PDF -Suggestions and Tips for Success in Calculus A - Calculus Success in 20 Minutes a Day2ndEdition[1]
Wait Loading...


PDF :1 PDF :2 PDF :3 PDF :4 PDF :5 PDF :6 PDF :7 PDF :8 PDF :9 PDF :10


Like and share and download

Calculus Success in 20 Minutes a Day2ndEdition[1]

Suggestions and Tips for Success in Calculus A

PDF Read Book \\ Calculus Success in 20 Minutes a Day data accessplan mysydneycbd nsw gov au 9781576858899 calculus calculus success in 20 minutes a day pdf PDF AP Calculus at JF has always been a very challenging and bedfordjfhs sharpschool

Related PDF

Read Book \\ Calculus Success in 20 Minutes a Day

[PDF] Read Book \\ Calculus Success in 20 Minutes a Day data accessplan mysydneycbd nsw gov au 9781576858899 calculus calculus success in 20 minutes a day pdf
PDF

AP Calculus at JF has always been a very challenging and

[PDF] AP Calculus at JF has always been a very challenging and bedfordjfhs sharpschool COURSE 20SYLLABUS pdf
PDF

To: All SAS Calculus Students

[PDF] To All SAS Calculus Students mdc edu main images AP instructions tcm6 32367 pdf
PDF

LearningExpress eBooks - State Library of Ohio - Ohiogov

[PDF] LearningExpress eBooks State Library of Ohio Ohio gov library ohio gov wp content LEL eBooks StLibOhio pdf
PDF

Calculus Success in 20 Minutes a Day pdf by L Editors - apdf684

[PDF] Calculus Success in 20 Minutes a Day pdf by L Editors a pdf 684a pdf 684 ecoolbooks calculus success in 20 pdf 1366337 pdf
PDF

Active Calculus - MIT Mathematics

[PDF] Active Calculus MIT Mathematicsmath mit edu seminars emes slides 2018 10 02 Boelkins pdf
PDF

The Role of Calculus in the Transition from High School to College

[PDF] The Role of Calculus in the Transition from High School to College macalester edu ~bressoud RoleOfCalculus UMLN pdf
PDF

1 COURSE SUPPLEMENT FOR Math 11A Calculus and

[PDF] 1 COURSE SUPPLEMENT FOR Math 11A Calculus and soe ucsc edu ~msmangel Math11ASupp pdf
PDF

Math 2144 -Calculus I - Oklahoma State University

[PDF] Math 2144 Calculus I Oklahoma State Universitymath okstate edu people nhoffman MATH2144 63421 Syllabus pdf
PDF

Suggestions and Tips for Success in Calculus A

Suggestions and Tips for Success in Calculus A *By failing to prepare, Most problems should take no more than 20 minutes if you have grasped the material
PDF

calculus tutoring book.pdf

The Calculus Tutoring Book By Carol Ash Robert B Ash - Ebooks

notendur hi is adl2 CalcI Complete pdf calculus I have included some material that I do not usually have time to cover in class and because this changes from semester to semester it is not noted here You will need to find one of your fellow class mates

Calculus With Analytic Geometry SM Yusuf (Solution Manual)

Course Number and Title: CALCULUS AND ANALYTICAL GEOMETRY-NS

eqexnaka files wordpress 2015 07 calculus Calculus And Analytic Geometry By Sm Yusuf Pdf Download EBook Calculus With Analytic Geometry SM Yusuf (Solution Manual) for freePublisher Imprint Ilmi Kitab Beautiful Code Download Free PDF eBook susotaran files wordpress 2015 07 solution Solution Manual Of Calculus With Analytic

Calculus

Time to conquer Calculus! - Photomath

PDF Calculus ocw mit edu ans7870 resources Strang Edited Calculus Calculus pdf PDF Calculus Volume 1 cloudfront d3bxy9euw4e147 cloudfront oscms CalculusVolume1 OP pdf PDF Calculus Volume 2 cloudfront d3bxy9euw4e147

  1. calculus pdf
  2. advanced calculus
  3. differential calculus tutorial pdf
  4. calculus for beginners pdf free download
  5. calculus textbook
  6. basic calculus problems with solutions pdf
  7. calculus 2 pdf
  8. calculus volume 1

Calcutta high court's lawyer list

Name & Address Name & Address SUPREME COURT BAR

PDF IN THE HIGH COURT AT CALCUTTA ORDINARY ORIGINAL CIVIL 164 100 79 153 kolkata CS 250 2010 04022019 J 236 pdf PDF SUPREME COURT OF INDIA LIST OF SENIOR ADVOCATES sci gov in pdf

  1. list of supreme court senior advocate
  2. supreme court top lawyers list
  3. list of senior advocates delhi high court 2018
  4. delhi high court senior advocates 2018
  5. supreme court advocate name list
  6. supreme court senior advocates contact
  7. list of senior advocates delhi high court 2017
  8. senior advocate delhi high court fees

Caldeira, Teresa - Ciudad de Muros

The Urban Question in the Twenty-first Century A Dialogue

cafedelasciudades ar carajillo imagenes4 Carajillo de > Democracia y muros nuevas articulaciones del espacio público Por Teresa Caldeira Traducción M Mayorga Texto traducido de la conferencia "Democracy and Walls New Articulations of the Public Space" pronunciada el 25 de mayo de 2003 en las jornadas "Ciudades

CALDEIRAS E VASOS SOB PRESSÃO - UFF

NR 13 - Caldeiras e Vasos de Pressão (113000-5) - IARUnicamp

PDF caldeiras e vasos de pressão DDS Online ddsonline br images caldeiras e vasos de pressao pdf PDF NR 13 Caldeiras e Vasos de Pressão IFBa ifba edu br professores Aula Caldeiras 20(NR13) pdf

Calder v. Bull, 3 U.S. 386 (1798)

the civil ex post facto clause - Wisconsin Law Review

PDF Calder v Bull, interpreting the Constitution as a social Dialnet dialnet unirioja es descarga articulo 2380130 pdf PDF Bicentennial of Calder v Bull In Defense of a Democratic Middle kb osu edu

CALDERA DE FLUIDO TÉRMICO - CE_INGLES

Índia só aceita a paz conquistando Bangla

PDF multisectorial componentes, equipos y sistemas de Automática y automatica robotica es Estadisticas R asp?C PDF Mejora la seguridad en Gamonal con la llegada de Gente Digital gentedigital es upload ficheros revistas 200901 377 pdf

Caldera de Marcet

PROPOSTA D'ADJUDICACIÓ - rubicat

upcommons upc edu bitstream handle 2099 1 7221 Plano de situación del edificio de Can Marcet 2 1 Tabla de sectorización del consumo energético destinado a climatización Combustible Sectores Energía Potencia Consumo ACS para las duchas de la policía Gas Policía Calefacción con caldera Gas Taller, policía y parte

Home back Next

s Success in 20 Minutes a Day2ndEdition[1]

Description

STUDY GUIDES/Mathematics

MASTER CALCULUS IN JUST 20 MINUTES A DAY

SKILL BUILDERS

CALCULUS ESSENTIALS INSIDE:

CALCULUS SUCCESS PRACTICE

A good knowledge of calculus is essential for success on many tests and applicable for a wide range of careers

Calculus Success in 20 Minutes a Day helps students refresh and acquire important calculus skills

This guide provides a thorough review that fits into any busy schedule

Each step takes just 20 minutes a day

! Pretest—Pinpoint your strengths and weaknesses Lessons—Master calculus essentials with hundreds of exercises

CALCULUS Success in 20 Minutes a Day

• Functions • Trigonometry • Graphs • Limits • Rates of change • Derivatives • Basic rules • Derivatives of sin(x) and cos(x) • Product and quotient rules • Chain rule • Implicit differentiation • Related rates • Graph sketching • Optimization • Antidifferentiation • Areas between curves • The fundamental theorem of calculus • Techniques of integration • and more

Posttest—Evaluate the progress you’ve made

❏ Packed with key calculus concepts including rates of change,

! Additional resources for preparing for important standardized tests

Visit LearningExpress’s Online Practice Center to:

Access additional calculus practice exercises

Receive immediate scoring and detailed answer explanations Focus your study with our customized diagnostic report,

and boost your overall score to guarantee success

Prepare for a Brighter Future

LearnATest

ADDED VALUE—Access to online practice with Instant Scoring

FREE Calculus Practice

❏ Includes hundreds of practice questions with detailed answer explanations

❏ Measure your progress with pre– and posttests

❏ Build essential calculus skills for success on the AP exams

2ND EDITION Completely Revised and Updated

McKibben

L EARNINGE XPRESS

Calc2e_00_i-x_FM

11/18/11

12:32 AM

CALCULUS SUCCESS in 20 Minutes a Day

Calc2e_00_i-x_FM

11/18/11

12:32 AM

Page ii

Calc2e_00_i-x_FM

11/18/11

12:32 AM

Page iii

CALCULUS SUCCESS in 20 Minutes a Day Second Edition Mark A

McKibben Christopher Thomas ®

Calc2e_00_i-x_FM

11/18/11

12:32 AM

Page iv

Copyright © 2012 LearningExpress,

All rights reserved under International and Pan-American Copyright Conventions

Published in the United States by LearningExpress,

New York

Library of Congress Cataloging-in-Publication Data: McKibben,

Calculus success in 20 minutes a day / Mark A

McKibben

Previous ed

: Calculus success in 20 minutes a day / Thomas,

Christopher

© 2006

ISBN 978-1-57685-889-9 1

Calculus—Problems,

Thomas,

Christopher,

Title: Calculus success in twenty minutes a day

T47 2012 515—dc23 2011030506 Printed in the United States of America 987654321 ISBN 978-1-57685-889-9 For information or to place an order,

contact LearningExpress at: 2 Rector Street 26th Floor New York,

NY 10006 Or visit us at: www

Calc2e_00_i-x_FM

11/18/11

12:32 AM

ABOUT THE AUTHOR

McKibben is a professor of mathematics and computer science at Goucher College in Baltimore,

Maryland

During his 12 years at this institution,

he has taught more than 30 different courses spanning the mathematics curriculum,

and has published two graduate-level books with CRC Press,

more than two dozen journal articles on differential equations,

and more than 20 supplements for undergraduate texts on algebra,

Christopher Thomas is a professor of mathematics at the Massachusetts College of Liberal Arts

He has taught at Tufts University as a graduate student,

Texas A&M University as a postdoctorate professor,

and the Senior Secondary School of Mozano,

His classroom assistant is a small teddy bear named ex

Calc2e_00_i-x_FM

11/18/11

12:32 AM

Page vi

Calc2e_00_i-x_FM

11/18/11

12:32 AM

Page vii

CONTENTS

INTRODUCTION

PRETEST

LESSON 1

Functions

LESSON 2

LESSON 3

Exponents and Logarithms

LESSON 4

Trigonometry

LESSON 5

Limits and Continuity

LESSON 6

Derivatives

LESSON 7

Basic Rules of Differentiation

LESSON 8

Rates of Change

LESSON 9

The Product and Quotient Rules

LESSON 10

Chain Rule

LESSON 11

Implicit Differentiation

LESSON 12

Related Rates

LESSON 13

Limits at Infinity

LESSON 14

Using Calculus to Graph

LESSON 15

Optimization

115 vii

Calc2e_00_i-x_FM

11/18/11

12:32 AM

Page viii

LESSON 16

The Integral and Areas under Curves

LESSON 17

The Fundamental Theorem of Calculus

LESSON 18

Antidifferentiation

LESSON 19

Integration by Substitution

LESSON 20

Integration by Parts

POSTTEST

SOLUTION KEY

GLOSSARY

ADDITIONAL ONLINE PRACTICE

Calc2e_00_i-x_FM

11/18/11

12:32 AM

Page ix

INTRODUCTION

f you have never taken a calculus course,

and now find that you need to know calculus—this is the book for you

If you have already taken a calculus course,

but felt like you never understood what the teacher was trying to tell you—this book can teach you what you need to know

If it has been a while since you have taken a calculus course,

and you need to refresh your skills—this book will review the basics and reteach you the skills you may have forgotten

Whatever your reason for needing to know calculus,

Calculus Success in 20 Minutes a Day will teach you what you need to know

Overcoming Math Anxiety Do you like math or do you find math an unpleasant experience

? It is human nature for people to like what they are good at

Generally,

people who dislike math have not had much success with math

If you have struggles with math,

Was it because the class went too fast

? Did you have a chance to fully understand a concept before you went on to a new one

? One of the comments students frequently make is,

“I was just starting to understand,

and then the teacher went on to something new

” That is why Calculus Success is self-paced

You work at your own pace

You go on to a new concept only when you are ready

When you study the lessons in this book,

the only person you have to answer to is you

You don’t have to pretend you know something when you don’t truly understand

You get to take the time you need to understand everything before you go on to the next lesson

You have truly learned something only when you thoroughly understand it

Take as much time as you need to understand examples

Check your work with the answers and if you don’t feel confident that you fully understand the lesson,

You might think you don’t want to take the time to go back over something again

making sure you understand a lesson

Calc2e_00_i-x_FM

11/18/11

12:32 AM

completely may save you time in the future lessons

Rework problems you missed to make sure you don’t make the same mistakes again

How to Use This Book Calculus Success teaches basic calculus concepts in 20 self-paced lessons

The book includes a pretest,

Before you begin Lesson 1,

The pretest will assess your current calculus abilities

You’ll find the answer key at the end of the pretest

Each answer includes the lesson number that the problem is testing

This will be helpful in determining your strengths and weaknesses

After taking the pretest,

Functions

Each lesson offers detailed explanations of a new concept

There are numerous examples with step-by-step solutions

As you proceed through a lesson,

you will find tips and shortcuts that will help

Each new concept is followed by a practice set of problems

The answers to the practice problems are in an answer key located at the end of the book

When you have completed all 20 lessons,

The posttest has the same format as the pretest,

but the questions are different

Compare the results of the posttest with the results of the pretest you took before you began Lesson 1

What are your strengths

? Do you still have weak areas

? Do you need to spend more time on some concepts,

or are you ready to go to the next level

Make a Commitment Success does not come without effort

If you truly want to be successful,

make a commitment to spend the time you need to improve your calculus skills

So sharpen your pencil and get ready to begin the pretest

Calc2e_00_1-14_Pre

11/18/11

12:34 AM

PRETEST

you may want to get an idea of what you know and what you need to learn

The pretest will answer some of these questions for you

The pretest consists of 50 multiple-choice questions covering the topics in this book

While 50 questions can’t cover every concept or skill taught in this book,

your performance on the pretest will give you a good indication of your strengths and weaknesses

If you score high on the pretest,

you have a good foundation and should be able to work through the book quickly

If you score low on the pretest,

This book will explain the key calculus concepts,

If you get a low score,

you may need to take more than 20 minutes a day to work through a lesson

However,

so you can spend as much time on a lesson as you need

You decide when you fully comprehend the lesson and are ready to go on to the next one

Take as much time as you need to complete the pretest

When you are finished,

check your answers with the answer key at the end of the pretest

Along with each answer is a number that tells you which lesson of this book teaches you about the calculus skills needed to answer that question

You will find the level of difficulty increases as you work your way through the pretest

Calc2e_00_1-14_Pre

11/18/11

12:34 AM

Calc2e_00_1-14_Pre

11/18/11

12:34 AM

– LEARNINGEXPRESS ANSWER SHEET –

Calc2e_00_1-14_Pre

11/18/11

12:34 AM

Calc2e_00_1-14_Pre

11/18/11

12:34 AM

What is the value of f(4) when f(x) = 3x2 – a

Use the following figure for questions 5 and 6

Simplify g(x + 3) when g(x) = x2 – 2x + 1

What is (f ° g)(x) when f(x) = x – x + 3

2 3 x 2 b

? x2  1 all real numbers except x = 1 all real numbers except x = 0 all real numbers except x = –1 and x = 1 all real numbers except x = –1,

What is the domain of h1x2  a

On what interval(s) is f(x) increasing

Which of the following is a point of inflection for f(x)

What is the equation of the straight line passing through (2,5) and (1,1)

Calc2e_00_1-14_Pre

11/18/11

12:34 AM

Simplify 642

4,096 9

Simplify 23

Solve for x when 3x = 15

Evaluate lim a

Evaluate sin  

Evaluate tan 

Evaluate lim

Evaluate lim xS2

Calc2e_00_1-14_Pre

11/18/11

12:34 AM

What is the slope of f 1x2  3x  2 at x  5

What is the slope of g1x2  x2  2x  1 at x  3

What is the derivative of y  x2  3cos1x2

 2x  3sin1x2  2x  3sin1x2  2x  3cos112  2x  3tan1x2

Differentiate f 1x2  ln1x2  ex  2

Differentiate h1x2  4x3  5x  1

The height of a certain plant is H(t) = 41  t inches after t  1 week

How fast is it growing after two weeks

Differentiate g1x2  x2sin1x2

g ′(x) = g ′(x) = g ′(x) = g ′(x) =

Calc2e_00_1-14_Pre

11/18/11

12:34 AM

Differentiate j(x) = a

Differentiate m(x) = 1x2  12 5

Compute

Differentiate y  tan1x2

Compute 25

Differentiate f 1x2  e4x 7

 8x  cos1y2  8xcos1y2  cos1y2  8x  8xsec1y2

Calc2e_00_1-14_Pre

11/18/11

12:34 AM

What is the slope of x  y  1 at  ,

3 3 3 3

If the radius of a circle is increasing at 4 feet per second,

how fast is the area increasing when the radius is 10 feet

The height of a triangle increases by 3 inches every minute while its base decreases by 1 inch every minute

How fast is the area changing when the triangle has a height of 10 inches and a base of 100 inches

It is increasing at 145 square inches per minute

It is increasing at 500 square inches per minute

It is decreasing at 1,500 square inches per minute

It is decreasing at 3,000 square inches per minute

Evaluate lim a

Evaluate lim a

Evaluate lim 1 3 b

Calc2e_00_1-14_Pre

11/18/11

12:34 AM

Page 10

Which of the following is the graph of 1

Calc2e_00_1-14_Pre

11/18/11

12:34 AM

Page 11

On what interval is g1x2  x4  6x2  5 concave down

What is

The surface area of a cube is increasing at a rate of 3 square inches per minute

How fast is an edge increasing at the instant when each side is 20 inches

A box with a square bottom and no top must contain 108 cubic inches

What dimensions will minimize the surface area of the box

× 2 in

× 27 in

× 8 in

× 3 in

× 6 in

× 3 in

× 4 in

20 1 3 9

2 3 10 12

If g1x2 is the area under the curve y  t3  4t between t  0 and t  x,

Evaluate a

Calc2e_00_1-14_Pre

11/18/11

12:34 AM

Page 12

Evaluate

Evaluate

81 2 44

Evaluate

Evaluate

Evaluate

Evaluate x(x 2 + 2) 5 dx

Calc2e_00_1-14_Pre

11/18/11

12:34 AM

Page 13

Evaluate

Evaluate

xcos1x2  sin1x2  c'1 2 x cos1x2  c'2 1 c

Calc2e_00_1-14_Pre

11/18/11

12:34 AM

Page 14

Answers 1

Lesson 1 Lesson 1 Lesson 1 Lesson 1 Lesson 2 Lesson 2 Lesson 2 Lesson 3 Lesson 3 Lesson 3 Lesson 4 Lesson 4 Lesson 5 Lesson 5 Lesson 5 Lessons 6,

Lesson 10 Lesson 11 Lessons 4,

L E S S O N

Calc2e_01_15-22

11/18/11

12:35 AM

Page 15

FUNCTIONS

alculus is the study of change

It is often important to know when a quantity is increasing,

and when it hits a high or low point

Much of the business of finance depends on predicting the high and low points for prices

In science and engineering,

it is often essential to know precisely how fast quantities such as temperature,

Calculus is the primary tool for calculating such changes

Numbers,

which are the focus of arithmetic,

The number 5 will always be 5

It never goes up or down

we need to introduce a new sort of mathematical object,

These objects,

Functions A function is a way of matching up one set of numbers with another

The first set of numbers is called the domain

For each of the numbers in the domain,

the function assigns exactly one number from the other set,

Calc2e_01_15-22

11/18/11

12:35 AM

Page 16

PARENTHESES HINT It is true that in algebra,

everyone is taught “parentheses mean multiplication

” This means that 5(2 + 7) = 5(9) = 45

If x is a variable,

However,

if f is the name of a function,

then f (2 + 7) = f (9) = the number to which f takes 9

The expression f (x) is pronounced “f of x” and not “f times x

” This can certainly be confusing

Mathematicians use parentheses to mean several different things and expect everyone to know the difference

For example,

the domain of the function could be the set of numbers {1,

Suppose the function takes 1 to 1,

4 to 2,

9 to 3,

25 to 5,

This could be illustrated by the following: 1S 1 4S 2 9S 3 25 S 5 100 S 10 Because we sometimes use several functions in the same discussion,

it makes sense to give them names

Let us call the function we just mentioned by the name Eugene

what does Eugene do with the number 4

?” The answer is “Eugene takes 4 to the number 2

” Mathematicians like to write as little as possible

instead of writing “Eugene takes 4 to the number 2,” we often write “Eugene(4)  2” to mean the same thing

Similarly,

we like to use names that are as short as possible,

g (for function when f is already being used),

The trigonometric functions in Lesson 4 all have threeletter names like sin and cos,

but even these are abbreviations

So let us save space and use f instead of Eugene

Because the domain is small,

it is easy to write out everything:

f 112 f 142 f 192 f 1252 f 11002

However,

It is much easier to find a pattern and use that pattern to describe the function

Our function f just happens to take each number of its domain to the square root of that number

Therefore,

we can describe f by saying: f(a number) = the square root of that number Of course,

anyone with experience in algebra knows that writing “a number” over and over is a waste of time

Why not just pick a variable to represent the number

? Just as f is a typical name for a function,

little x is often used for a variable name

Using both,

here is a nice way to represent our function f: f(x) =

This tells us that putting a number into the function f is the same as putting it into

Example Find the value of g(3) if g1x2  x2  2

Calc2e_01_15-22

11/18/11

12:35 AM

Page 17

Solution Replace each occurrence of x with 3

Example Find the value of h(4) if h(t) = t3  2t2 + 5

Solution Replace each occurrence of t with –4

h(–4) = (–4)3 – 2(–4)2 + 5 Simplify

h(–4) = –64 – 2(16) + 5 = –64 – 32 + 5 = –91

Suppose that after t seconds,

a rock thrown off a bridge has height s1t2  16t2  20t  100 feet off the ground

What is the height above the ground after 3 seconds

Suppose that the profit on making and selling x x x2 cookies is P (x) = − − 10 dollars

Plugging Variables into Functions Variables can be plugged into functions just as easily as numbers can

the result can’t be simplified as much

Example When multiplying,

an even number of negatives results in a positive number,

whereas an odd number of negatives results in a negative number

Simplify f(w) if f(x) =

Solution Replace each occurrence of x with w

Practice

That is all we can say without knowing more about w

Find the value of f 152 when f 1x2  2x  1

Example

Find the value of g132 when g1x2  x3  x2  x  1

Solution

Find the value of h a b when h1t2  t2 

Find the value of f 172 when f 1x2  2

Simplify g1a  52 if g1t2  t2  3t  1

Replace each occurrence of t with (a  5)

g1a  52  1a  52 2  31a  52  1

Multiply out 1a  52 2 and 31a  52

Find the value of m  −  when m(t) = –5t 3

Find the value of h1642 when 3 h1x2  2x  2 x

Remember to FOIL (first,

last) to get (a + b)2 = a2 + 2ab + b2

Simplify

Calc2e_01_15-22

11/18/11

12:35 AM

Page 18

Example

f 1x  a2  f 1x2 Simplify if f 1x2  x2

g (2 x) − g (x) when g (t ) = 14

f ( x + a) − f ( x ) when f (x) = −x 2 + 5 a

g (x + 2) − g (x) when g (x) = x 3 2

Solution Start with what needs to be simplified

f 1x  a2  f 1x2 a Use f 1x2  x2 to evaluate f 1x  a2 and f 1x2

8 − 6t t

Composition of Functions

Multiply out 1x  a2 2

Now that we can plug anything into functions,

we can plug one function in as the input of another function

This is called composition

The composition of function f with function g is written f  g

This means to plug g into f like this:

Cancel the x2 and the x2

2xa  a2 a

( f o g )(x ) = f (g (x)) It may seem that f comes first in ( f o g )(x) ,

This means that the function g acts on the x first

Factor out an a

Example Cancel an a from the top and bottom

Practice

If f(x) = x + 2x and g1x2  4x  7,

then what is the composition ( f o g )(x)

Solution Start with the definition of composition

Simplify the following

f 1y2 when f 1x2  x2  3x  1 10

f 1x  a2 when f 1x2  x  3x  1

Use g1x2  4x  7

f (x + h) − f (x) when f (x) = 1 h 2x 12

Replace each occurrence of x in f with 4x  7

( f o g )(x) = 4 x + 7 + 2(4 x + 7) Simplify

( f o g )(x) = 4 x + 7 + 8 x + 14

Calc2e_01_15-22

11/18/11

12:35 AM

Page 19

Practice

Conversely,

we compute: (g o f )(x) = g ( f (x)) Use f(x) =

Using f 1x2 

(g o f )(x) = g ( x + 2 x) Replace each occurrence of x in g with

(g o f )(x) = 4( x + 2 x) + 7 Simplify

simplify the following compositions

This shows that the order in which you compute a composition matters

We can form the composition of more than two functions

Just apply the functions,

working your way from the one closest to x outward

Domains

Example

If f (x) = x + 1 ,

then 2x − 3 what is (f ° g ° h)(x)

Solution Start with the definition of composition

(f ° g ° h)(x) = f (g(h(x))) Use h(x) = 4x

(f ° g ° h)(x) = f (g(4x)) Compute g(4x) by replacing each occurrence of x in g with 4x

substitute this into the composition

(f ° g ° h)(x) = f (g(4x)) = f (2 – 4x)

Replace every occurrence of x in f with 2 – 4x

(f ° g ° h)(x) = f (2 – 4x) = (2 − 4 x) + 1 2(2 − 4 x) − 3 Simplify

When an expression is used to describe a function f (x),

it is convenient to think of the domain as the set of all numbers that can be substituted into the expression and get a meaningful output

This set is called the domain

The range of the function is the set of all possible numbers produced by evaluating f at the numbers in its domain

In the beginning of the lesson,

we considered the function: f(x) =

However,

we left out a crucial piece of information: the domain

The domain of this function consisted of only the numbers 1,

Usually,

the domain of a function is not given explicitly like this

In such situations,

it is assumed that the domain is as large as it possibly can be,

Calc2e_01_15-22

11/18/11

12:35 AM

Page 20

it contains all real numbers that,

when plugged into the function,

Specifically,

including a number in the domain cannot violate one of the following two fundamental prohibitions: ■ Never divide by zero

■ Never take an even root of a negative number

Example

What is the domain of f 1x2 

Solution We must never let the denominator x  2 be zero,

Therefore,

the domain of this function consists of all real numbers except 2

The prohibition against even roots (like square roots) of negative numbers is less severe

An even root of a negative number is an imaginary number

Useful mathematics can be done with imaginary numbers

However,

we will avoid them in this book

Example What is the domain of g(x) = 3x + 2

Solution The numbers in the square root must not be negative,

The domain consists 3 2 of all numbers greater than or equal to 

It is only when numbers are less than zero that even roots become imaginary

Example

Solution To avoid dividing by zero,

To avoid an even root of a negative number,

4  x  0,

A nice way of representing certain collections of real numbers is interval notation,

as follows: COLLECTION OF REAL NUMBERS

INTERVAL NOTATION