THE ONTARIO CURRICULUM: PROPOSED REVISIONS
OCTOBER 2005
Mathematics: Grade 12
Advanced Functions
Side-by-Side
The comparison charts comparing the original to the proposed revisions are intended as a guide to assist you with the review. In general:
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Grey cells indicate the expectation has been moved from this location
An expectation in the left column, and nothing to the right of it (or the word deleted) means the expectation has been deleted
An expectation in the right column, and nothing to the left of it, indicates the expectation is new
When expectations are written side by side, the left column shows the original expectation and the right column shows the proposed
revision for that expectation
In some cases, notes in bold italics have been added for clarification
Please Note: Original expectations that are not associated with a revised expectation have not necessarily been removed or deleted.
Some overall expectations have been incorporated into specific expectations. Some specific expectations may have been combined, or moved to
another section of the program.
Original
Revised
Course Code: MCB 4U Name: Advanced Revised: Advanced Functions
Functions and Introductory Calculus
Grade: 12
Program Area: Mathematics
Revised: Polynomial and Rational Functions
Strand: Advanced Functions
Section: Overall Expectations
determine, through investigation, the
characteristics of the graphs of polynomial
functions of various degrees;
demonstrate facility in the algebraic manipulation
of polynomials;
Unchanged: Overall Expectations
determine the characteristics of a polynomial
function of various degrees given its graph and make
connections to its algebraic representation;
determine the characteristics of a polynomial
function of various degrees, given its algebraic
representation and make connections to its graph;
demonstrate an understanding of the nature of
Moved to Grade 12 Advanced Functions - MCB 4U,
exponential growth and decay;
Strand: Trigonometric, Exponential and Logarithmic
Functions, Section: Overall Expectations
define and apply logarithmic functions;
Deleted.
demonstrate an understanding of the operation of Deleted.
the composition of functions.
Specific Expectations—Section: Investigating
the Graphs of Polynomial Functions
determine, through investigation, using graphing
calculators or graphing software, various
properties of the graphs of polynomial functions
(e.g., determine the effect of the degree of a
polynomial function on the shape of its graph; the
effect of varying the coefficients in the
polynomial function; the type and the number of
x-intercepts; the behaviour near the x-intercepts;
the end behaviours; the existence of symmetry);
describe the nature of change in polynomial
functions of degree greater than two, using finite
differences in tables of values;
compare the nature of change observed in
polynomial functions of higher degree with that
observed in linear and quadratic functions;
sketch the graph of a polynomial function whose
equation is given in factored form;
determine an equation to represent a given graph
of a polynomial function, using methods
appropriate to the situation (e.g., using the zeros
of the function; using a trial-and-error process on
a graphing calculator or graphing software; using
Revised: Investigating the Graphs of Polynomial
and Rational Functions
Unchanged: determine, through investigation, using
graphing calculators or graphing software, various
properties of the graphs of polynomial functions
(e.g., determine the effect of the degree of a
polynomial function on the shape of its graph; the
effect of varying the coefficients in the polynomial
function; the type and the number of x-intercepts;
the behaviour near the x-intercepts; the end
behaviours; the existence of symmetry);
Unchanged: describe the nature of change in
polynomial functions of degree greater than two,
using finite differences in tables of values;
compare the nature of change observed in
polynomial functions of higher degree with that
observed in linear and quadratic functions (e.g.,
compare the graphs of f(x) = x^4 and f(x) = x²;
Unchanged: sketch the graph of a polynomial
function whose equation is given in factored form;
Unchanged: determine an equation to represent a
given graph of a polynomial function, using methods
appropriate to the situation (e.g., using the zeros of
the function; using a trial-and-error process on a
graphing calculator or graphing software; using
Page 2 of 13
Draft of Proposed Senior Mathematics Curriculum
finite differences).
finite differences).
draw, using technology, the graph of rational
functions (e.g., f(x) = 1/x, g(x) = 3/(x - 3), h(x) = (x 2)/(x²-4) ) and identify through investigation, the key
features of the graph (e.g., vertical vs. horizontal
asymptotes, domain, range, positive/negative
intervals, increasing/decreasing intervals);
sketch the graph of a rational function given its
equation by considering the key features of the
function f(x) = 1/x;
Specific Expectations—Section: Manipulating
Algebraic Expressions
demonstrate an understanding of the remainder
theorem and the factor theorem;
factor polynomial expressions of degree greater
than two, using the factor theorem;
Revised: Investigating the Algebra of Polynomial
Functions
Unchanged: demonstrate an understanding of the
remainder theorem and the factor theorem;
factor polynomial expressions of degree greater than
two, using the factor theorem (e.g., x³ + 2x² - 1x - 2
and x^4 - 6x³ + 4x² + 6x - 5);
determine, by factoring, the real or complex roots determine, by factoring, the real roots of polynomial
of polynomial equations of degree greater than
equations of degree greater than two (e.g., 2x³ - 3x²
two;
+ 8x - 12 = 0) and verify graphically using
technology;
determine the real roots of non-factorable
determine the real roots of non-factorable
polynomial equations by interpreting the graphs polynomial equations (e.g., π x³- 4x² - 3x +π = 0) by
of the corresponding functions, using graphing
interpreting the graphs of the corresponding
calculators or graphing software;
functions, using graphing calculators or graphing
software;
write the equation of a family of polynomial
write the equation of a family of polynomial
functions, given the real or complex zeros [e.g., a functions, given the zeros [e.g., a polynomial
polynomial function having non-repeated zeros 5, function having non-repeated zeros 5, -3, and -2 will
3, and 2 will be defined by the equation
be defined by the function f(x) = k(x- 5)(x + 3)(x +
f(x) = k(x5)(x + 3)(x + 2),
2), for any real number k) and find the specific
for k is an element of a set aleph];
equation when given additional information;
verify, by investigation with technology (e.g.,
dynamic geometry software), that the output values
of a polynomial function can only change sign at a
zero;
describe intervals and distances, using absolutevalue notation;
solve factorable polynomial inequalities;
solve linear and factorable polynomial inequalities,
by determining intercepts and representing the
solutions on number lines (e.g., x^4 - 5x² + 4 < 0);
solve non-factorable polynomial inequalities by solve non-factorable polynomial inequalities (e.g., x³
graphing the corresponding functions, using
- x² + 3x - 9 ≥ 0) by graphing the corresponding
graphing calculators or graphing software and
functions (e.g., f(x) = x³ - x² + 3x - 9, using graphing
identifying intervals above and below the x-axis; calculators or graphing software and identifying
Page 3 of 13
Draft of Proposed Senior Mathematics Curriculum
intervals above and below the x-axis;
solve problems involving the abstract extensions solve problems involving the abstract extensions of
of algorithms (e.g., a problem involving the
concepts related to polynomials (e.g., problems
nature of the roots of polynomial equations: If h involving the nature of the roots of polynomial
and k are the roots of the equation
equations, problems involving the factor theorem);
3x^2 + 28x - 20 = 0,
Sample problem: For what values of k does the
find the equation whose roots are h + k and hk; a function f(x) = x³ + 6x² + kx - 4 give the same
problem involving the factor theorem: For what remainder when divided by x - 1 and x + 2?
values of k does the function
f(x) = x^3 + 6x^2 + kx - 4
give the same remainder when divided by either x
- 1 or x + 2?).
compare and describe, through investigation, the
algebraic and graphical behaviour of even and odd
polynomial functions (e.g., examining the values of
the function for very large positive and negative
values of x, the number of real roots, symmetry etc.)
Sample problem: Under what conditions will an
even function have an even number of zeros?
solve equations and inequalities involving simple
rational functions, graphically, and algebraically,
using zeros, asymptotes and the values of the
function between these intervals;
Specific Expectations—Section: Understanding Deleted.
the Nature of Exponential Growth and Decay
identify, through investigations, using graphing
calculators or graphing software, the key
properties of exponential functions of the form
a^x (a > 0, a != 1)
and their graphs (e.g., the domain is the set of the
real numbers; the range is the set of the positive
real numbers; the function either increases or
decreases throughout its domain; the graph has
the x-axis as an asymptote and has y-intercept =
1);
describe the graphical implications of changes in
the parameters a, b, and c in the equation
y = ca^x + b;
compare the rates of change of the graphs of
exponential and non-exponential functions (e.g.,
those with equations
y = 2x,
y = x^2,
y = x^(1/2), and
y = 2^x);
Page 4 of 13
Draft of Proposed Senior Mathematics Curriculum
describe the significance of exponential growth or
decay within the context of applications
represented by various mathematical models (e.g.,
tables of values, graphs);
pose and solve problems related to models of
exponential functions drawn from a variety of
applications, and communicate the solutions with
clarity and justification.
Specific Expectations—Section: Defining and Deleted.
Applying Logarithmic Functions
define the logarithmic function
log to the base a of x (a > 1)
as the inverse of the exponential function a^x, and
compare the properties of the two functions;
express logarithmic equations in exponential
form, and vice versa;
simplify and evaluate expressions containing
logarithms;
solve exponential and logarithmic equations,
using the laws of logarithms;
solve simple problems involving logarithmic
scales (e.g., the Richter scale, the pH scale, the
decibel scale).
Specific Expectations—Section: Understanding Unchanged: Understanding the Composition of
the Composition of Functions
Functions
Moved from this location.
identify composition as an operation in which two Moved to Grade 12 Advanced Functions - MCB 4U,
functions are applied in succession;
Strand: Trigonometric, Exponential and Logarithmic
Functions, Section: Connecting Functions, Moved w
Section
demonstrate an understanding that the
Moved to Grade 12 Advanced Functions - MCB 4U,
composition of two functions exists only when the Strand: Trigonometric, Exponential and Logarithmic
range of the first function overlaps the domain of Functions, Section: Connecting Functions, Moved w
the second;
Section
determine the composition of two functions
Deleted.
expressed in function notation;
decompose a given composite function into its
Deleted.
constituent parts;
describe the effect of the composition of inverse Moved to Grade 12 Advanced Functions - MCB 4U,
functions [i.e.,
Strand: Trigonometric, Exponential and Logarithmic
f(f^(-1)(x)) = x].
Functions, Section: Connecting Functions, Moved w
Page 5 of 13
Draft of Proposed Senior Mathematics Curriculum
Section
Strand: New strand added.
New: Trigonometric, Exponential and
Logarithmic Functions
Specific Expectations—Section: New section
added.
Revised: Overall Expectations
(Expectation moved here; formerly AFV.03)
demonstrate an understanding of the nature of
exponential growth and decay;
Specific Expectations—Section: New section
added.
extend an understanding of exponential functions
and the related logarithmic functions to solve
problems involving exponential growth and decay;
extend an understanding of trigonometric functions
using radian measure and solve related problems;
consolidate their understanding of the characteristics
of functions by considering compound functions and
the composition of functions
Revised: Exponential and Logarithmic Functions
solve exponential equations by finding a common
base (e.g., 4^x = 8^(x+3), 2^(x+2) - 2^x = 12).
evaluate numerical expressions involving logarithms
(e.g., log 10(29), log3(25), log10(400) - log(10) 4),
using a calculator,;
determine, through investigation, the laws of
logarithms and use them to simplify and evaluate
logarithmic expressions;
write a logarithmic statement in exponential form,
and vice versa;
solve exponential and logarithmic equations using
the laws of logarithms
define the logarithmic function f(x) = log b( x )(b >
0, b/=1) as the inverse of the exponential function
f(x) = b^x;
compare the properties of the exponential and
logarithmic functions;
pose and solve problems related to models of
exponential and logarithmic functions drawn from a
variety of applications (e.g., exponential growth and
decay, the Richter scale, the pH scale, the decibel
scale)
Specific Expectations—Section: New section
added.
Revised: Trigonometric Functions
define radian measure and develop the relationship
between radian and degree measure;
represent, in applications, radian measure in exact
form, as an expression involving π (e.g., π/3, 2π) and
in approximate form as a rational number (e.g. 1.05)
Page 6 of 13
Draft of Proposed Senior Mathematics Curriculum
determine the exact values of the sine, cosine, and
tangent of the special angles 0, π/6, π/4, π/3, π/2, and
their multiples;
demonstrate facility in the use of radian measure in
graphing (e.g., f(x) = cos(x), g(x)=2sin(x+ π/3)
demonstrate facility in the use of the reciprocal
trigonometric ratios (i.e., cosecant, secant and
tangent)
sketch the graph of the tangent function and the
reciprocal trigonometric functions and identify the
key features of their graphs (e.g., state the domain,
range, and period and identify and explain the
occurrence of asymptotes)
demonstrate an understanding of the development of
the compound angle and double angle formulae and
the formulae to determine exact trigonometric values
(e.g., determine the exact value of sin(pi/12));
solve, with and without graphing technology, linear
and quadratic trigonometric equations between 0 and
2π, and over R (the Real numbers);
determine, through investigation using graphing
technology, whether or not two trigonometric
expressions are equivalent;
prove trigonometric identities using a variety of
relationships, including the reciprocal relationships
and the compound angle formulae;
pose and solve problems related to models of
trigonometric functions drawn from a variety of
applications (e.g., tides, length of day, oscillating
spring) with justification, with and without
technology;
Specific Expectations—Section: Understanding Revised: Connecting Functions
the Composition of Functions
Moved to this location.
describe, through investigation using a variety of
tools and strategies, some of the properties of
compound functions (e.g., f(x) = x sin x, g(x) = x²
+2^x);
(Expectation moved here; formerly AF5.01)
demonstrate an understanding of composition as an
identify composition as an operation in which two operation in which two functions are applied in
functions are applied in succession;
succession and determine the composition of two
functions expressed in functional notation
(Expectation moved here; formerly AF5.02)
demonstrate an understanding of the domain and
demonstrate an understanding that the
range of the composition of two functions (i.e.,
composition of two functions exists only when the f(g(x)) is defined for those x for which g(x) is
range of the first function overlaps the domain of defined and included in the domain of f(x))
Page 7 of 13
Draft of Proposed Senior Mathematics Curriculum
the second;
(Expectation moved here; formerly AF5.05)
describe the effect of the composition of inverse
functions [i.e.,
f(f^(-1)(x)) = x].
describe, using a variety of representations and an
understanding of the inverse as a reverse process, the
effect of the composition of inverse functions [i.e.,
f(f^(-1)(x)) = x].
compare and contrast, through investigation, the
characteristics (e.g., symmetry, asymptotes,
intercepts, domain and range, increasing/decreasing,
critical points) of functions (i.e., linear, quadratic,
trigonometric, exponential, logarithmic, polynomial,
rational) using a variety of representations (e.g.,
tables of values, function machines, graphs and
algebraic representations);
Strand: Underlying Concepts of Calculus Revised: Rates of Change
Section: Overall Expectations
Unchanged: Overall Expectations
determine and interpret the average and
instantaneous rates of change of given functions
(Expectation moved here; formerly CCV.03)
demonstrate an understanding of the relationship
demonstrate an understanding of the relationship between the shape of a graph and the rate of change
between the derivative of a function and the key of the dependent variable.
features of its graph.
determine and interpret the rates of change of
Unchanged: determine and interpret the rates of
functions drawn from the natural and social
change of functions drawn from the natural and
sciences;
social sciences;
demonstrate an understanding of the graphical
definition of the derivative of a function;
demonstrate an understanding of the relationship Moved to Grade 12 Advanced Functions - MCB 4U,
between the derivative of a function and the key Strand: Rates of Change, Section: Overall
features of its graph.
Expectations
Specific Expectations—Section: Understanding Unchanged: Understanding Rates of Change
Rates of Change
pose problems and formulate hypotheses
pose problems and formulate hypotheses regarding
regarding rates of change within applications
rates of change within applications drawn from
drawn from the natural and social sciences;
mathematics (e.g., rate of change of the area of a
circle as the radius increases) and from the real
world (e.g., inflation rates, cycling up a hill,
infection rates) Sample Problem: Given that the
bacteria count in a sample is 1 000 000 at 1:00 pm,
and 250 000 at 3:00 pm, pose and solve a problem
involving the rate of change of the bacterial
population.)
calculate and interpret average rates of change
calculate and interpret average rates of change from
from various models (e.g., equations, tables of
various representations (e.g., equations, tables of
values, graphs) of functions drawn from the
values, graphs) of functions drawn from the natural
Page 8 of 13
Draft of Proposed Senior Mathematics Curriculum
natural and social sciences;
and social sciences; Enter Your Revision:
estimate and interpret instantaneous rates of
estimate instantaneous rates of change given various
change from various models (e.g., equations,
representations (e.g., algebraic, graphical);
tables of values, graphs) of functions drawn from
the natural and social sciences;
interpret the meaning of the instantaneous rates of
change; Sample Problem: If the instantaneous rate of
change is given by a speedometer as 60 km/h,
interpret the meaning);
explain the difference between average and
demonstrate an understanding of the difference
instantaneous rates of change within applications between average and instantaneous rates of change
and in general;
using relevant applications (e.g., for a given average
velocity over an interval, there must be at least one
point in time in that interval, where the average is
the instantaneous velocity) Sample Problem: How
can you determine the average rate of change and
the instantaneous rate of change from a table of
values?
make inferences from models of applications and Unchanged: make inferences from models of
compare the inferences with the original
applications and compare the inferences with the
hypotheses regarding rates of change.
original hypotheses regarding rates of change.
Specific Expectations—Section: Understanding Revised: Interpreting Rates of Change on a
the Graphical Definition of the Derivative
Graph
demonstrate an understanding that the slope of a Unchanged: demonstrate an understanding that the
secant on a curve represents the average rate of slope of a secant on a curve represents the average
change of the function over an interval, and that rate of change of the function over an interval, and
the slope of the tangent to a curve at a point
that the slope of the tangent to a curve at a point
represents the instantaneous rate of change of the represents the instantaneous rate of change of the
function at that point;
function at that point;
demonstrate an understanding that the slope of the demonstrate, through investigation, an
tangent to a curve at a point is the limiting value understanding that the slope of the tangent to a curve
of the slopes of a sequence of secants;
at a point can be approximated by the slope of a
secant;
demonstrate an understanding that the
demonstrate an understanding that the instantaneous
instantaneous rate of change of a function at a
rate of change of a function at a point can be
point is the limiting value of a sequence of
approximated by average rates of change;
average rates of change;
demonstrate an understanding that the derivative determine when the rate of change is increasing or
of a function at a point is the instantaneous rate of decreasing from the shape of the graph of a function;
change or the slope of the tangent to the graph of
the function at that point.
demonstrate an understanding of the concept of
acceleration (e.g., rate of change of velocity,
accelerating cost), graphically, numerically (i.e.,
second differences), algebraically, verbally;
Page 9 of 13
Draft of Proposed Senior Mathematics Curriculum
(Expectation moved here; formerly CC3.03)
sketch, by hand, the graph of the derivative of a
given graph.
sketch, by hand, the graph of the rate of change of a
function, (i.e. using the slopes of the tangents), given
the graph of the function;
sketch the graph of a function, given the graph of its
rate of change (e.g., given a velocity-time graph,
sketch the position-time graph );
(Expectation moved here; formerly CC3.01)
describe the key features of a given graph of a
describe the key features of a given graph of a
function, including intervals of increase and
function, including intervals of increase and
decrease, local and absolute extrema, endpoints,
decrease, critical points, points of inflection, and points of inflection, and intervals of concavity;
intervals of concavity;
(Expectation moved here; formerly CC3.02)
Unchanged: identify the nature of the rate of change
of a given function, and the rate of change of the rate
of change, as they relate to the key features of the
graph of that function;
Specific Expectations—Section: Using Calculus Revised: Using Concepts of Rate of Change in
Techniques to Analyse Models of Functions
Applications and Modelling
Moved to this location.
(Expectation moved here; formerly DA6.01)
determine the key features of a mathematical model
determine the key features of a mathematical
of an application drawn from the natural or social
model of an application drawn from the natural or sciences, using concepts of rate of change;
social sciences, using the techniques of
differential calculus;
(Expectation moved here; formerly DA6.02)
Unchanged: compare the key features of a
mathematical model with the features of the
application it represents;
(Expectation moved here; formerly DA6.03)
predict future behaviour by extrapolating from a
predict future behaviour within an application by function used to model a relationship in an
extrapolating from a mathematical model of a
application and determine when it is appropriate;
function;
(Expectation moved here; formerly DA6.04)
pose questions related to an application and answer
pose questions related to an application and
them by analysing mathematical models, using the
answer them by analysing mathematical models, concept of rate of change
using the techniques of differential calculus;
demonstrate an understanding of how functions are
used to model situations in the real world, using a
variety of tools and strategies (e.g., dynamic
statistical software, dynamic geometry software,
spreadsheets, models);
Specific Expectations—Section: Connecting
Derivatives and Graphs
describe the key features of a given graph of a
function, including intervals of increase and
decrease, critical points, points of inflection, and
intervals of concavity;
Unchanged: Connecting Derivatives and Graphs
Moved to Grade 12 Advanced Functions - MCB 4U,
Strand: Rates of Change, Section: Interpreting Rates
of Change on a Graph
Page 10 of 13
Draft of Proposed Senior Mathematics Curriculum
identify the nature of the rate of change of a given Moved to Grade 12 Advanced Functions - MCB 4U,
function, and the rate of change of the rate of
Strand: Rates of Change, Section: Interpreting Rates
change, as they relate to the key features of the
of Change on a Graph
graph of that function;
sketch, by hand, the graph of the derivative of a Moved to Grade 12 Advanced Functions - MCB 4U,
given graph.
Strand: Rates of Change, Section: Interpreting Rates
of Change on a Graph
Strand: Derivatives and Applications
Deleted.
Section: Overall Expectations
demonstrate an understanding of the firstprinciples definition of the derivative;
determine the derivatives of given functions,
using manipulative procedures;
determine the derivatives of exponential and
logarithmic functions;
solve a variety of problems, using the techniques
of differential calculus;
sketch the graphs of polynomial, rational, and
exponential functions;
analyse functions, using differential calculus.
Specific Expectations—Section: Understanding
the First-Principles Definition of the Derivative
determine the limit of a polynomial, a rational, or
an exponential function;
demonstrate an understanding that limits can give
information about some behaviours of graphs of
functions [e.g.,
lim as x approaches 5 (x^2-25)/(x-5)
predicts a hole at (5, 10)];
identify examples of discontinuous functions and
the types of discontinuities they illustrate;
determine the derivatives of polynomial and
simple rational functions from first principles,
using the definitions of the derivative function,
f’(x) = lim as h approaches 0[f(x+h)-f(x)]/h
and
f’(a) = lim as x approaches a[f(x) - f(a)]/(x-a);
identify examples of functions that are not
differentiable.
Specific Expectations—Section: Determining
Derivatives
justify the constant, power, sum-and- difference,
product, quotient, and chain rules for determining
Page 11 of 13
Draft of Proposed Senior Mathematics Curriculum
derivatives;
determine the derivatives of polynomial and
rational functions, using the constant, power,
sum-and-difference, product, quotient, and chain
rules for determining derivatives;
determine second derivatives;
determine derivatives, using implicit
differentiation in simple cases (e.g.,
4x^2 + 9y^2 = 36).
Specific Expectations—Section: Determining
the Derivatives of Exponential and
Logarithmic Functions
identify
e as lim as n approaches infinity(1 + 1/n)^n and
approximate the limit, using informal methods;
define ln x as the inverse function of e^x;
determine the derivatives of the exponential
functions a^x and e^x and the logarithmic
functions
log base a of x and ln x;
determine the derivatives of combinations of the
basic polynomial, rational, exponential, and
logarithmic functions, using the rules for sums,
differences, products, quotients, and compositions
of functions.
Specific Expectations—Section: Using
Differential Calculus to Solve Problems
determine the equation of the tangent to the graph
of a polynomial, a rational, an exponential, or a
logarithmic function, or of a conic;
solve problems of rates of change drawn from a
variety of applications (including distance,
velocity, and acceleration) involving polynomial,
rational, exponential, or logarithmic functions;
solve optimization problems involving
polynomial and rational functions;
solve related-rates problems involving polynomial
and rational functions.
Specific Expectations—Section: Sketching the
Graphs of Polynomial, Rational, and
Exponential Functions
determine, from the equation of a rational
function, the intercepts and the positions of the
Page 12 of 13
Draft of Proposed Senior Mathematics Curriculum
vertical and the horizontal or oblique asymptotes
to the graph of the function;
determine, from the equation of a polynomial, a
rational, or an exponential function, the key
features of the graph of the function (i.e., intervals
of increase and decrease, critical points, points of
inflection, and intervals of concavity), using the
techniques of differential calculus, and sketch the
graph by hand;
determine, from the equation of a simple
combination of polynomial, rational, or
exponential functions (e.g.,
f(x) = e^x/x),
the key features of the graph of the combination
of functions, using the techniques of differential
calculus, and sketch the graph by hand;
sketch the graphs of the first and second
derivative functions, given the graph of the
original function;
sketch the graph of a function, given the graph of
its derivative function.
Specific Expectations—Section: Using Calculus
Techniques to Analyse Models of Functions
Moved from this location.
determine the key features of a mathematical
Moved to Grade 12 Advanced Functions - MCB 4U,
model of an application drawn from the natural or Strand: Rates of Change, Section: Using Concepts of
social sciences, using the techniques of
Rate of Change in Applications and Modelling,
differential calculus;
Moved w Section
compare the key features of a mathematical model Moved to Grade 12 Advanced Functions - MCB 4U,
with the features of the application it represents; Strand: Rates of Change, Section: Using Concepts of
Rate of Change in Applications and Modelling,
Moved w Section
predict future behaviour within an application by Moved to Grade 12 Advanced Functions - MCB 4U,
extrapolating from a mathematical model of a
Strand: Rates of Change, Section: Using Concepts of
function;
Rate of Change in Applications and Modelling,
Moved w Section
pose questions related to an application and
Moved to Grade 12 Advanced Functions - MCB 4U,
answer them by analysing mathematical models, Strand: Rates of Change, Section: Using Concepts of
using the techniques of differential calculus;
Rate of Change in Applications and Modelling,
Moved w Section
communicate findings clearly and concisely,
Deleted.
using an effective integration of essay and
mathematical forms.
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Draft of Proposed Senior Mathematics Curriculum