Calculus12 Cc

8/8/2019 Calculus12 Cc

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Advanced Functions and Introductory Calculus, Grade12, University Preparation (MCB4U)

This course builds on students experience with functions and introduces the basic concepts and skills ofcalculus. Students will investigate and apply the properties of polynomial, exponential, and logarithmic

functions; broaden their understanding of the mathematics associated with rates of change; and develop

facility with the concepts and skills of differential calculus as applied to polynomial, rational, exponential, and

logarithmic functions. Students will apply these skills to problem solving in a range of applications.

Prerequisite: Functions and Relations, Grade 11, University Preparation, or Functions, Grade 11,

University/College Preparation

Advanced Functions

Overall Expectations

By the end of this course, students will: determine, through investigation, the characteristics of the graphs of polynomial functions of various

degrees;

demonstrate facility in the algebraic manipulation of polynomials; demonstrate an understanding of the nature of exponential growth and decay; define and apply logarithmic functions; demonstrate an understanding of the operation of the composition of functions.

Specific Expectations

Chapter Page(s)

Investigating the Graphs of Polynomial Functions

By the end of this course, students will:

determine, through investigation, using graphingcalculators or graphing software, various properties of thegraphs 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 ofx-

intercepts; the behaviour near thex-intercepts; the end

behaviours; the existence of symmetry);

1

2

6-7, 9-10, 15-22, 25-29

42-47

describe the nature of change in polynomial functions ofdegree greater than two, using finite differences in tables

of values;

2 97-100

compare the nature of change observed in polynomialfunctions of higher degree with that observed in linear and

quadratic functions;

2 42-47

sketch the graph of a polynomial function whose equationis given in factored form;

2 84-87, 91-96

determine an equation to represent a given graph of apolynomial 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 finite differences).

2 101-103, 105


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Manipulating Algebraic Expressions


demonstrate an understanding of the remainder theoremand the factor theorem;

2 54-70

factor polynomial expressions of degree greater than two,using the factor theorem;

2 62-70

determine, by factoring, the real or complex roots ofpolynomial equations of degree greater than two;

2 71-82

determine the real roots of non-factorable polynomialequations by interpreting the graphs of the corresponding

functions, using graphing calculators or graphing software;

2 74, 76, 78-81

write the equation of a family of polynomial functions,given the real or complex zeros [e.g., a polynominal

function having non-repeated zeros 5, 3, and 2 will be

defined by the equation (x) = k(x 5)(x + 3)(x + 2), for k

];

2 78-79, 81

describe intervals and distances, using absolute-valuenotation;

1 7-8, 11-14, 18-22

solve factorable polynomial inequalities; 2 87-91, 93-96 solve non-factorable polynomial inequalities by graphing

the corresponding functions, using graphing calculators or

graphing software and identifying intervals above and

below thex-axis;

2 83-84, 90-96

solve problems involving the abstract extensions ofalgorithms (e.g., a problem involving the nature of the

roots of polynomial equations: Ifh and kare the roots of

the equation 3x2 + 28x 20 = 0, find the equation whose

roots are h + kand hk; a problem involving the factor

theorem: For what values ofkdoes the function (x) =x3 +

6x2 + kx 4 give the same remainder when divided by

eitherx 1 orx + 2?).

2 48-53, 59-61, 64-65, 67-

70, 73-75, 77, 79-81

Understanding the Nature of Exponential Growth and DecayBy the end of this course, students will:

identify, through investigations, using graphing calculatorsor graphing software, the key properties of exponential

functions of the form ax (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 thex-axis as an asymptote and hasy-intercept = 1);

7 414-415, 417-421

describe the graphical implications of changes in theparameters a, b, and c in the equationy = cax + b;

7 414-421

compare the rates of change of the graphs of exponentialand non-exponential functions (e.g., those with equationsy

= 2x,y =x2,y = andy = 2x);

7 464-465

describe the significance of exponential growth or decaywithin the context of applications represented by various

mathematical models (e.g., tables of values, graphs);

7 452-454, 458-461, 467-

476

pose and solve problems related to models of exponentialfunctions drawn from a variety of applications, and

communicate the solutions with clarity and justification.

7 447-448, 452-454, 456-

458, 461, 467-472, 475-

476


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Defining and Applying Logarithmic Functions


define the logarithmic function logax (a > 1) as the inverseof the exponential function ax, and compare the properties

of the two functions;

7 422-429

express logarithmic equations in exponential form, andvice versa; 7 422, 424-425, 427-429 simplify and evaluate expressions containing logarithms; 7 423-424, 428, 430-435 solve exponential and logarithmic equations, using the

laws of logarithms;

7 436-443

solve simple problems involving logarithmic scales (e.g.,the Richter scale, the pH scale, the decibel scale).

7 443-448

Understanding the Composition of Functions

By the end of this course, student will:

identify composition as an operation in which twofunctions are applied in succession;

5 268-269, 273-274

demonstrate an understanding that the composition of twofunctions exists only when the range of the first function

overlaps the domain of the second;

5 270-271, 273-275

determine the composition of two functions expressed infunction notation;

5 269-276

decompose a given composite function into its constituentparts;

5 273, 276

describe the effect of the composition of inverse functions[i.e., (f1(x)) =x].

5 272-273, 275

Underlying Concepts of Calculus


By the end of this course, students will: determine and interpret the rates of change of functions drawn from the natural and social sciences; demonstrate an understanding of the graphical definition of the derivative of a function; demonstrate an understanding of the relationship between the derivative of a function and the key features

of its graph.

Specific Expectations

Chapter Page(s)

Understanding Rates of Change


pose problems and formulate hypotheses regarding rates ofchange within applications drawn from the natural and

social sciences;

3

4

159-160, 172

231, 327-238, 245-252

calculate and interpret average rates of change fromvarious models (e.g, equations, tables of values, graphs) of

functions drawn from the natural and social sciences;

3 161-165, 167-170, 172

estimate and interpret instantaneous rates of change fromvarious models (e.g, equations, tables of values, graphs) of

functions drawn from the natural and social sciences;

3 161-172

explain the difference between average and instantaneousrates of change within applications and in general;

3 161-164, 167-172

make inferences from models of applications and comparethe inferences with the original hypotheses regarding rates

3 159-160, 166-167, 169,

171-172


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Chapter Page(s)

of change.

Understanding the Graphical Definition of the Derivative


demonstrate an understanding that the slope of a secant ona curve represents the average rate of change of the

function over an interval, and that the slope of the tangentto a curve at a point represents the instantaneous rate of

change of the function at that point;

3 121-122, 126, 128-129,

159-160, 166, 168-170,

172

demonstrate an understanding that the slope of the tangentto a curve at a point is the limiting value of the slopes of a

sequence of secants;

3 120-129

demonstrate an understanding that the instantaneous rate ofchange of a function at a point is the limiting value of a

sequence of average rates of change;

3 130-139, 159-164, 166-

172

demonstrate an understanding that the derivative of afunction at a point is the instantaneous rate of change or

the slope of the tangent to the graph of the function at that

point.

4 186, 191, 193-196, 222-

223, 225-239, 245-252

Connecting Derivatives and GraphsBy the end of this course, students will:

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;

6 312-315, 317-323, 326-

341, 362-374

identify the nature of the rate of change of a givenfunction, and the rate of change of the rate of change, as

they relate to the key features of the graph of that function;

4

6

222-223, 225-226, 232,

234-239

322-329, 331-341, 377,

379, 381, 386-387, 389-

392

sketch, by hand, the graph of the derivative of a givengraph.

4 188-189, 195

Derivatives and Applications



demonstrate an understanding of the first-principles 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 ExpectationsChapter Page(s)

Understanding the First-Principles Definition of the Derivative


determine the limit of a polynomial, a rational, or anexponential function;

3

7

140-141, 143-156

415-416, 418-421, 477


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demonstrate an understanding that limits can giveinformation about some behaviours of graphs of functions

[e.g.,5

25lim

2

5

x

x

x

predicts a hole at (5, 10)];

3 140-145, 147, 153-156

identify examples of discontinuous functions and the typesof discontinuities they illustrate; 3 141-146, 148, 150, 152-156

determine the derivatives of polynomial and simplerational functions from first principles, using the

definitions of the derivative function, identify examples of

functions that are not differentiable.

(x + h) - (x)

and'(x) =lim

h 0

h

(x) - (a)

;'(a) =lim

x a

x - a

4 187-188, 190-191, 193-

196

Determining Derivatives


justify the constant, power, sum-and- difference, product,quotient, and chain rules for determining derivatives;

4

5

197-198, 200, 203, 206,

209-211, 214

277, 283

determine the derivatives of polynomial and rationalfunctions, using the constant, power, sum-and-difference,

product, quotient, and chain rules for determining

derivatives;

4

5

197-220

278-283

determine second derivatives; 4 221-226 determine derivatives, using implicit differentiation in

simple cases (e.g., 4x2 + 9y2 = 36).

5 285-290, 300

Determining the Derivatives of Exponential and Logarithmic

Functions


identify e as and approximate the limit, using informalmethods;

lim

n

7 456-458, 461

define lnx as the inverse function ofex; 7 452-453 determine the derivatives of the exponential functions a

x

and ex and the logarithmic functions logax and lnx;7 449-453, 458-459, 463-

464, 468

determine the derivatives of combinations of the basicpolynomial, rational, exponential, and logarithmic

functions, using the rules for sums, differences, products,

quotients, and compositions of functions.

7 452, 545-456, 458-460,

465-466, 468-470


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Using Differential Calculus to Solve Problems


determine the equation of the tangent to the graph of apolynomial, a rational, an exponential, or a logarithmic

function, or of a conic;

3

7

5

124, 127, 134, 137-139

456, 459, 469, 477

288, 290

solve problems of rates of change drawn from a variety ofapplications (including distance, velocity, and

acceleration) involving polynomial, rational, exponential,

or logarithmic functions;

3

7

4

161-172

471-476

227-239, 245-252

solve optimization problems involving polynomial andrational functions;

6 375-394

solve related-rates problems involving polynomial andrational functions.

5 291-299

Sketching the Graphs of Polynomial, Rational, and Exponential

Functions

By the end of this course, students will: determine, from the equation of a rational function, the

intercepts and the positions of the vertical and the

horizontal or oblique asymptotes to the graph of the

function;

6 342-374, 398

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;

6

7

314-318, 321-322, 324-

329, 331-341, 362-374

416-417, 419-421

determine, from the equation of a simple combination ofpolynomial, rational, or exponential functions (e.g., (x) =

), the key features of the graph of the combination of

functions, using the techniques of differential calculus, and

sketch the graph by hand;

7 454-455, 466-467

sketch the graphs of the first and second derivativefunctions, given the graph of the original function;

4 224-225

sketch the graph of a function, given the graph of itsderivative function.

4 189-190, 196

Using Calculus Techniques to Analyse Models of Functions


determine the key features of a mathematical model of anapplication drawn from the natural or social sciences,

using the techniques of differential calculus;

throughout the

book

compare the key features of a mathematical model with thefeatures of the application it represents;

throughout the

book

predict future behaviour within an application byextrapolating from a mathematical model of a function;

throughout the

book

pose questions related to an application and answer themby analysing mathematical models, using the techniques of

differential calculus;

throughout the

book

communicate findings clearly and concisely, using aneffective integration of essay and mathematical forms.

throughout the

book


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Calculus12 Cc