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Grade 12 Calculus (45S)

A Half Course for Independent Study

Grade 12 calculus (45s)

A Half Course for

Independent Study

2008

Manitoba Education, Citizenship and Youth

Manitoba Education, Citizenship and Youth Cataloguing in Publication Data

515 Grade 12 calculus (45S) : a half course for independent

study

Previously published as : Senior 4 calculus

(45S) : a half course for distance education delivery.

ISBN-13: 978-0-7711-3723-5

ISBN-10: 0-7711-3723-0

1. Calculus—Study and teaching (Secondary).

2. Calculus—Study and teaching (Secondary)—Manitoba.

3. Calculus—Programmed instruction. I. Manitoba. Dept. of

Education and Training. II. Title : Senior 4 calculus (45S) :

a half course for education delivery

Copyright © 1995, 2008, the Crown in Right of Manitoba as represented by the Minister of

Education and Training. Manitoba Education and Training, School Programs Division,

1970 Ness Avenue, Winnipeg, Manitoba R3J 0Y9.

Every effort has been made to provide proper acknowledgement of original sources and to

comply with copyright law. If cases are identified where this has not been done, please

notify Manitoba Education and Training to correct any omissions.

This document was originally published as Senior 4 Calculus (45S): A Half Course forDistance Education Delivery.

Grade 12 Calculus Acknowledgements iii

Acknowledgements

Manitoba Education and Training gratefully acknowledges the contributions of the following

individuals in the development of Grade 12 Calculus (45A): A Half Course for Independent

Study.

Course Writer

Ray Brown Brandon, MB

Course Editor

Katharine Tetlock Courseware Developer/ Math/Science Technology Unit

Project Manager Program Development Unit

Members of the Development Team

Ray Brown Brandon, MB

Paul Cuthbert School Programs Division Manitoba Education and Training

Katharine Tetlock School Programs Division Manitoba Education and Training

Manitoba Education and Training Staff

School Programs Division

Lee-Ila Bothe Consultant Technical Support Unit

Program Development Branch

Paul Cuthbert Project Leader Math/Science Technology Unit


Lynn Harrison Desktop Publisher Technical Support Unit


Michael Hartley Publications Editor Technical Support Unit


Duncan McCaig Project Manager Math/Science Technology Unit


Katharine Tetlock Project Manager Math/Science Technology Unit


Grade 12 Calculus

Introduction

Welcome to Grade 12 Calculus! You are embarking upon the

study of an exciting and extremely useful branch of

mathematics.

The general objectives of Grade 12 Calculus as listed in the

Manitoba Curriculum Guide (1991) are to

• develop an understanding of the properties of functions

and their role in many different areas of mathematics

• increase the student’s awareness of the scope of

mathematics

• prepare the student for further courses in calculus and

related fields

• develop skills that will increase a student’s ability to

simplify algebraic and numeric expressions

• assist students to see that the mathematics of calculus

has many practical applications

Course Structure

The Grade 12 Calculus Independent Study course is a self-

contained learning package. No additional resources (e.g.,

textbooks) are required.

In this course, you will be introduced to the concepts of

limits, derivative of a function, and the many applications of

derivatives.

The course is divided into four units

Unit 1 — Limits

Unit 2 — Derivatives

Unit 3 — Applications of Derivatives

Unit 4 — Integration

Each unit is composed of three parts

Part 1 — Tutorial, with examples

Part 2 — Answer Keys (Answers Only)

Part 3 — Answer Keys (Complete Solutions)

Grade 12 Calculus Introduction i

Evaluation

There are seven self-tests for the student to test his or her

progress

Self-Test 1 — to be written at the mid-point of Unit 1.

Self-Test 2 — to be written at the end of Unit 1.






At the completion of Grade 12 Calculus, you will write the

Final Examination.

Grade 12 Calculus is a challenging but rewarding learning

experience.

Guide Graphics

Throughout this course you will notice different graphics in

the margin. These graphics help guide you throughout the

course. Below is a description of them.

Note: Prompts you to take special note of

what is being stated.

Key words: Draws your attention to

important words throughout this course.

Assignment: Indicates that the

following work must be completed before

continuing.

ii Introduction Grade 12 Calculus

Grade 12 CaLCULUS (45S)

Unit 1

Limits

Contents

Unit 1 — Limits

Topic 0 — Review of Algebraic Skills

Objectives

1: To Use Laws of Exponents and Perform

Operations with Polynomials 1

2: To Factor Polynomials 7

3: To Perform Operations with Rational Expression 11

4: To Rationalize Numerators or Denominators 15

5: To Determine the Slope of a Line 19

6: To Find Equations of Oblique Lines and Lines

Parallel to an Axis 23

7: To Use Functional Notation 29

Topic 1 — Limits

Objectives

1: To Understand the Concept of Limit 31

2: To Understand the Mathematical Notation

for Limits 33

Topic 2 — Limit Theorems and Evaluating Limits

Objectives

1: To Verify the Six Limit Theorems 39

2: To Evaluate Limits 43

Topic 3 — Evaluating Special Types of Limits

Objectives

1: To Evaluate Limits at Infinity 55

2: To Evaluate One-sided Limits 59

3: To Determine Infinite Limits 67

4: To Determine the Asymptotes and Graph

of a Function Using Limits 75

Topic 4 — Continuity

Objectives

1: To Understand the Concept of Continuity 91

2: To Use the Algebraic Definition of the Function

Together with the Limit Concept 97

Grade 12 Calculus Contents 3

Notes

4 Contents Grade 12 Calculus

Topic 0— Review of Algebraic Skills

To be successful in Grade 12 Calculus, you must be able to use

the algebraic skills (manipulating algebraic expressions,

analytical geometry of the straight line, and simple graphing)

developed during the study of your Grade 10 and Grade 11

mathematics courses. You must be sufficiently proficient in the

use of these skills that you can concentrate on the calculus

concept you are learning without becoming “bogged down” in the

algebra. Please work carefully through the following objectives

of topic 0. Later, when you are working your way through a new

concept, your algebraic skills—an asset rather than a liability—

will provide the confidence you need for the successful

conclusion of the task.

Objective 1: To Use Laws of Exponents and Perform

Operations with Polynomials

1. The multiplication of powers: xm× x

n= x

m+n

Examples:

Note: When the

powers are

multiplied, the

exponents are

added.

x x x

x

x x x

x

x x x

x

x

2 5 2 5

7

0 5 0 2 0 2 0 5

0 7

2

12

15

12

15

710

×

×

×

. . . .

.

××

×

x x

x

x x x

x

5 2 5

3

2 5 2 5

7

Grade 12 Calculus Unit 1, Topic 0, Objective 1 5

6 Unit 1, Topic 0, Objective 1 Grade 12 Calculus

2. The division of powers: xm x

n= x

m–n

Examples:

3. The power of a power: (xm)n

= xmn

Examples:

4. The power of a product: (a × b)m

= am× b

m

Examples:

Note: When

finding the

power of a

product, the

product of

powers is

determined.

x y x y

x x

x

x x

× ×

× ×

× ×

2 2 2

4 4 4

4

3 3 3

3 3

81

3 3

27xx3

Note: When

finding the

power of a

power, the

exponents are

multiplied.

x x

x x

x x

x x

2 3 6

2 3 6

2 3 6

2 3 6

Note: When the

powers are

divided, the

exponents are

subtracted.

x

xx

x

x x x

x

x x x

x

x x x

5

2

5 2

2 5 2 5

3

2 5 2 5

12

15

12

15

310

x3


5. The power of a quotient:

Examples:

6. The exponent of a power is zero: a0

= 1

Examples:

Note: When the

exponent of a

power is zero,

the power

always has a

value of one.

5 1

1

2 1

2 1

27 1

0

0

0

0

0

x

x

xy

x

y

x

y

m m

m

Note: When

finding the

power of a

quotient, the

quotient of

powers is

determined.

2

3

2

3

16

81

2

3

2

3

8

27

3

4

3

4 4

4

3 3

3

2

x

y

x22

2

2

2

4

9

16

y

x

y

7. The exponent is negative:

Examples:

8. The exponent is a fraction:

Examples:

Note: When the

exponent is a

fraction, the

numerator (p) is

the power to

which the qth

root of x (square

root, cube root,

4th root, etc.—

depending on

the value of q) is

taken.

8 8 2 4

811

81

1

81

1

3

1

27

625 625

1

23

34

34

13

32

2

43

3

3

225 5

5 5

3

3

×

x x or x xpq

pqq

ppq

Note: When the

exponent is

negative,

determine the

reciprocal of the

power and

change the

negative to a

positive

exponent.

21

2

1

8

21

2

1

8

21

2

1

32

21

3

3

3

3

5

5

5

3 2

xx

x

x y22

1

4

3 2

6 2

x y

x y

xx

n

n

1


Assignment

Complete each of the following exercises.

1. Remove the negative and/or zero exponents and simplify the

following:

2. Rewrite the following using exponents:

3. Simplify the following expression and leave the answer with

positive exponents:

4. Simplify the following:

Do you remember these? a b a a b ab b

a b a a b ab b

3 2 2 2 3

3 3 2 2 3

3 3

3 3

and

a)

b)

x x

x x x

3 3 2 5 2

2 2 1 5

2

3 2

3 4

6

2 3 0 2

1 2

xy x y

x y

a)

b)

x

x

43

45

a)

b)

2

125

27

16

56

13

6

4 3 0

2 6

x y

x y z

x y


Notes


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