Mathematics 31 (Senior High) /1 (1995) - Alberta Education · PDF fileMathematics 31 (Senior High) /1 (1995) A. COURSE OVERVIEW RATIONALE To set goals and make informed choices, students

Mathematics 31 (Senior High) /1 (1995)

A. COURSE OVERVIEW

RATIONALE

To set goals and make informed choices, studentsneed an array of thinking and problem-solvingskills. Fundamental to this is an understanding ofmathematical techniques and processes that willenable them to apply the basic skills necessary toaddress everyday mathematical situations, as wellas acquire higher order skills in logical analysesand methods for making valid inferences.

A knowledge of mathematics is essential to awell-educated citizenry. However, the need forand use of mathematics in the life of the averagecitizen are changing. Emphasis has shifted fromthe memorization of mathematical formulasand algorithms toward a dynamic view ofmathematics as a precise language of terms andsymbolic notation, used to describe, reason,interpret and explore. There is still a need forthe logical development of concepts and skills as abasis for the appropriate use of mathematicalinformation to solve problems. Problem-solvingstrategies, combined with techniques such asestimation and simulation, and incorporated withmodern technology, are the tools with whichmathematical problems are solved.

Students need procedural competence withnumbers and operations, together with a facility inbasic algebraic operations and an understanding offundamental mathematical concepts. They also

need to understand the connections among relatedconcepts and be familiar with their use in relevantapplications. Students must be able to solveproblems, using the mathematical processesdeveloped, and be confident in their ability toapply known mathematical skills and concepts inthe acquisition of new mathematical knowledge.In addition, the ability of technology to providequick and accurate computation and manipulation,to enhance conceptual understanding and tofacilitate higher order thinking, should berecognized and used by students.

Mathematics programs must anticipate thechanging needs of society, and provide studentswith the mathematical concepts, skills andattitudes necessary to cope with the challenges ofthe future.

The majority of students who are enrolled insenior high school mathematics have not yetcompletely internalized formal thinking withregard to mathematics. They are expected toacquire much abstract understanding in seniorhigh school mathematics courses. The content ofeach course in the mathematics program isdesigned to be cognitively appropriate andpresented in a way that is consistent with thestudents’ abilities to understand.

The senior high school mathematics programincludes the course sequences:

MATHEMATICS 31

Mathematics 31 (Senior High) /2(1995)

• Mathematics 10–20–30 and Mathematics 31• Mathematics 13–23–33• Mathematics 14–24• Mathematics 16–26.

Mathematics 31 is generally taken afterMathematics 30; however, Mathematics 31 andMathematics 30 may be taken concurrently.

Mathematics 10–20–30 and Mathematics 31 aredesigned for students who have achieved theacceptable standard in Mathematics 9, and whoare intending to pursue studies beyond high schoolat a university or in a mathematics-intensiveprogram at a technical school or college.Mathematics 10–20–30 emphasizes the theoreticaldevelopment of topics in algebra, geometry,trigonometry and statistics up to a level acceptablefor entry into such programs.

Having successfully achieved the acceptablestandard in each of Mathematics 10,Mathematics 20 and Mathematics 30, studentswill have met or exceeded the basic mathematicsentry requirements for most university oruniversity transfer programs, and will have met orexceeded all mathematics entry requirements fortechnical school or college programs. Asignificant number of students would benefit bytaking a course in calculus, thus improving theiropportunity for success in higher level courses inmathematics at university, such as those requiredby mathematics and physics honours programs, orby engineering and business programs.

Mathematics 31 emphasizes the theoretical andpractical development of topics in the algebra offunctions, trigonometry, differential calculus andintegral calculus up to a standard acceptable forentry into all first-year programs in mathematics,science, engineering and business. The course isdesigned to bridge the gap between theMathematics 10–20–30 course sequence and thecalculus course sequences offered bypost-secondary institutions.

GOALS

As a culmination of the Mathematics 10–20–30and Mathematics 31 course sequence, the specificpurposes of the Mathematics 31 course of studiesare as follows:

• to develop an understanding of the algebra offunctions and transformations, together withtheir graphs, and to apply theseunderstandings in different areas ofmathematics

• to develop a fluency in algebraic computationsinvolving rational expressions, inequalities,absolute values and trigonometric functions

• to introduce the principal concepts andmethods of differential and integral calculus

• to develop skills in problem solving andreasoning, using calculus concepts andprocedures as the context

• to apply the methods of calculus to varioussimple applications in the physical andbiological sciences, in engineering, inbusiness and in the social sciences.

PHILOSOPHY

Following the principles elaborated in theCurriculum and Evaluation Standards for SchoolMathematics, National Council of Teachers ofMathematics, 1989, the major focuses of theMathematics 31 course of studies are onattitudes, problem solving, communication,reasoning, connections and technology. In thisdocument, the term mathematical processes isused to make reference to these focuses. Furtherelaborations of these focuses are given below.

Attitudes

Students are expected to develop an appreciationfor the beauty and power of mathematics as amethod of solving problems and as an expressionof human thought. From this appreciation, theyshould respect evidence and data, showopen-mindedness toward the points of view ofothers, endeavour to be creative in thinking,persevere in the face of difficulties, be curious intheir view of the world, and show a desire tounderstand. They should show respect for andappreciation of the roles of science andtechnology in the learning and doing ofmathematics.


Problem Solving

Students develop a deep understanding ofmathematical concepts and procedures when theysolve problems related to practical contexts, tocontexts within mathematics itself, and to otherschool subjects. Students are expected to decideon which concepts and procedures are likely to beuseful in solving a particular problem, plan theappropriate application of concepts andprocedures to the situation, carry out the plan, andevaluate the reasonableness of the solution andsolution procedure when the task is complete.They should then be expected to modify thesolution procedure to solve a closely relatedproblem, or apply the solution procedure to a newsituation.

Communication

As well as being expected to construct solutions toproblems, students are expected to communicateboth solutions and solution procedures. They areexpected to communicate the representations usedfor concepts, any approximations or limitationsthat are inherent in the solution, the reliability ofthe final solution, and the reasons why particularmethods were used. These communications maybe oral or written. Students should be able to usethe language and symbolism of mathematicsappropriately.

Reasoning

A major part of mathematics is reasoning.Students are expected to analyze situations, makeassumptions, make and test conjectures, drawconclusions and justify solution procedures. Thelevel of reasoning may vary from speculative toformally rigorous, and students are expected to beable to reason at all of these levels, depending onthe context and the purpose set in the problem.

Connections

Students are expected to make connections amongmathematics concepts, between mathematics andother school subjects, and between mathematicsand everyday life. They are required especially tolink new learning to the mathematics they alreadyknow, and to apply both old and new mathematics

to a wide variety of problem situations. They areexpected to make connections between algebraicand geometric representations, and betweennumerical and symbolic methods. In addition,they are expected to be able to communicate andjustify the connections made in a problemsolution.

Technology

Students are expected to be aware of thecapabilities of various technologies, such asscientific calculators, including those withgraphing capabilities; graphing computer softwareand productivity software, such as wordprocessing, spreadsheet, database and symbolmanipulation programs. From this awareness,students are expected to use technologyappropriate to the task at hand. Besides the use oftechnology for precise solutions, students areexpected to provide estimates, approximations andsketches, using pencil and paper methods, with notechnological aids.

EXPECTATIONS

Students are expected to be mathematicallyliterate at the conclusion of their senior highschool mathematics education. Mathematicalliteracy refers to students’ abilities andinclinations to manage the demands of their worldthrough the use of mathematical concepts andprocedures to solve problems, communicate andreason. More specifically, students will beexpected to:

• have developed the concepts, skills andattitudes that will enable the acquisition ofmathematical knowledge beyond theconclusion of secondary education

• have developed the concepts, skills andattitudes that will ensure success inmathematical situations that occur in futureeducational endeavours, employment andeveryday life

• have achieved understanding of the basicmathematical concepts, and developed theskills and attitudes needed to becomeresponsible and contributing members ofsociety


• apply basic mathematical concepts and skillsin practical situations

• have developed critical and creative thinkingskills

• communicate mathematical ideas effectively

• understand how mathematics can be used toinvestigate, interpret and make decisions inhuman affairs

• understand how mathematics can be used inthe analyses of natural phenomena

• understand the connections and interplayamong various mathematical concepts andbetween mathematics and other disciplines

• understand and appreciate the positivecontributions of mathematics, as a science andas an art, to civilization and culture.

.


B. LEARNER EXPECTATIONS

COURSE FOCUS

The Mathematics 31 course is designed tointroduce students to the mathematical methods ofcalculus. The course acts as a link between theoutcomes of the Mathematics 10–20–30 programand the requirements of the mathematicsencountered in post-secondary programs. Thecourse builds on existing skills in working withfunctions and expands this knowledge to include astudy of limits in preparation for a study ofdifferential and integral calculus. The methods ofcalculus are applied to problems encountered inthe areas of science, engineering, business andother fields of endeavour.

The focus of the course is to examine functionsthat describe changing situations as opposed to themore static situations encountered in previousmathematics courses. Emphasis is placed onusing graphical methods to illustrate the way inwhich changing functions behave.

COURSE STRUCTURE

Mathematics 31 is designed in a required–electiveformat. The required component is intended totake the larger proportion of the instructional time.There are eight units available in the electivecomponent, of which one or more units areintended to take the remainder of the instructionaltime.

The time given to the required component, and thenumber of elective units covered, will vary,depending on local needs. In general, moststudents will do one or two elective units;however, some students must do as many as fourin order to integrate the requirements of externalagencies into the Mathematics 31 course.

Required Component

The four sections of the required component areas follows:

• precalculus and limits• derivatives and derivative theorems• applications of derivatives• integrals, integral theorems and integral

applications.

Elective Component

The eight possible units available in the electivecomponent are as follows:

• calculus of exponential and logarithmicfunctions

• numerical methods• volumes of revolution• applications of calculus to physical sciences

and engineering• applications of calculus to biological sciences• applications of calculus to business and

economics• calculus theorems• further methods of integration.

COURSE LEARNER EXPECTATIONS

The learner expectations found on pages 8 through51 are the required student outcomes at the end ofthe Mathematics 31 course. They do not serve asa required sequence for instruction. Thesequence for instruction will vary, depending onthe mathematical background of the students, thelearning resources being used, and the preferencesof individual teachers.

The learner expectations are divided into coursegeneral learner expectations and related specificlearner expectations.

The course general learner expectations forMathematics 31, on pages 8 and 9, provide abridge between the philosophy statements for allsenior high school mathematics courses and therelated specific learner expectations for eachsection or unit of study.

The related specific learner expectations refer toparticular areas of study, such as limits andantiderivatives. They are on pages 10 through 35for the required component, and on pages 36through 51 for the elective component.

Learner expectations are organized into fiveconcept strands, each with three outcomecolumns. The strands are identified asMathematical Processes, Measurement andGeometry, Algebra of Functions and Limits,Calculus of Derivatives and Integrals, and DataManagement. The outcomes are organized under


Conceptual Understanding, ProceduralKnowledge and Problem-solving Contexts.

Although itemized separately, the outcomes aremeant to be developed together, using anappropriate balance for the content being studied,and applying the learnings in an appropriatecontext.

Strands

The strand, Mathematical Processes, describesproblem-solving, communication and reasoningstrategies that apply to many areas ofmathematics. Examples include the constructionand use of mathematical models and the properuse of approximation techniques for difficultcalculations.

In the context of the Mathematics 31 course,mathematical processes include the making andtesting of conjectures, the comparison ofnumerical and algebraic solutions, the connectionsbetween similar solution procedures andcompletely different problems, and the differencesbetween intuitive and rigorous proofs.

The strand, Measurement and Geometry,describes the mathematics of spatial reasoning.Measurement involves associating numericalvalues to various geometrical properties of one-,two- or three-dimensional figures while thegeometry involves the properties of one-, two- orthree-dimensional figures, and the geometricalrepresentation of domains, limits, derivatives andintegrals.

In the context of the Mathematics 31 course,measurement and geometry involve thegeneralizations of the concepts of slope, curve,line and average to contexts characterized bychange, and to contexts where the traditionaldescriptions give misleading interpretations.

The strand, Algebra of Functions and Limits,describes the generalization of patterns, and itsdescription in ordered pair, symbolic andgeometrical forms.

In the context of the Mathematics 31 course, thealgebra of functions and limits represents anextension of algebraic concepts to enablealgebraic models to be used in contexts wherefunctions may not exist, or may be discontinuous.

The strand, Calculus of Derivatives andIntegrals, describes the extension of algebra tosituations that model change.

In the context of the Mathematics 31 course, thecalculus of derivatives and integrals refers only tothe differential and integral calculus of relationsand functions of a single, real variable. Theemphasis is more on the use and application ofderivatives, and less on the use and application ofintegrals.

The strand, Data Management, describes themathematics of information processing, from thedescription of simple data sets to the making ofpredictions based on models constructed from thedata.

In the context of the Mathematics 31 course, datamanagement is confined to the generalization ofthe concepts of average value and equation root,together with the development of tools that cancompute numerical approximations to any desireddegree of accuracy. The calculus of randomvariables is not part of Mathematics 31, thoughthe calculus methods developed in Mathematics31 could be used in the context of a randomvariable and its associated probability distribution.

Outcomes

The Conceptual Understanding outcomes referto the major concepts involved in the developmentof understanding the algebra of functions, theestablishment of limits, the differential calculusand the integral calculus.

Conceptual expectations are characterized by theuse of such action verbs as defining,demonstrating, describing, developing,establishing, explaining, giving, identifying,illustrating, linking, recognizing and representing.


The Procedural Knowledge outcomes refer tothe development of fluency in carrying outprocedures related to the general learnerexpectations. Examples of such procedures arethose used for the computation of limits,derivatives and integrals, the determination ofmaxima and minima, and the computation ofaverage values of functions over intervals.

Procedural expectations are characterized by theuse of such action verbs as approximating,calculating, constructing, differentiating,estimating, factoring, locating, rationalizing,simplifying, sketching, solving, using andverifying.

The Problem-solving Contexts outcomes refer tothe construction of models describing applicationsin a broad range of contexts. They also refer tothe combination of algorithms to determine morecomplicated combinations of limits, derivativesand integrals, and to the use of appropriatetechnology to approximate difficult models or tocarry out long, computational procedures. Theyalso refer to the investigation of phenomena, thecollection and interpretation of data in variouscontexts, and the transfer of knowledge frommathematics to other areas of human endeavour.

Problem-solving expectations are characterized bythe use of such action verbs as adapting,analyzing, combining, communicating,comparing, connecting, constructing, deriving,evaluating, fitting, investigating, justifying,modelling, proving, reconstructing andtranslating.


COURSE GENERAL LEARNER EXPECTATIONS BY STRAND

STRANDS CONCEPTUAL UNDERSTANDING

Students will demonstrate understanding of the concepts ofMathematics 31, by:

Mathematical Processes

Mastery of course expectations enablesstudents to construct and solve modelsdescribing mathematical situations in a broadrange of contexts, and to use the appropriatetechnology to approximate difficult models andcarry out long calculations.

• identifying and giving examples of the strengths andlimitations of the use of technology in the computationof limits, derivatives and integrals

• giving examples of the differences between intuitive and

rigorous proofs in the context of limits, derivatives andintegrals.

Measurement and Geometry

Mastery of course expectations enablesstudents to demonstrate the conceptualunderpinnings of calculus to determine limits,derivatives, integrals, rates of change andaverages, using geometrical representations.

• describing the relationships among functions afterperforming translations, reflections, stretches andcompositions on a variety of functions

• linking displacement, velocity and acceleration of an

object moving in a straight line with nonuniformvelocity.

Algebra of Functions and Limits

Mastery of course expectations enablesstudents to translate among symbolic, diagramand graphical representations of situations thatdescribe both continuous and discrete functionsof one real variable.

• giving examples of the limits of functions andsequences, both at finite and infinite values of theindependent variable.

Calculus of Derivatives and Integrals

Mastery of course expectations enablesstudents to determine limits, derivatives,integrals and rates of change.

• connecting the derivative with a particular limit, andexpressing this limit in situations like secant and tangentlines to a curve

• relating the derivative of a complicated function to thederivative of simpler functions

• relating the zeros of the derivative function to thecritical points on the original curve

• recognizing that integration can be thought of as aninverse operation to that of finding derivatives.

Data Management

Mastery of course expectations enables studentsto use calculus concepts to describe datadistributions and random variables at anintroductory level.

• describing the connections between the operation ofintegration and the finding of areas and averages.


PROCEDURAL KNOWLEDGE PROBLEM-SOLVING CONTEXTS

Students will demonstrate competence in theprocedures of Mathematics 31, by:

Students will demonstrate problem-solving skills inMathematics 31, by:

• combining and modifying familiar solutionprocedures to form a new solution procedure to arelated problem.

• using the connections between a given problem and

either a simpler or equivalent problem, or a previouslysolved problem, to solve the given problem

• producing approximate answers to complexcalculations by simplifying the models used.

• drawing the graphs of functions by applyingtransformations to the graphs of known functions

• calculating the displacement, velocity andacceleration of an object moving in a straight linewith nonuniform velocity.

• constructing geometrical proofs at an intuitive level

that generalize the concepts of slope, area, averageand rate of change.

• expressing final algebraic and trigonometricanswers in a variety of equivalent forms, with theform chosen to be the most suitable form for thetask at hand

• computing limits of functions, using definitions,limit theorems and calculator/computer methods.

• constructing mathematical models for situations in a

broad range of contexts, using algebraic andtrigonometric functions of a single real variable.

• computing derivatives of functions, usingdefinitions, derivative theorems and calculator/computer methods

• computing definite and indefinite integrals ofsimple functions, using antiderivatives, integraltheorems and calculator/computer methods.

• determining the optimum values of a variable in

various contexts, using the concepts of maximum andminimum values of a function

• using the concept of critical values to sketch thegraphs of functions, and comparing these sketches tocomputer-generated plots of the same functions

• connecting a derivative and the appropriate rate ofchange, and using this connection to relatecomplicated rates of change to simpler ones.

• calculating the mean value of a function over aninterval.

• fitting mathematical models to situations described by

data sets.

Mathematics 31 (Senior High) /10(1995) Precalculus and Limits (Required)

RELATED SPECIFIC LEARNER EXPECTATIONS FOR REQUIRED COMPONENT

PRECALCULUS AND LIMITS (Required)

Overview

The first parts of this Mathematics 31 section aredesigned to reinforce and extend algebraic andtrigonometric skills. The algebra of functionsintroduced in Mathematics 20 and applied toexponential, logarithmic, trigonometric and polynomialfunctions in Mathematics 30 is extended to the generalfunction. Algebraic and geometric representations oftransformations are used. The skills acquired are anecessary adjunct to all work in the Mathematics 31course. The last part concentrates on the numerical andgeometrical development of the concepts of limits andlimit theorems.

Student Outcomes Summary

After completing this section, students will be expectedto have acquired reliability and fluency in the algebraicskills of factoring, operations with radicals and radicalexpressions, coordinate geometry, and transformation offunctions. They will also be expected to solve linear,quadratic and absolute value inequalities, and to solvetrigonometric equations and identities. They will beexpected to express the concept of a limit numericallyand geometrically, and to calculate limits, using firstprinciples and the limit theorems.

GENERAL LEARNER EXPECTATIONS CONCEPTUAL UNDERSTANDING

Students are expected to understand thatfunctions, as well as variables, can becombined, using operations, such as additionand multiplication, and demonstrate this, by:

Students will demonstrate conceptual understandingof the algebra of functions, by:

• illustrating different notations that describefunctions and intervals

• expressing, in interval notation, the domain andrange of functions

• expressing the sum, product, difference andquotient, algebraically and graphically, given anytwo functions

• expressing, algebraically and graphically, thecomposition of two or more functions

• illustrating the solution sets for linear, quadraticand absolute value inequalitiesP x a

P x a

( )

( )

≥

≤

ax bx c d2 + + ≥

• illustrating the difference between the conceptsof equation and identity in trigonometriccontexts.

• describing the relationship amongfunctions after performing translations,reflections, stretches and compositions ona variety of functions

• drawing the graphs of functions by

applying transformations to the graphs ofknown functions

• expressing final algebraic and

trigonometric answers in a variety ofequivalent forms, with the form chosen tobe the most suitable form for the task athand

• constructing mathematical models for

situations in a broad range of contexts,using algebraic and trigonometricfunctions of a single real variable.

Mathematics 31 (Senior High) /11Precalculus and Limits (Required) (1995)


Students will demonstrate competence in theprocedures associated with the algebra offunctions, by:

Students will demonstrate problem-solving skills,by:

• using open, closed and semi-open intervalnotation

• finding the sum, difference, product,quotient and composition of functions

• modelling problem situations, using sums,

differences, products and quotients of functions

• investigating the connections between thealgebraic form of a function f (x) and thesymmetries of its graph.

• solving inequalities of the typesP x a

P x a

( )

( )

≥

≤

P x

Q xa

( )

( )≥

P x

Q xa

( )

( )≥ ax bx c d2 + + ≥

• using the following trigonometric identities:• primary and • double and half

reciprocal ratio angle• sum and difference • Pythagorean

sin (A±B)cos (A±B)

to simplify expressions and solveequations, express sums and differences asproducts, and rewrite expressions in avariety of equivalent forms.



Students are expected to understand thatfunctions can be transformed, and thesetransformations can be representedalgebraically and geometrically, anddemonstrate this, by:

Students will demonstrate conceptual understandingof the transformation of functions, by:

• describing the relationship amongfunctions after performing translations,reflections, stretches and compositions on avariety of functions

• drawing the graphs of functions byapplying transformations to the graphs ofknown functions

• expressing final algebraic andtrigonometric answers in a variety ofequivalent forms, with the form chosen tobe the most suitable form for the task athand

• constructing mathematical models forsituations in a broad range of contexts,using algebraic and trigonometric functionsof a single real variable.

• describing the similarities and differencesbetween the graphs of y = f(x) andy = af[k(x+c)]+d, where a, k, c and d are realnumbers

• describing the effects of the reflection of thegraphs of algebraic and trigonometric functionsacross any of the lines y = x, y = 0, or x = 0

• describing the effects of the parameters a, b, cand d on the trigonometric functionf (x) = a sin [b(x+c)]+d

• describing the relationship between parallel andperpendicular lines

• describing the condition for tangent, normal andsecant lines to a curve

• linking two problem conditions to a system oftwo equations for two unknowns.



Students will demonstrate competence in theprocedures associated with the transformationof functions, by:


• sketching the graph of, and describingalgebraically, the effects of any translation,reflection or dilatation on any of thefollowing functions or their inverses:• linear, quadratic or cubic polynomial• absolute value• reciprocal• exponential• step

• sketching and describing, algebraically, the

effects of any combination of translation,reflection or dilatation on the followingfunctions:• f (x) = a sin [b(x+c)]+d• f (x) = a cos [b(x+c)]+d• f (x) = a tan [b(x+c)]

• translating problem conditions into equation or

inequality form.

• finding the equation of a line, given anytwo conditions that serve to define it

• solving systems of linear–linear, linear–

quadratic or quadratic–quadratic equations.



Students are expected to understand that finalanswers may be expressed in differentequivalent forms, and demonstrate this, by:

Students will demonstrate conceptual understandingof equivalent forms, by:

• giving examples of the differences betweenintuitive and rigorous proofs in the contextof limits, derivatives and integrals

• expressing final algebraic andtrigonometric answers in a variety ofequivalent forms, with the form chosen tobe the most suitable form for the task athand.

• describing what it means for two algebraic ortrigonometric expressions to be equivalent.



Students will demonstrate competence in theprocedures associated with the construction ofequivalent forms, by:


• factoring expressions with integral andrational exponents, using a variety oftechniques

• rationalizing expressions containing a

numerator or a denominator that contains aradical

• simplifying rational expressions, using any

of the four basic operations.

• illustrating the difference between verification

and proof in the comparison of two algebraic ortrigonometric expressions.



Students are expected to understand thatdescriptions of change require a carefuldefinition of limit and the precise use of limittheorems, and demonstrate this, by:

Students will demonstrate conceptual understandingof limits and limit theorems, by:

• explaining the concept of a limit

• giving examples of functions with limits, with left-hand or right-hand limits, or with no limit

• giving examples of bounded and unboundedfunctions, and of bounded functions with no limit

• explaining, and giving examples of, continuousand discontinuous functions

• defining the limit of an infinite sequence and aninfinite series

• explaining the limit theorems for sum,difference, multiple, product, quotient and power

• illustrating, using suitable examples, the limit

theorems for sum, difference, multiple, product,quotient and power.

• identifying and giving examples of thestrengths and limitations of the use oftechnology in the computation of limits,derivatives and integrals


• giving examples of the limits of functionsand sequences, both at finite and infinitevalues of the independent variable

• computing limits of functions, usingdefinitions, limit theorems and calculator/computer methods.



Students will demonstrate competence in theprocedures associated with limits and limittheorems, by:


• determining the limit of any algebraicfunction as the independent variableapproaches finite or infinite values forcontinuous and discontinuous functions

• sketching continuous and discontinuous

functions, using limits, intercepts andsymmetry

• calculating the sum of an infiniteconvergent geometric series

• using definitions and limit theorems todetermine the limit of any algebraicfunction as the independent variableapproaches a fixed value

• using definitions and limit theorems to

determine the limit of any algebraicfunction as the independent variableapproaches ± ∞ .

• proving and illustrating geometrically, thefollowing trigonometric limits:

lim sin lim

sinx

x

x x

x

x→ →=

0 01or

AND

lim cos

x

x

x→−

=0

10

• using the basic trigonometric limits, combinedwith limit theorems, to determine the limits ofmore complex trigonometric expressions

• comparing numerical and algebraic processes

for the determination of algebraic andtrigonometric limits

Mathematics 31 (Senior High) /18(1995) Derivatives and Derivative Theorems (Required)

DERIVATIVES AND DERIVATIVETHEOREMS (Required)

Overview

The concept of the slope of a line can be extended togive students an intuitive feel for the meaning of slopeat a point on a curve. By examining slopes of secants,and using the ideas of limits, the derivative can be putinto a well-defined form. Derivative theorems are usedto find only the derivatives of algebraic andtrigonometric functions.


After completing this section, students will be expectedto find derivatives of algebraic and trigonometricfunctions and determine equations of tangents atspecific points. For selected functions, they will also beexpected to prove these derivatives, using firstprinciples. Students use the power, chain, product andquotient rules to calculate the derivatives of thesefunctions. They will also be expected to use implicitdifferentiation to find the derivatives of certainalgebraic and trigonometric relations.


Students are expected to understand that thederivative of a function is a limit that can befound, using first principles, and demonstratethis, by:

Students will demonstrate conceptual understandingof derivatives, by:

• showing that the slope of a tangent line is a limit • explaining how the derivative of a polynomial

function can be approximated, using a sequenceof secant lines

• explaining how the derivative is connected to the

slope of the tangent line

• recognizing that f (x) = xn can be differentiatedwhere n∈R

• identifying the notations ′ ′f x ydy

dx( ), , and as

alternative notations for the first derivative of afunction

• explaining the derivative theorems for sum and

difference ( ) ( ) ( ) ( )f g x f x g x± ′ = ′ ± ′ • explaining the sense of the derivative theorems for

sum and difference, using practical examples.


• connecting the derivative with a particularlimit, and expressing this limit in situationslike secant and tangent lines to a curve

• computing derivatives of functions, usingdefinitions, derivative theorems andcalculator/computer methods.

Mathematics 31 (Senior High) /19Derivatives and Derivative Theorems (Required) (1995)


Students will demonstrate competence in theprocedures associated with derivatives, by:


• finding the slopes and equations of tangentlines at given points on a curve, using thedefinition of the derivative

• estimating the numerical value of the

derivative of a polynomial function at apoint, using a sequence of secant lines

• deriving ′f x( ) for polynomial functions up to

the third degree, using the definition of thederivative

• using the definition of the derivative to find

′ = + = − =f x f x ax c n nn( ) ( ) ( ) .for where or1 12

• using the definition of the derivative todetermine the derivative of f (x) = xn wheren is a positive integer

• differentiating polynomial functions, using

the derivative theorems for sum anddifference

• differentiating functions that are singleterms of the form xn where n is rational.



Students are expected to understand thatderivatives of more complicated functions canbe found from the derivatives of simplerfunctions, using derivative theorems, anddemonstrate this, by:

Students will demonstrate conceptual understandingof derivative theorems, by:

• demonstrating that the chain, power, product andquotient rules are aids to differentiatecomplicated functions

• identifying implicit differentiation as a tool to

differentiate functions where one variable isdifficult to isolate

• explaining the relationship between implicit

differentiation and the chain rule

• comparing the sum, difference, product andquotient theorems for limits and derivatives

• explaining the derivation of the derivative

theorems for product and quotient

[ ]

( ) ( ) ( ) ( ) ( ) ( )

( )( ) ( ) ( ) ( )

( )

fg x f x g x f x g x

f

gx

f x g x f x g x

g x

′ = ′ + ′′

=′ − ′

2

• explaining the derivative theorems for productand quotient, using practical examples

• illustrating second, third and higher derivativesof algebraic functions

• describing the second derivative geometrically.


• relating the derivative of a complicatedfunction to the derivative of simplerfunctions

• expressing final algebraic andtrigonometric answers in a variety ofequivalent forms, with the form chosen tobe the most suitable form for the task athand

• computing derivatives of functions, usingdefinitions, derivative theorems andcalculator/ computer methods.



Students will demonstrate competence in theprocedures associated with derivativetheorems, by:


• finding the derivative of a polynomial,power, product or quotient function

• applying the chain rule in combination with

the product and quotient rule • using the technique of implicit

differentiation

• writing final answers in factored form • finding the slope and equations of tangent

lines at given points on a curve

• finding the second and third derivatives offunctions.

• deriving the quotient rule from the productrule

• showing that equivalent forms of the

derivative of a rational function can be foundby using the product and the quotient rules

• determining the derivative of a function

expressed as a product of more than twofactors

• determining the second derivative of an

implicitly defined function AND, in one or more of the following, by: • showing the derivative for a relation found by

both implicit and explicit differentiation to bethe same

• finding the equations of tangent lines to the

standard conics.



Students are expected to understand thattrigonometric functions have derivatives, andthese derivatives obey the same derivativetheorems as algebraic functions, anddemonstrate this, by:

Students will demonstrate conceptual understandingof the derivatives of trigonometric functions, by:




• computing derivatives of functions, usingdefinitions, derivative theorems andcalculator/computer methods.

• demonstrating that the three primarytrigonometric functions have derivatives at allpoints where the functions are defined

• explaining how the derivative of a trigonometric

function can be approximated, using a sequenceof secant lines.



Students will demonstrate competence in theprocedures associated with derivatives oftrigonometric functions, by:


• calculating the derivatives of the threeprimary and three reciprocal trigonometricfunctions

• estimating the numerical value of the

derivative of a trigonometric function at apoint, using a sequence of secant lines

• using the power, chain, product and

quotient rules to find the derivatives ofmore complicated trigonometric functions

• using the derivative of a trigonometric

function to calculate its slope at a point,and the equation of the tangent at thatpoint.

• using the definition of the derivative to find the

derivative for the sine and cosine functions • explaining why radian measure has to be used

in the calculus of trigonometric functions.

Mathematics 31 (Senior High) /24(1995) Applications of Derivatives (Required)

APPLICATIONS OF DERIVATIVES(Required)

Overview

Inside the area of pure mathematics, graph sketching ofpolynomial functions, rational algebraic functions andsimple trigonometric functions are looked at.Derivatives are used to solve maximum and minimumproblems in economic and geometrical contexts and tosolve related rate problems in geometrical and algebraiccontexts.


After completing this section, students will be expectedto provide systematic sketches of graphs of polynomial,rational, algebraic and trigonometric functions of onevariable. They will be expected to solve maximum–minimum and related rate problems in economic andgeometric contexts. They will be expected to constructmaximum–minimum and related rate models ingeometric contexts, and to use the techniques ofderivatives to solve problems based on the modelsconstructed.


Students are expected to understand thatcalculus is a powerful tool in determiningmaximum and minimum points and insketching of curves, and demonstrate this, by:

Students will demonstrate conceptualunderstanding of maxima and minima, by:

• identifying, from a graph sketch, locations atwhich the first and second derivative are zero

• illustrating under what conditions symmetry

about the x-axis, y-axis or the origin will occur • explaining how the sign of the first derivative

indicates whether or not a curve is rising orfalling; and by explaining how the sign of thesecond derivative indicates the concavity of thegraph

• illustrating, by examples, that a first derivative ofzero is one possible condition for a maximum ora minimum to occur

• explaining circumstances wherein maximum and

minimum values occur when ′f x( ) is not zero • illustrating, by examples, that a second derivative

of zero is one possible condition for an inflectionpoint to occur

• explaining the differences between local maxima

and minima and absolute maxima and minima inan interval

• explaining when finding a maximum value isappropriate and when finding a minimum valueis appropriate.

• relating the zeros of the derivative functionto the critical points on the original curve

• using the connections between a givenproblem and either a simpler or equivalentproblem, or a previously solved problem,to solve the given problem

• constructing mathematical models forsituations in a broad range of contexts,using algebraic and trigonometric functionsof a single real variable

• determining the optimum values of avariable in various contexts, using theconcepts of maximum and minimum valuesof a function

• using the concept of critical values tosketch the graphs of functions, andcomparing these sketches to computer-generated plots of the same functions

• fitting mathematical models to situationsdescribed by data sets.

Mathematics 31 (Senior High) /25Applications of Derivatives (Required) (1995)


Students will demonstrate competence in theprocedures associated with maxima andminima, by:


• sketching the graphs of the first and secondderivative of a function, given its algebraicform or its graph

• using zeros and intercepts to aid in graph

sketching

• using the first and second derivatives tofind maxima, minima and inflection pointsto aid in graph sketching

• determining vertical, horizontal and

oblique asymptotes, and domains andranges of a function

• finding intervals where the derivative isgreater than zero or less than zero in orderto predict where the function is increasingor decreasing

• verifying whether or not a critical point is a

maximum or a minimum

• using a given model, in equation or graphform, to find maxima or minima that solvea problem.

• employing a systematic calculus procedure tosketch algebraic and trigonometric functions

• comparing and contrasting graphs plotted on a

calculator and graphs sketched, using asystematic calculus procedure

• calculating maxima and minima for such

quantities as volumes, areas, perimeters andcosts

• illustrating the connections among geometric,

economic or motion problems, the modellingequations of these problems, the resultingcritical points on the graphs and theirsolutions, using derivatives

AND, in one or more of the following, by: • constructing a mathematical model to

represent a geometric problem, and using themodel to find maxima and/or minima

• constructing a mathematical model to

represent a problem in economics, and usingthe model to find maximum profits orminimum cost

• constructing a mathematical model to

represent a motion problem, and using themodel to find maximum or minimum time ordistance.

Mathematics 31 (Senior High) /26(1995) Applications of Derivatives (Required)


Students are expected to understand thatcomplicated rates of change can be related tosimpler rates, using the chain rule, anddemonstrate this, by:

Students will demonstrate conceptual understanding ofrelated rates, by:

• illustrating how the chain rule can be used torepresent the relationship between two or morerates of change

• explaining the clarity that Leibnitz’s notation

gives to expressing related rates

• illustrating the time rate of change of a functiony = f (x) or a relation R(x, y) = 0.


• combining and modifying familiarsolution procedures to form a new solutionprocedure to a related problem

• using the connections between a givenproblem and either a simpler or equivalentproblem, or a previously solved problem,to solve the given problem

• constructing geometrical proofs at anintuitive level that generalize the conceptsof slope, area, average and rate of change

• expressing the connection between aderivative and the appropriate rate ofchange, and using this connection to relatecomplicated rates of change to simplerones.

Mathematics 31 (Senior High) /27Applications of Derivatives (Required) (1995)


Students will demonstrate competence in theprocedures associated with related rates, by:


• using the chain rule to find the derivativeof a function with respect to an externalvariable, such as time

• using Leibnitz’s notation to illustrate

related rates

• constructing a mathematical model to represent

time rates of change of linear measures, areas,volumes, surface areas, etc.

• solving related rate problems that use models

containing primary trigonometric functions.

• constructing a chain rule of related rates,using appropriate variables

• calculating related rates for the timederivatives of areas, volumes, surface areasand relative motion

• calculating related rates of change withrespect to time, given an equation thatmodels electronic circuits or otherengineering situations

• using the chain rule to derive anacceleration function in terms of position,given a velocity function expressed interms of position.


Integrals, Integral Theorems andIntegral Applications (Required)

INTEGRALS, INTEGRAL THEOREMS ANDINTEGRAL APPLICATIONS (Required)

Overview

Corresponding to the operation of finding derivatives isthe inverse operation of finding the antiderivatives. Thearea under a curve can be defined as the limit of the sumof small rectangular areas and can be found from thisdefinition. The fundamental theorem of calculus linksthe solution of the area problem to the finding of specialantiderivatives. As in the case of derivatives, the use ofintegral theorems makes the finding of antiderivativesand areas much easier. Derivatives and integrals can beinterrelated through the context of nonuniform motionin a straight line.


After completing this section, students will be expectedto find the antiderivatives of any polynomial function,together with the antiderivatives of the simplest rational,algebraic and trigonometric functions. They will beexpected to use antiderivatives to find areas undercurves, and to use the connection between derivativesand integrals to relate displacement, velocity andacceleration. Conceptually, students will be expected toillustrate the properties of antiderivatives, definiteintegrals and the fundamental theorem of calculus, usinggeometric representations.


Students are expected to understand that theoperation of finding a derivative has an inverseoperation of finding an antiderivative, anddemonstrate this, by:

Students will demonstrate conceptual understandingof antiderivatives, by:

• explaining how differentiation can have aninverse operation

• showing that many different functions can havethe same derivative

• representing, on the same grid, a family of curves

that form a sequence of functions, all having thesame derivative.

• giving examples of the differencesbetween intuitive and rigorous proofs inthe context of limits, derivatives andintegrals

• recognizing that integration can be

thought of as an inverse operation to thatof finding derivatives

• computing definite and indefinite

integrals of simple functions, usingantiderivatives, integral theorems andcalculator/computer methods

• using the connections between a given

problem and either a simpler or equivalentproblem, or a previously solved problem,to solve the given problem.




Students will demonstrate competence in theprocedures associated with antiderivatives, by:


• finding the antiderivatives of polynomials,rational algebraic functions andtrigonometric functions

• finding the family of curves whose firstderivative has been given

• solving separable first order differential

equations for general and specificsolutions.

• finding the antiderivatives of rationalfunctions and polynomial powers bycomparison and inspection methods

• determining antiderivatives for polynomial,rational and trigonometric functions.

Mathematics 31 (Senior High) /30

(1995) Integrals, Integral Theorems andIntegral Applications (Required)


Students are expected to understand that thearea under a curve can be expressed as the limitof a sum of smaller rectangles, and demonstratethis, by:

Students will demonstrate conceptual understandingof area limits, by:

• defining the area under a curve as a limit of thesums of the areas of rectangles

• establishing the existence of upper and lower

bounds for the area under a curve.


• giving examples of the differences

between intuitive and rigorous proofs inthe context of limits, derivatives andintegrals

• constructing geometrical proofs at an

intuitive level that generalize the conceptsof slope, area, average and rate of change.




Students will demonstrate competence in theprocedures associated with area limits, by:


• sketching the area under a curve(polynomials, rational, trigonometric) overa given interval, and approximating thearea as the sum of individual rectangles.

• communicating the similarities and differences

between definite integrals and areas undercurves.




Students are expected to understand that thearea under a curve can be related to theantiderivative of the function defining thecurve; and in turn, can use the antiderivative todetermine the areas under and between curves,and demonstrate this, by:

Students will demonstrate conceptual understandingof definite integrals, by:

• identifying the indefinite integral f x dx( )∫ as a

sum of an antiderivative F x( ) and a constant c

• explaining how the definite integral betweenfixed limits a and b is a number whose value isF(b) – F(a)

• explaining the connection between the numericalvalues of the area and the definite integral forfunctions f of a constant sign and a variable sign

• illustrating the following properties of integrals

cf x dx c f x dx

f x g x dx f x dx g x dx

a

b

a

b

a

b

a

b

a

b

( ) ( )

[ ( ) ( )] ( ) ( )

=

± = ±

∫∫∫∫∫

• describing the sense of the fundamental theoremof calculus

• explaining how the fundamental theorem ofcalculus relates the area limit to theantiderivative of a function describing a curve.


• recognizing that integration can be

thought of as an inverse operation to thatof finding derivatives

• describing the connections between the

operation of integration with the findingof areas and averages



• calculating the mean value of a function

over an interval • constructing geometrical proofs at an





Students will demonstrate competence in theprocedures associated with definite integrals,by:


• calculating the definite integral forpolynomial, rational and trigonometricfunctions

• determining the area between a curve andthe x-axis over a given domain

• determining the area between a curve andthe x-axis:• if f(x) has a constant sign over a given

interval• if f(x) has a change in sign over a given

interval

• determining the area between curves over agiven interval

• determining the area between intersectingcurves.

• relating the value of the integral between x = aand x = b to the area between the curve and thex-axis over the interval [a, b]

• using integration theorems, such as those listed

below, to simplify definite integrals

dxxfdxxfdxxf

dxxfdxxf

c

b

b

a

c

a

a

b

b

a

)()()(

)()(

∫∫∫

∫

+=

−=∫

• using integral theorems to simplify morecomplicated integrals

• illustrating the conditions necessary for a

function to be differentiable, or able to beintegrated.




Students are expected to understand thatvelocity and acceleration are the first andsecond derivatives of displacement with respectto time; and that once one of the threefunctions is known, the other two can be foundby finding derivatives or antiderivatives, anddemonstrate this, by:

Students will demonstrate conceptual understandingof the relationships among displacement, velocityand acceleration, by:

• describing the motion of a body, using sketchesof the first and second derivatives

• explaining the difference between a stationarypoint and a turning point in the context of linearmotion

• illustrating the concepts of derivative andantiderivative in the context of displacement,velocity and acceleration.

• linking displacement, velocity andacceleration of an object moving in astraight line with nonuniform velocity

• calculating the displacement, velocity and

acceleration of an object moving in astraight line with nonuniform velocity







Students will demonstrate competence in theprocedures associated with displacement,velocity and acceleration, by:


• estimating an instantaneous velocity, usingslopes of secant lines to represent averagevelocities

• finding the first and second derivatives of aposition function to get instantaneousvelocity and instantaneous accelerationfunctions, where the position function is analgebraic function or a trigonometricfunction of time

• using antiderivatives of acceleration andvelocity functions to get velocity anddisplacement functions.

• solving problems associated with distance,velocity and acceleration whose models arerestricted to those of the forms

′ = ′′ =y t f t y t f t( ) ( ) ( ) ( )and • deriving the following kinematic equations,

starting from the expression a = constant:• v = at + v0

• v2 = v02 + 2ad

• d = ½at2 + v0 t + d0

• determining the equations of velocity and

acceleration in simple harmonic motion, startingfrom the displacement equation:• x = A cos (kt + c).

Mathematics 31 (Senior High) /36(1995) Calculus of Exponential and Logarithmic Functions (Elective)

RELATED SPECIFIC LEARNER EXPECTATIONS FOR ELECTIVE COMPONENT

CALCULUS OF EXPONENTIAL ANDLOGARITHMIC FUNCTIONS (Elective)

Overview

The calculus of exponential and logarithmic functions hasmany important applications in modelling naturalphenomena, such as natural growth and decay, andproblems involving return to equilibrium, such as thosefound in thermal and chemical reactions. The study ofnatural logarithms and the natural base exponentialfunction is key in the development of derivatives andintegrals for all such functions, as well as in theirapplications. This elective, together with the electiveApplications of Calculus to Biological Sciences, can betaught as an integrated whole.


After completing this elective section, students will beexpected to represent symbolically and graphically,differentiate and integrate, exponential and logarithmicfunctions. They will also be expected to solve problemswhere the modelling equation is given, and to constructequations to model a situation with given parameters.


Students are expected to understand thatexponential and logarithmic functions havelimits, derivatives and integrals that obey thesame theorems as do algebraic andtrigonometric functions, and demonstrate this,by:

Students will demonstrate conceptual understandingof the calculus of exponential and logarithmicfunctions, by:

• defining exponential and logarithmic functionsas inverse functions

• explaining the special properties of the number e,together with a definition of e as a limit

• illustrating that the derivative of an exponentialor logarithmic function may be derived from thedefinition of the derivative

• illustrating that base-e exponential andlogarithmic functions form a convenientframework within which the calculus of similarfunctions in any base may be developed

• illustrating how exponential and logarithmicfunctions may be used to model certain naturalproblems involving growth, decay and return toequilibrium.

• computing limits of functions, usingdefinitions, limit theorems and calculator/computer methods

• computing derivatives of functions, using

definitions, derivative theorems andcalculator/computer methods




situations in a broad range of contexts,using algebraic and trigonometricfunctions of a single real variable

• fitting mathematical models to situations

described by data sets.

Mathematics 31 (Senior High) /37Calculus of Exponential and Logarithmic Functions (Elective) (1995)


Students will demonstrate competence in theprocedures associated with the calculus ofexponential and logarithmic functions, by:


• estimating the values of the limits e and ex

• approximating the slopes of y = ex andy = ln x for some specific value of x

• finding the derivative and antiderivative of

the base-e exponential function • using limit theorems to evaluate the limits

of simple exponential and logarithmicfunctions

• finding the derivative of the natural

logarithmic function

• finding the derivatives of logarithmicfunctions having bases other than e

• finding the derivatives and antiderivativesof exponential functions having bases otherthan e

• computing a

b

a

b kx

xdx e dx∫ ∫1

and

• solving the differential equations′ = ′ = −y ky and y k y y( ).0

• evaluating maxima and minima of givenfunctions involving exponential andlogarithmic functions

• finding areas bounded by exponential,logarithmic or reciprocal functions

AND, in one or more of the following, by:

• relating natural growth, natural decay andreturn to equilibrium to the differentialequations ′ = ′ = −y ky y k y yor ( )0

• solving natural growth and decay problems,starting from the equations

′ = ′ = −y ky y k y yor ( )0

• fitting exponential models to observed data

• calculating distance, velocity and accelerationfor falling bodies, with air resistance presentas part of the model

• connecting the integral a

b dx

px q∫ +

with the integral a

b dx

x∫ .

Mathematics 31 (Senior High) /38(1995) Numerical Methods (Elective)

NUMERICAL METHODS (Elective)

Overview

Limits and derivatives are sometimes difficult tocompute using algebraic methods, but often can becalculated more easily using numerical approximationsand estimates. In the case of definite integrals, mostfunctions cannot be integrated exactly, and numericalmethods are necessary. This unit has a balance betweenintuitive methods based on guess-and-test, and formalprocedures, such as Simpson’s rule for definiteintegrals.


After completing this elective section, students will beexpected to compute numerical approximations forlimits, equation roots and definite integrals. They willbe expected to relate these procedures to thefundamental concepts of calculus studied in the requiredsections of Mathematics 31, and to apply these conceptsto a variety of relations, functions and equations.Finally, they will be expected to evaluate the reliabilityof a numerical procedure and to communicate theresults of the evaluation.


Students are expected to understand that manylimits, derivatives, equation roots and definiteintegrals can be found numerically, anddemonstrate this, by:

Students will demonstrate conceptual understandingof the principles of numerical analysis, by:

• describing the difference between an exactsolution and an approximate solution

• identifying when a particular numerical method

is likely to give poor results

• explaining the difference between iterative andnoniterative procedures

• explaining the basis of the Newton–Raphson

procedure for determining the roots of f (x) = 0

• describing the basis of a limit, derivative,equation root or integral procedure in geometricterms

• connecting the number of subdivisions of therange of integration with the accuracy of theestimate for the integral

• showing that all numerical integration formulasare procedures that interpolate between the lowerand upper Riemann sums for the integral.


• computing limits of functions, using

definitions, limit theorems and calculator/computer methods

• computing derivatives of functions, using

definitions, derivative theorems andcalculator/computer methods


integrals of simple functions, usingantiderivatives, integral theorems andcalculator/computer methods.

Mathematics 31 (Senior High) /39Numerical Methods (Elective) (1995)


Students will demonstrate competence in theprocedures associated with numerical methods,by:

Students will demonstrate problem-solving skills, inone or more of the following:

• estimating the value of a limit bysystematic trial and error

• calculating the numerical value of the

derivative at a point on a curve, whether ornot there is a defining formula for the curve

• solving the equation f (x) = 0 by systematictrial and error, and by the Newton–Raphson method

• calculating the upper and lower Riemannsums for a definite integral

• calculating the value of a definite integral,using the midpoint rule

• calculating the value of a definite integral,using the trapezoidal rule

• calculating the value of a definite integral,using Simpson’s rule.

• comparing the errors in computing definiteintegrals when using different procedures

• writing computer software for the computationof limits, equation roots or definite integrals

• reconstructing limit processes so thatnumerical evaluations can be efficient andreliable

• evaluating the reliability of a numericalprocedure for finding a limit, an equation rootor a definite integral.

Mathematics 31 (Senior High) /40(1995) Volumes of Revolution (Elective)

VOLUMES OF REVOLUTION (Elective)

Overview

Volumes of solids may be calculated using tripleintegrals, but for volumes formed by revolving thegraphs of simple functions about the x- or y-axis, asingle integration will suffice. By using limits to find aninfinite sum of cylindrical volumes, a value for avolume of revolution can be found.


After completing this elective section, students will beexpected to find volumes of revolutions of the graphs ofsimple polynomial and trigonometric functions by usingthe “disc” method. Students will be expected to relatethis procedure to that of finding the area between thegraph of a function and the x-axis by the limit of thesum of a set of rectangles whose widths approach zero(a Riemann sum).


Students are expected to understand thatvolumes of revolution may be considered as thelimiting sum of smaller volumes and can berelated to definite integrals, and demonstratethis, by:

Students will demonstrate conceptual understandingof volumes of revolution, by:

• identifying the solid generated by the rotation ofthe graph of a function, either between twoboundary values, or between two graphs

• explaining the connection between the volume of

revolution and the volume of a cylindrical disc • demonstrating how the formula for the volume of

revolution by the disc method could begenerated.

• describing the connections between theoperation of integration and the finding ofareas and averages

• combining and modifying familiar

solution procedures to form a newsolution procedure to a related problem




problem and either a simpler or equivalentproblem, or a previously solved problem,to solve the given problem.

Mathematics 31 (Senior High) /41Volumes of Revolution (Elective) (1995)


Students will demonstrate competence in theprocedures associated with volumes ofrevolution, by:

Students will demonstrate problem-solving skills, inone or both of the following, by:

• using the relationship V f x dxa

b= ∫π [ ( )]2

to find the volume of revolution betweenthe boundaries of a and b for polynomialand trigonometric functions

• finding the volume of revolution between

two polynomial or trigonometric functionsby first finding the intersection points ofthe graphs of the two functions.

• deriving formulas for the volume of a cylinder,cone and sphere

• revolving the graph of a function about ahorizontal or vertical line, other than the x- ory-axis, and finding the resulting volume ofrevolution.


Applications of Calculus toPhysical Sciences and Engineering (Elective)

APPLICATIONS OF CALCULUS TOPHYSICAL SCIENCES AND ENGINEERING(Elective)

Overview

Calculus is used extensively in physical sciences and inengineering. The study of energy, forces and motioncan be represented in terms of differential equations thatinvolve algebraic and trigonometric functions of time.These differential equation models, together with theirsolutions, can be used to make hypotheses andpredictions.


After completing this elective section, students will beexpected to construct differential equation models forsimple physical science and engineering situations, andto solve those models that involve algebraic andtrigonometric functions of time.


Students are expected to understand that mostof the important equations of physics aredifferential equations, and calculus providesthe most efficient solution method, anddemonstrate this, by:

Students will demonstrate conceptual understandingof the links among calculus, the physical sciencesand engineering, by:

• illustrating situations in which differentialequations are required to represent problems

• developing one or more differential equations inthe areas of linear motion, simple harmonicmotion, work, hydrostatic force, moments ofinertia, radioactive decay or similar situations

• showing that the concept of mean value can beapplied to situations where the quantity varieswith time, or where a range of values exists in asystem of multiple bodies.

• linking displacement, velocity andacceleration of an object moving in astraight line with nonuniform velocity

• describing the connections between the

operation of integration with the findingof areas and averages

• calculating the displacement, velocity and

acceleration of an object moving in astraight line with nonuniform velocity


over an interval • producing approximate answers to

complex calculations by simplifying themodels used






Applications of Calculus toPhysical Sciences and Engineering (Elective)


Students will demonstrate competence in theprocedures associated with the application ofcalculus to the physical sciences andengineering, by:

Students will demonstrate problem-solving skills, inone of the following, by:

• solving differential equations of type′′ =y t f t( ) ( )

• solving differential equations of type

′′ = −y t k y( ) 2

• calculating the work done by anynonuniform force f (x) using

W f x dxa

b= ∫ ( )

• determining root–mean–square values forsinusoidal functions.

• determining the half-life, the decay rate, andthe activity as a function of time for aradioisotope

• analyzing the motion and energy in anoscillating spring system (Hooke’s law)

• analyzing hydrostatic forces on the surface ofsubmerged objects

• determining the moment of inertia for rigidbodies

• determining the centre of mass for individualbodies and for systems of bodies

• integrating Newton’s second law whenexpressed in the form mx t f x′′ =( ) ( ).

Mathematics 31 (Senior High) /44(1995) Applications of Calculus to Biological Sciences (Elective)

APPLICATIONS OF CALCULUS TOBIOLOGICAL SCIENCES (Elective)

Overview

Many biological problems are concerned with rates ofgrowth, rates of decay, and rates of transport of energyand materials across boundaries. Mathematicalsolutions to many of these problems can be found byconstructing differential equation models that havesolutions involving exponential and logarithmicfunctions. This elective, together with the electiveCalculus of Exponential and Logarithmic Functions, canbe taught as an integrated whole.


After completing this elective section, students will beexpected to construct differential equation models forsimple biological situations, and to solve those modelsthat involve exponential and logarithmic functions.


Students are expected to understand that manyimportant biological applications of calculusare connected with models involving thesolution of the differential equation

′ =f x kf x( ) ( ), and demonstrate this, by:

Students will demonstrate conceptual understandingof the links between calculus and the biologicalsciences, by:

• defining exponential and logarithmic functionsas inverse functions

• explaining the special properties of the number e,

together with a definition of e as a limit • illustrating that the derivative of an exponential

or logarithmic function may be derived from thedefinition of the derivative

• illustrating how differential equations may beused to model certain biological problemsinvolving growth, decay and movement across aboundary.

• combining and modifying familiarsolution procedures to form a newsolution procedure to a related problem


problem and either a simpler or equivalentproblem, or a previously solved problem,to solve the given problem

• producing approximate answers to






Mathematics 31 (Senior High) /45Applications of Calculus to Biological Sciences (Elective) (1995)


Students will demonstrate competence in theprocedures associated with the application ofcalculus to the biological sciences, by:


• estimating the values of the limits e and ex

• using limit theorems to evaluate the limits

of simple exponential and logarithmicfunctions

• approximating the slopes of y = ex and

y = ln x for some specific value of x • finding the derivative of the natural

logarithmic function

• finding the derivative and antiderivative ofthe base-e exponential function

• using methods of guess-and-test, andcomparing coefficients to solve thedifferential equations

′ = ′ = −y ky y k y yand ( )0

• verifying that y = Aekx satisfies thedifferential equation ′ =y ky.

• relating natural growth, natural decay andreturn to equilibrium to the differentialequations ′ = ′ = −y ky y k y yor ( )0

• solving natural growth and decay problemsstarting from the equations

′ = ′ = −y ky y k y yor ( )0

AND, either, by:

• fitting differential equation models toobserved biological data

OR, both of the following, by:

• relating growth subject to limits to the logisticequation ′ = −y ky L y( )

• solving logistic equation models, andestimating values for the parameters k and L,from experimental data.

Mathematics 31 (Senior High) /46(1995) Applications of Calculus to Business and Economics (Elective)

APPLICATIONS OF CALCULUS TOBUSINESS AND ECONOMICS (Elective)

Overview

Although there are far too many variables andinfluences present to make a complete analysis ofeconomic situations, many of them can be simplified toa point where Mathematics 31 students can begin tounderstand the applications of calculus to this area ofstudy.


After completing this elective section, students will beexpected to analyze an economic model based on theneed to maximize revenue and profit while minimizingcost.


Students are expected to understand thatcalculus may be used as a tool to analyzesituations in business and economics thatinvolve revenue, profit and cost, anddemonstrate this, by:

Students will demonstrate conceptual understandingof the links among calculus, business and economics,by:

• explaining how calculus procedures frequentlyarise in models used in business and economics

• explaining how both maximum and minimumvalues are important in the making of businessand economic decisions.

• relating the zeros of the derivativefunction to the critical points on theoriginal curve

• producing approximate answers to




• determining the optimum values of a

variable in various contexts, using theconcepts of maximum and minimumvalues of a function

• using the concept of critical values to

sketch the graphs of functions, andcomparing these sketches to computer-generated plots of the same functions



Mathematics 31 (Senior High) /47Applications of Calculus to Business and Economics (Elective) (1995)


Students will demonstrate competence in theprocedures associated with the application ofcalculus to business and economics, by:

Students will demonstrate problem-solving skills, inone or both of the following, by:

• sketching polynomial, exponential andtrigonometric functions of one variable

• using a given revenue, profit or costfunction to calculate and justify optimumvalues

• finding the maximum of a revenue or profitfunction that is expressed as a function ofprice or number sold

• finding the minimum of a cost function thatis expressed as a function of price ornumber sold.

• determining a revenue, profit or cost functionfrom a problem situation that can be modelled,using polynomial functions

• modelling the business cycle, usingtrigonometric functions.

Mathematics 31 (Senior High) /48(1995) Calculus Theorems (Elective)

CALCULUS THEOREMS (Elective)

Overview

The simple geometric representations of limit,continuity, derivative and integral work well for mostfunctions used in problem applications. However, it ispossible to give examples of misleading conclusionswhen these representations are being used. Thiselective section shows these examples and leads thestudent into a deeper knowledge, and the proofs, of thebasic limit, derivative and integral theorems.


After completing this elective section, students will beexpected to illustrate the differences between intuitiveproofs and rigorous proofs in the context of proof,construct a counterexample to a hypothesis, andconstruct a rigorous proof of one of the simpler limit,derivative or integral theorems.


Students are expected to understand that limit,derivative and integral theorems can be provedat different levels of rigor, from intuitive toanalytic, and demonstrate this, by:

Students will demonstrate conceptual understandingof the nature of proof in the context of limit,derivative and integral theorems, by:

• proving the equivalence of the product and thequotient rule for derivatives

• comparing the nature of intuitive and rigorousproofs knowing the conditions under which atheorem is true

• explaining what is an example, and what is acounterexample

• illustrating the intermediate value theorem,Rolle’s theorem, the mean value theorem and thefundamental theorem of calculus, by examplesand counterexamples, using both graphical andalgebraic formulations.





Mathematics 31 (Senior High) /49Calculus Theorems (Elective) (1995)


Students will demonstrate competence in theprocedures associated with the construction ofproofs of limit, derivative and integraltheorems, by:


• using specific functions to illustrate theequivalence of the product and quotientrules

• locating an error in a given proof of acalculus theorem

• verifying the mean value theorem and thefundamental theorem of calculus forspecific examples.

• composing rigorous proofs for derivativetheorems, starting from the correspondinglimit theorems

AND, in one of the following, by:

• deriving formulas for the derivatives ofcomplicated functions, using the basic rules;e.g., functions, such as

yf x g x

h xy f g h x= =( ) ( )

( )( ( ( )))or

• proving that a differentiable function satisfiesthe mean value theorem

• constructing, and justifying, an iterativesolution procedure for the equation f(x) = c,using the intermediate value theorem.

Mathematics 31 (Senior High) /50(1995) Further Methods of Integration (Elective)

FURTHER METHODS OF INTEGRATION(Elective)

Overview

Certain algebraic and trigonometric functions cannot beintegrated using the usual methods involving the powerrule. It is therefore necessary to look at other methods;these being integration by substitution, integration byparts, or integration by partial fractions.


After completing this elective section, students will beexpected to recognize which particular method ofintegration is needed. They will also be expected to useand combine these methods appropriately in evaluatingwith indefinite and definite integrals.


Students are expected to understand thatintegration by substitution, by parts, and bypartial fractions are procedures necessary whendealing with certain algebraic andtrigonometric functions, and demonstrate this,by:

Students will demonstrate conceptual understandingof the methods of integration, by:

• identifying integrals that cannot be evaluated,using the antiderivatives of polynomial ortrigonometric functions

• showing how substitutions into a definite integralchange both the function to be integrated and thelimits of integration

• recognizing when it would be appropriate to useintegration by substitution, by parts or by partialfractions.

• recognizing that integration can bethought of as an inverse operation to thatof finding derivatives

• expressing final algebraic and

trigonometric answers in a variety ofequivalent forms, with the form chosen tobe the most suitable form for the task athand



• using the connections between a givenproblem and either a simpler or equivalentproblem, or a previously solved problem,to solve the given problem.

Mathematics 31 (Senior High) /51Further Methods of Integration (Elective) (1995)


Students will demonstrate competence in theprocedures associated with methods ofintegration, by:


• using a change of variable to integrate bysubstitution

• using a trigonometric substitution tointegrate algebraic functions containing

terms, such as a x2 2−

• using a trigonometric identity as a first stepin an integration by substitution

• using integration by parts to integrate aproduct

• writing a rational function as a sum ofpartial fractions

• using integration by partial fractions tointegrate rational algebraic functions.

• deriving the formula for integration by parts,using the formula for the derivative of aproduct

• combining two or more methods to integrate arational algebraic function

• adapting integration methods forantiderivatives to the evaluation of definiteintegrals.

Documents

Mathematics 31 (Senior High) /1 (1995) - Alberta Education · PDF fileMathematics 31 (Senior High) /1 (1995) A. COURSE OVERVIEW RATIONALE To set goals and make informed choices, students