distribution graphically using a probability histogram (i.e., a histogram on which each rectangle has a base of width 1, centred on the value of the discrete random variable, and a height equal to the probability associated with the value of the random variable), and make connections between the frequency histogram and the probability histogram (e.g., by comparing their shapes)

MDM.B.1.4: recognize conditions (e.g., independent trials) that give rise to a random variable that follows a binomial probability distribution, calculate the probability associated with each value of the random variable, represent the distribution numerically using a table and graphically using a probability histogram, and make connections to the algebraic representation P(X=x)= (n factorial x factorial)(p to the x power)((1 - p) to the (n - x) power)

MDM.B.1.5: recognize conditions (e.g., dependent trials) that give rise to a random variable that follows a hypergeometric probability distribution, calculate the probability associated with each value of the random variable (e.g., by using a tree diagram; by using combinations), and represent the distribution numerically using a table and graphically using a probability histogram

MDM.B.1.6: compare, with technology and using numeric and graphical representations, the probability distributions of discrete random variables (e.g., compare binomial distributions with the same probability of success for increasing numbers of trials; compare the shapes of a hypergeometric distribution and a binomial distribution)

MDM.B.1.7: solve problems involving probability distributions (e.g., uniform, binomial, hypergeometric), including problems arising from real-world applications

MDM.B.2: demonstrate an understanding of continuous probability distributions, make connections to discrete probability distributions, determine standard deviations, describe key features of the normal distribution, and solve related problems from a variety of applications.

MDM.B.2.1: recognize and identify a continuous random variable (i.e., a variable that assumes values from the infinite number of possible outcomes in a continuous sample space), and distinguish between situations that give rise to discrete frequency distributions (e.g., counting the number of outcomes for drawing a card or tossing three coins) and situations that give rise to continuous frequency distributions (e.g., measuring the time taken to complete a task or the maximum distance a ball can be thrown)

MDM.B.2.2: recognize standard deviation as a measure of the spread of a distribution, and determine, with and without technology, the mean and standard deviation of a sample of values of a continuous random variable

MDM.B.2.3: describe challenges associated with determining a continuous frequency distribution (e.g., the inability to capture all values of the variable, resulting in a need to sample; uncertainties in measured values of the variable), and recognize the need for mathematical models to represent continuous frequency distributions

MDM.B.2.4: represent, using intervals, a sample of values of a continuous random variable numerically using a frequency table and graphically using a frequency histogram and a frequency polygon, recognize that the frequency polygon approximates the frequency distribution, and determine, through investigation using technology (e.g., dynamic statistical software, graphing calculator), and compare the effectiveness of the frequency polygon as an approximation of the frequency distribution for different sizes of the intervals

MDM.B.2.5: recognize that theoretical probability for a continuous random variable is determined over a range of values (e.g., the probability that the life of a lightbulb is between 90 hours and 115 hours), that the probability that a continuous random variable takes any single value is zero, and that the probabilities of ranges of values form the probability distribution associated with the random variable

MDM.B.2.6: recognize that the normal distribution is commonly used to model the frequency and probability distributions of continuous random variables, describe some properties of the normal distribution (e.g., the curve has a central peak; the curve is symmetric about the mean; the mean and median are equal; approximately 68% of the data values are within one standard deviation of the mean and approximately 95% of the data values are within two standard deviations of the mean), and recognize and describe situations that can be modelled using the normal distribution (e.g., birth weights of males or of females, household incomes in a neighbourhood, baseball batting averages)

MDM.B.2.7: make connections, through investigation using dynamic statistical software, between the normal distribution and the binomial and hypergeometric distributions for increasing numbers of trials of the discrete distributions (e.g., recognizing that the shape of the hypergeometric distribution of the number of males on a 4-person committee selected from a group of people more closely resembles the shape of a normal distribution as the size of the group from which the committee was drawn increases)

#### MDM.C: Organization of Data for Analysis

MDM.C.1: demonstrate an understanding of the role of data in statistical studies and the variability inherent in data, and distinguish different types of data;

MDM.C.1.3: distinguish different types of statistical data (i.e., discrete from continuous, qualitative from quantitative, categorical from numerical, nominal from ordinal, primary from secondary, experimental from observational, microdata from aggregate data) and give examples (e.g., distinguish experimental data used to compare the effectiveness of medical treatments from observational data used to examine the relationship between obesity and type 2 diabetes or between ethnicity and type 2 diabetes)

MDM.C.2: describe the characteristics of a good sample, some sampling techniques, and principles of primary data collection, and collect and organize data to solve a problem.

MDM.C.2.1: determine and describe principles of primary data collection (e.g., the need for randomization, replication, and control in experimental studies; the need for randomization in sample surveys) and criteria that should be considered in order to collect reliable primary data (e.g., the appropriateness of survey questions; potential sources of bias; sample size)

MDM.C.2.2: explain the distinction between the terms population and sample, describe the characteristics of a good sample, explain why sampling is necessary (e.g., time, cost, or physical constraints), and describe and compare some sampling techniques (e.g., simple random, systematic, stratified, convenience, voluntary)

MDM.C.2.3: describe how the use of random samples with a bias (e.g., response bias, measurement bias, non-response bias, sampling bias) or the use of non-random samples can affect the results of a study

#### MDM.D: Statistical Analysis

MDM.D.1: analyse, interpret, and draw conclusions from one-variable data using numerical and graphical summaries;

MDM.D.1.1: recognize that the analysis of one-variable data involves the frequencies associated with one attribute, and determine, using technology, the relevant numerical summaries (i.e., mean, median, mode, range, interquartile range, variance, and standard deviation)

MDM.D.2: analyse, interpret, and draw conclusions from two-variable data using numerical, graphical, and algebraic summaries;

MDM.D.2.2: determine the positions of individual data points within a one-variable data set using quartiles, percentiles, and z-scores, use the normal distribution to model suitable onevariable data sets, and recognize these processes as strategies for one-variable data analysis

MDM.D.1.3: generate, using technology, the relevant graphical summaries of one-variable data (e.g., circle graphs, bar graphs, histograms, stem-and-leaf plots, boxplots) based on the type of data provided (e.g., categorical, ordinal, quantitative)

MDM.D.1.5: interpret statistical summaries (e.g., graphical, numerical) to describe the characteristics of a one-variable data set and to compare two related one-variable data sets (e.g., compare the lengths of different species of trout; compare annual incomes in Canada and in a third-world country; compare Aboriginal and non-Aboriginal incomes); describe how statistical summaries (e.g., graphs, measures of central tendency) can be used to misrepresent one-variable data; and make inferences, and make and justify conclusions, from statistical summaries of one-variable data orally and in writing, using convincing arguments

MDM.D.2.1: recognize that the analysis of two-variable data involves the relationship between two attributes, recognize the correlation coefficient as a measure of the fit of the data to a linear model, and determine, using technology, the relevant numerical summaries (e.g., summary tables such as contingency tables; correlation coefficients)

MDM.D.2.3: generate, using technology, the relevant graphical summaries of two-variable data (e.g., scatter plots, side-by-side boxplots) based on the type of data provided (e.g., categorical, ordinal, quantitative)

MDM.D.3: demonstrate an understanding of the applications of data management used by the media and the advertising industry and in various occupations.

MDM.D.3.1: interpret statistics presented in the media (e.g., the UN's finding that 2% of the world's population has more than half the world's wealth, whereas half the world's population has only 1% of the world's wealth), and explain how the media, the advertising industry, and others (e.g., marketers, pollsters) use and misuse statistics (e.g., as represented in graphs) to promote a certain point of view (e.g., by making a general statement based on a weak correlation or an assumed cause-and-effect relationship; by starting the vertical scale at a value other than zero; by making statements using general population statistics without reference to data specific to minority groups)

MDM.D.3.2: assess the validity of conclusions presented in the media by examining sources of data, including Internet sources (i.e., to determine whether they are authoritative, reliable, unbiased, and current), methods of data collection, and possible sources of bias (e.g., sampling bias, non-response bias, cultural bias in a survey question), and by questioning the analysis of the data (e.g., whether there is any indication of the sample size in the analysis) and conclusions drawn from the data (e.g., whether any assumptions are made about cause and effect)

#### MDM.E: Culminating Data Management Investigation

MDM.E.1: design and carry out a culminating investigation that requires the integration and application of the knowledge and skills related to the expectations of this course;

MDM.E.1.1: pose a significant problem of interest that requires the organization and analysis of a suitable set of primary or secondary quantitative data (e.g., primary data collected from a student-designed game of chance, secondary data from a reliable source such as E-STAT), and conduct appropriate background research related to the topic being studied

MDM.E.1.2: design a plan to study the problem (e.g., identify the variables and the population; develop an ethical survey; establish the procedures for gathering, summarizing, and analysing the primary or secondary data; consider the sample size and possible sources of bias)

Correlation last revised: 5/15/2009

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