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

s Success in 20 Minutes a Day2ndEdition[1]

Description

STUDY GUIDES/Mathematics

- ! 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

! 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,

- optimization,
- antidifferentiation,
- techniques of integration,
- and much more

! Additional resources for preparing for important standardized tests

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

and boost your overall score to guarantee success

❏ 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

L EARNINGE XPRESS

Calc2e_00_i-x_FM

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CALCULUS SUCCESS in 20 Minutes a Day

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Calc2e_00_i-x_FM

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CALCULUS SUCCESS in 20 Minutes a Day Second Edition Mark A

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Copyright © 2012 LearningExpress,

All rights reserved under International and Pan-American Copyright Conventions

New York

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

- —2nd ed

: Calculus success in 20 minutes a day / Thomas,

© 2006

Calculus—Problems,

- exercises,

Thomas,

- 1973– Calculus success in 20 minutes a day

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,

- learnatest

Calc2e_00_i-x_FM

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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,

- trigonometry,
- statistics,
- and calculus

Texas A&M University as a postdoctorate professor,

and the Senior Secondary School of Mozano,

- as a Peace Corps volunteer

His classroom assistant is a small teddy bear named ex

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CONTENTS

PRETEST

LESSON 4

Limits and Continuity

LESSON 6

Derivatives

LESSON 8

LESSON 9

The Product and Quotient Rules

Chain Rule

LESSON 11

Related Rates

Limits at Infinity

Using Calculus to Graph

115 vii

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Page viii

- – CONTENTS –

LESSON 18

Antidifferentiation

LESSON 19

Integration by Parts

POSTTEST

ADDITIONAL ONLINE PRACTICE

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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

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

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

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

people who dislike math have not had much success with math

- ask yourself why

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

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

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

- do it again

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

- however,

making sure you understand a lesson

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- – INTRODUCTION –

completely may save you time in the future lessons

The book includes a pretest,

- a posttest,
- 20 lessons,
- each covering a new topic,
- and a glossary

- take the pretest

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

This will be helpful in determining your strengths and weaknesses

- move on to Lesson 1,

Functions

you will find tips and shortcuts that will help

- you learn a concept

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

- take the posttest

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

? 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 to spend the time you need to improve your calculus skills

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PRETEST

- efore you begin Lesson 1,

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

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

- don’t despair

- step by step

If you get a low score,

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

- this is a self-paced program,

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

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

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– LEARNINGEXPRESS ANSWER SHEET –

Calc2e_00_1-14_Pre

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- – PRETEST –

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

Use the following figure for questions 5 and 6

- y 6 5 y = f(x)

- x2 + 2x + 4 b
- x2 – 2x + 4 c
- x2 + 4x + 16 d
- x2 + 4x + 13 3

- 2 and g(x) = x

2 3 x 2 b

- 2x 3 x a
- x2 2 3x d
- x 3

? 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,

- and x=1

- 3 2 1 –3 –2 –1 –1

- (∞,1) and (5,∞) b
- (1,5) c
- (1,6) d
- (5,∞) 6

- 5) (1,6) (3,3) (5,1)

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

- y 2x 5 b
- y 2x 1 c
- y 2x 9 d
- y 2x 3

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- – PRETEST –

Simplify 642

4,096 9

Simplify 23

- ln152 c
- xS4 x2 1

- ln1152 ln132
- π 11

- 3 1 a
- 2 1 b
- 3π 12

- 4
- xS1 x2 1

- undefined

- ∞ 1 c
- 4 −3 d

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- – PRETEST –

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

- ? dy dx dy b
- dx dy c
- dx dy d

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

- f ¿1x2 ln1x2 ex

- 1 ex x 1 d
- f ¿1x2 ex x
- h ′(x) = 12x2
- f ¿1x2
- h ′(x) = 12x2 5 c
- h ′(x) = 12x2 5x d
- h ′(x) = 12x2 5x
- f ¿1x2 ln1x2 ex

How fast is it growing after two weeks

- 5 inches per week b
- 10 inches per week c
- 21 inches per week d
- 31 inches per week

Differentiate g1x2 x2sin1x2

g ′(x) = g ′(x) = g ′(x) = g ′(x) =

- 2xcos1x2 2x cos1x2 2xsin1x2 x2cos1x2 2xsin1x2cos1x2

Calc2e_00_1-14_Pre

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- – PRETEST –

Differentiate j(x) = a

- j ′(x) = 0

- m ′(x) = 10x b
- m ′(x) = 12x2 5
- j ′(x) = x 1 ln1x2 c
- j ′(x) = x2 ln1x2 1 d
- j ′(x) = x2
- m ′(x) = 51x2 12 4 d
- m ′(x) = 10x1x2 12 4 27

- dy = sec2 1x2 a
- dx dy = –cot(x) b
- 2 2 dy cos 1x2 sin 1x2 = cos2 1x2 dx
- dy = sin1x2cos1x2 dx
- dy x2 dx
- 3x2 y dy dx 2y x
- dy 3x2 dx 1 2y
- dy 3x 2 − x = dx 2y

Differentiate f 1x2 e4x 7

- f ′(x) = e8x 2
- f ′(x) = e4x 7 2
- f ′(x) = 8xe
- f ′(x) = 14x2 72e4x 8 2
- dy if y2 xy x3 5
- dy dx dy b
- dx dy c
- dx dy d
- dy if sin1y2 4x2

8x cos1y2 8xcos1y2 cos1y2 8x 8xsec1y2

Calc2e_00_1-14_Pre

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- – PRETEST –
- 1 3 29

- ? 2 2 2

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

- 20p square feet per second b
- 80p square feet per second c
- 100p square feet per second d
- 400p square feet per second 31

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

- 4x 2 − 5x + 2
- x →∞ 1 − x2

Evaluate lim a

- 4 4 2 undefined 4x 5 + 6x + 4
- x →−∞ x 3 + 10 x − 1

Evaluate lim a

- ∞ ∞ 4 4 ln(x)
- x →−∞ 3x + 2

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- – PRETEST –

- ? y x2 a
- –2 –3
- –1 –1
- –3 y 3 2
- –1 –1
- 2 –2 1 –3 –2
- –1 –1 –2 –3

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- – PRETEST –

- (1,12) b
- (6,5) c
- (− 3 ,
- f 1x2 dx

- y y = f(x) 4 (4,3) 3

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

- inch per minute 80 3 b
- inch per minute 20 c
- 80 inches per minute d
- 24,000 inches per minute 38

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

× 27 in

× 8 in

× 3 in

× 4 in

- g1x2 dx 4 ,
- g1x2 dx 5 and
- g1x2 dx
- what is

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,

- what is g¿1x2
- x3 4x b
- 3x2 4 1 c
- x4 2x 4 d

Evaluate a

- 8x 52 dx
- 6x 8 6x 8 c'x3 4x2 5x x3 4x2 5x c

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Page 12

- – PRETEST –
- 2x dx

Evaluate

81 2 44

- ln x − 1 + c
- 1 ln(x 2 − 1) + c'2 1 ln x 2 − 1 + c'2
- 4 sin1x3 2 c'3 4 c
- x3sin1x3 2 c'3 4 d
- x2sin1x3 2 c'3 b
- 4x cos1x 2 dx
- 4sin1x3 2 c
- sin1x2 c
- 1 5x e c 5
- sin1x2 dx
- sin1x2 c
- e5 c'1 d
- e5 c'5
- cos1x2 c
- 1 2 2x 1 3 3x
- e5x c
- cos1x2 c

- 1 2 (x + 2)6 + c'6
- 5(x 2 + 2) 4 + c'6
- x2 1 3 c
- 2 3 x + 2 x + c'1 2 6 d
- 12 (x + 5) + c

Calc2e_00_1-14_Pre

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- – PRETEST –

- xln1x2 dx
- 1 2 x ln1x2 c'2 b
- xln1x2 ln1x2 c'a
- x2ln1x2 x2 c'4 1 1 d
- x2ln1x2 x2 c'2 4

Evaluate

- xsin1x2 dx

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

- x2cos1x2 c'2 d
- xcos(x) – cos(x) + c'b

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Page 14

- – PRETEST –

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,

- 7 Lessons 6,
- 7 Lesson 7 Lesson 8 Lesson 8 Lesson 8 Lesson 9 Lessons 8,
- 9 Lesson 9 Lesson 10

- 11 Lesson 11 Lesson 12 Lesson 12 Lesson 13 Lesson 13 Lesson 13 Lesson 14 Lesson 14 Lesson 12 Lesson 16 Lesson 16 Lesson 16 Lesson 17 Lesson 18 Lesson 18 Lesson 18 Lesson 19 Lesson 19 Lesson 19 Lesson 19 Lesson 20 Lesson 20

L E S S O N

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alculus is the study of change

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

- when it is decreasing,

and when it hits a high or low point

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

- and speed are changing

which are the focus of arithmetic,

- do not change

The number 5 will always be 5

It never goes up or down

we need to introduce a new sort of mathematical object,

- something that can change

These objects,

- the centerpiece of calculus,
- are functions

For each of the numbers in the domain,

the function assigns exactly one number from the other set,

- the range

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Page 16

everyone is taught “parentheses mean multiplication

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

If x is a variable,

- then x(2 + 7) = x(9) = 9x

However,

if f is the name of a function,

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

” This can certainly be confusing

- as you gain experience,
- it will become second nature

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

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

- and the range could be {1,

Suppose the function takes 1 to 1,

4 to 2,

9 to 3,

25 to 5,

- and 100 to 10

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

- we can ask,
- “Hey,

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,

- such as f (for function),

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

- and so on

but even these are abbreviations

Because the domain is small,

it is easy to write out everything:

f 112 f 142 f 192 f 1252 f 11002

- 1 2 3 5 10

- if the domain were large,
- this would get very tedious

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

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

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

- f(25) = 25 = 5 and f(f) =

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

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- – FUNCTIONS –

- g(3) 32 2 Simplify
- g(3) 9 2 11

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

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

- 000 How much profit is made on selling 100 cookies

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

- though,

the result can’t be simplified as much

an even number of negatives results in a positive number,

whereas an odd number of negatives results in a negative number

- x + 2x 2 + 2

Solution Replace each occurrence of x with w

- f(w) = w + 2w 2 + 2

Practice

That is all we can say without knowing more about w

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

Solution

Find the value of h a b when h1t2 t2

Find the value of f 172 when f 1x2 2

g1a 52 1a 52 2 31a 52 1

Multiply out 1a 52 2 and 31a 52

- g1a 52 a2 10a 25 3a 15 1
- 1 5

- 5 6

- (a + b)2 ≠ a2 + b2

- outside,
- inside,

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

- g1a 52 a2 7a 11 17

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- – FUNCTIONS –

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

- h1x a2 h1x2 when h(x) = –2x + 1 a

g (x + 2) − g (x) when g (x) = x 3 2

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

- 1x a2 2 x2 a

8 − 6t t

Composition of Functions

Multiply out 1x a2 2

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

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

This means to plug g into f like this:

- x2 2xa a2 x2 a

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) ,

- reading from left to right,
- but actually,
- the g is closer to the x

Factor out an a

- 12x a2a a

Practice

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

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

- (f ° g)(x) = f(g(x))

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

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

- ( f o g )(x) = f (4 x + 7)

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

- 8 x ) when g1t2 6t t

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

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

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12:35 AM

- – FUNCTIONS –

Practice

Conversely,

- to evaluate (g ° f )(x),

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

(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

- (g o f )(x) = 4 x + 8 x + 7
- g1x2 x3 2x2 1 ,
- and h(x) = x x

simplify the following compositions

- (f ° g)(x) 18
- (g ° f )(x) 19
- (f ° h)(t) 20
- (f ° f )(z) 21
- (h ° h)(w)

- ! In general,
- ( f ° g)(x) ≠ (g ° f )(x)

Just apply the functions,

- one at a time,

working your way from the one closest to x outward

- (g ° h)(16) 23
- (h ° f ° g)(x) 24
- (f ° h ° f )(2x)

If f (x) = x + 1 ,

- g(x) = 2 – x,
- and h(x) = 4x,

then 2x − 3 what is (f ° g ° h)(x)

(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

- g(4x) = 2 – 4x Next,

substitute this into the composition

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

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

- 3 − 4x (f ° g ° h)(x) = 1 − 8x

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

we considered the function: f(x) =

we left out a crucial piece of information: the domain

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

- and 100
- we should have written f(x) =
- x if x 1,

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,

- meaning that 19

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12:35 AM

- – FUNCTIONS –

it contains all real numbers that,

when plugged into the function,

- produce another real number

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

- so x cannot be 2

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

An even root of a negative number is an imaginary number

- for the sake of simplicity,

we will avoid them in this book

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

- 2 so 3x 2 0 ,
- thus x

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

- 3 Do note that it is perfectly okay to take the square root of zero,
- since 0 = 0

- 4 − x Find the domain of k(x) = 2
- x + 5x + 6

Solution To avoid dividing by zero,

- we need x2 5x 6 0,
- so 1x 321x 22 0,
- thus x 3 and x 2

4 x 0,

- so x 4
- the domain of k is x 4 ,
- x 3 ,

as follows: COLLECTION OF REAL NUMBERS

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