Calculus for Dummies

Want to read this on your NOOK? Request as NOOK Book from the publisher

Thank you for requesting this book as a NOOK book from the publisher.

Customers Who Bought This Also Bought

More About This Book

Overview
Product Details
Related Subjects
Meet the Author
Read an Excerpt
Table of Contents

Overview

The mere thought of having to take a required calculus course is enough to make legions of students break out in a cold sweat. Others who have no intention of ever studying the subject have this notion that calculus is impossibly difficult unless you happen to be a direct descendant of Einstein.

Well, the good news is that you can master calculus. It's not nearly as tough as its mystique would lead you to think. Much of calculus is really just very advanced algebra, geometry, and trig. It builds upon and is a logical extension of those subjects. If you can do algebra, geometry, and trig, you can do calculus.

Calculus For Dummies is intended for three groups of readers:

Students taking their first calculus course – If you're enrolled in a calculus course and you find your textbook less than crystal clear, this is the book for you. It covers the most important topics in the first year of calculus: differentiation, integration, and infinite series.
Students who need to brush up on their calculus to prepare for other studies – If you've had elementary calculus, but it's been a couple of years and you want to review the concepts to prepare for, say, some graduate program, Calculus For Dummies will give you a thorough, no-nonsense refresher course.
Adults of all ages who'd like a good introduction to the subject – Non-student readers will find the book's exposition clear and accessible. Calculus For Dummies takes calculus out of the ivory tower and brings it down to earth.

This is a user-friendly math book. Whenever possible, the author explains the calculusconcepts by showing you connections between the calculus ideas and easier ideas from algebra and geometry. Then, you'll see how the calculus concepts work in concrete examples. All explanations are in plain English, not math-speak. Calculus For Dummies covers the following topics and more:

Real-world examples of calculus
The two big ideas of calculus: differentiation and integration
Why calculus works
Pre-algebra and algebra review
Common functions and their graphs
Limits and continuity
Integration and approximating area
Sequences and series

Don't buy the misconception. Sure calculus is difficult – but it's manageable, doable. You made it through algebra, geometry, and trigonometry. Well, calculus just picks up where they leave off – it's simply the next step in a logical progression.

Read More Show Less

Product Details

ISBN-13: 9780764524981
Publisher: Wiley, John & Sons, Incorporated
Publication date: 8/29/2003
Series: For Dummies Series
Edition number: 1
Pages: 384
Sales rank: 67,083
Product dimensions: 7.36 (w) x 9.14 (h) x 0.74 (d)

Meet the Author

Mark Ryan has taught pre-math through calculus for more than a decade. He is a member of the National Council of Teachers of Mathematics.

Read More Show Less

Read an Excerpt

Calculus For Dummies

By Mark Ryan

John Wiley & Sons

Copyright © 2003

Mark Ryan
All right reserved.

ISBN: 0-7645-2498-4

Chapter One

What Is Calculus?

* * *

In This Chapter

* You're only on page 1 and you've got a calc test already

* Calculus - it's just souped-up regular math

* Zooming in is the key

* The world before and after calculus

* * *

"My best day in Calc 101 at Southern Cal was the day I had to cut class to get
a root canal."

- Mary Johnson

"I keep having this recurring dream where my calculus professor is coming
after me with an axe."

- Tom Franklin, Colorado College sophomore

"Calculus is fun, and it's so easy. I don't get what all the fuss is about."

- Sam Einstein, Albert's great grandson

In this chapter, I answer the question "What is calculus?" in plain English,
and I give you real-world examples of how calculus is used. After reading
this and the following two short chapters, you will understand what calculus
is all about. But, here's a twist, why don't you start out on the wrong foot by
briefly checking out what calculus is not.

What Calculus Is Not

No sense delaying theinevitable. Ready for your first calculus test? Answer
True or False.

T F Unless you actually enjoy wearing a pocket protector, you've got no
business taking calculus.

T F Studying calculus is hazardous to your health.

T F Calculus is totally irrelevant.

False, false, false! There's this mystique about calculus that it's this ridiculously
difficult, incredibly arcane subject that no one in their right mind would sign up
for unless it was a required course.

Don't buy into this misconception. Sure calculus is difficult - I'm not going to
lie to you - but it's manageable, doable. You made it through algebra, geometry,
and trigonometry. Well, calculus just picks up where they leave off - it's
simply the next step in a logical progression.

And calculus is not a dead language like Latin, spoken only by academics. It is
the language of engineers, scientists, and economists - okay, so it's a couple
steps removed from your everyday life and unlikely to come up at a cocktail
party. But the work of those engineers, scientists, and economists has a huge
impact on your day-to-day life - from your microwave oven, cell phone, TV,
and car to the medicines you take, the workings of the economy, and our
national defense. At this very moment, something within your reach or within
your view has been impacted by calculus.

So What Is Calculus Already?

Calculus is basically just very advanced algebra and geometry. In one sense,
it's not even a new subject - it takes the ordinary rules of algebra and geometry
and tweaks them so that they can be used on more complicated problems.
(The rub, of course, is that darn other sense in which it is a new and more difficult
subject.)

Look at Figure 1-1. On the left is a man pushing a crate up a straight incline.
On the right, the man is pushing the same crate up a curving incline. The
problem, in both cases, is to determine the amount of energy required to
push the crate to the top. You can do the problem on the left with regular
math. For the one on the right, you need calculus (assuming you don't know
the physics shortcuts).

For the straight incline, the man pushes with an unchanging force, and the
crate goes up the incline at an unchanging speed. With some simple physics
formulas and regular math (including algebra and trig), you can compute
how many calories of energy are required to push the crate up the incline.
Note that the amount of energy expended each second remains the same.

For the curving incline, on the other hand, things are constantly changing. The
steepness of the incline is changing - and not just in increments like it's one
steepness for the first 10 feet then a different steepness for the next 10 feet - it's
constantly changing. And the man pushes with a constantly changing force - the
steeper the incline, the harder the push. As a result, the amount of energy
expended is also changing, not every second or every thousandth of a second,
but constantly changing from one moment to the next. That's what makes it a
calculus problem. By this time, it should come as no surprise to you that calculus
is described as "the mathematics of change." Calculus takes the regular
rules of math and applies them to fluid, evolving problems.

For the curving incline problem, the physics formulas remain the same, and
the algebra and trig you use stay the same. The difference is that - in contrast
to the straight incline problem, which you can sort of do in a single shot - you've
got to break up the curving incline problem into small chunks and do
each chunk separately. Figure 1-2 shows a small portion of the curving incline
blown up to several times its size.

When you zoom in far enough, the small length of the curving incline becomes
practically straight. Then, because it's straight, you can solve that small chunk
just like the straight incline problem. Each small chunk can be solved the same
way, and then you just add up all the chunks.

That's calculus in a nutshell. It takes a problem that can't be done with regular
math because things are constantly changing - the changing quantities
show up on a graph as curves - it zooms in on the curve till it becomes
straight, and then lets regular math finish off the problem.

What makes calculus such a fantastic achievement is that it actually zooms in
infinitely. In fact, everything you do in calculus involves infinity in one way or
another, because if something is constantly changing, it's changing infinitely
often from each infinitesimal moment to the next.

Real-World Examples of Calculus

So, with regular math you can do the straight incline problem; with calculus
you can do the curving incline problem. Here are some more examples.

With regular math you can determine the length of a buried cable that runs
diagonally from one corner of a park to the other. With calculus you can
determine the length of a cable hung between two towers that has the shape
of a catenary (which is different, by the way, from a simple circular arc or a
parabola). Knowing the exact length is of obvious importance to a power
company planning hundreds of miles of new electric cable. See Figure 1-3.

You can calculate the area of the flat roof of a home with regular math. With
calculus you can compute the area of a complicated, nonspherical shape like
the dome of the Houston Astrodome. Architects designing such a building
need to know the dome's area to determine the cost of materials and to figure
the weight of the dome (with and without snow on it). The weight, of course,
is needed for planning the strength of the supporting structure. Check out
Figure 1-4.

With regular math and some simple physics, you can calculate by how
much a quarterback must lead his receiver to complete a pass. Note that
the receiver runs in a straight line and at a constant speed. But when NASA,
in 1975, calculated the necessary "lead" for aiming the Viking I at Mars, it
needed calculus because both the Earth and Mars travel on elliptical orbits
(of different shapes) and the speeds of both are constantly changing - not to
mention the fact that on its way to Mars, the spacecraft is affected by the
different and constantly changing gravitational pulls of the Earth, the moon,
Mars, and the sun. See Figure 1-5.

You see many real-world applications of calculus throughout this book. The
differentiation problems in Part IV all involve the steepness of a curve - like
the steepness of the curving incline in Figure 1-1. In Part V, you do integration
problems like the cable-length problem shown back in Figure 1-3. These
problems involve breaking up something into little sections, calculating each
section, and then adding up the sections to get the total. More about this in
Chapter 2.

(Continues...)

Excerpted from Calculus For Dummies
by Mark Ryan
Copyright © 2003 by Mark Ryan.
Excerpted by permission.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

Read More Show Less

	Introduction	1
	About This Book	1
	Conventions Used in This Book	2
	How to Use This Book	2
	Foolish Assumptions	3
	How This Book Is Organized	3
	Icons Used in This Book	5
	Where to Go from Here	6
Part I	An Overview of Calculus	7
Chapter 1	What Is Calculus?	9
	What Calculus Is Not	9
	So What Is Calculus Already?	10
	Real-World Examples of Calculus	12
Chapter 2	The Two Big Ideas of Calculus: Differentiation and Integration	15
	Defining Differentiation	15
	Investigating Integration	18
	Sorting Out Infinite Series	19
Chapter 3	Why Calculus Works	23
	The Limit Concept: A Mathematical Microscope	23
	What Happens When You Zoom In	24
	Two Caveats--or Precision, Preschmidgen	26
Part II	Warming Up with Calculus Prerequisites	29
Chapter 4	Pre-Algebra and Algebra Review	31
	Fine-Tuning Your Fractions	31
	Absolute Value: Absolutely Easy	36
	Empowering Your Powers	36
	Rooting for Roots	37
	Logarithms--This Is Not an Event at a Lumberjack Competition	39
	Factoring Schmactoring, When Am I Ever Going to Need It?	40
	Solving Quadratic Equations	42
Chapter 5	Funky Functions and Their Groovy Graphs	47
	What Is a Function?	47
	What Does a Function Look Like?	52
	Common Functions and Their Graphs	54
	Inverse Functions	60
	Shifts, Reflections, Stretches, and Shrinks	61
Chapter 6	The Trig Tango	65
	Studying Trig at Camp SohCahToa	65
	Two Special Right Triangles	66
	Circling the Enemy with the Unit Circle	68
	Graphing Sine, Cosine, and Tangent	74
	Inverse Trig Functions	75
	Identifying with Trig Identities	76
Part III	Limits	77
Chapter 7	Limits and Continuity	79
	Take It to the Limit--Not	79
	Linking Limits and Continuity	89
	The 33333 Limit Mnemonic	92
Chapter 8	Evaluating Limits	95
	Easy Limits	95
	The "Real Deal" Limit Problems	97
	Evaluating Limits at Infinity	106
Part IV	Differentiation	111
Chapter 9	Differentiation Orientation	113
	Differentiating: It's Just Finding the Slope	114
	The Derivative: It's Just a Rate	119
	The Derivative of a Curve	122
	The Difference Quotient	124
	Average Rate and Instantaneous Rate	130
	To Be or Not to Be? Three Cases Where the Derivative Does Not Exist	131
Chapter 10	Differentiation Rules--Yeah, Man, It Rules	133
	Basic Differentiation Rules	134
	Differentiation Rules for Experts--Oh, Yeah, I'm a Calculus Wonk	139
	Differentiating Implicity	146
	Getting into the Rhythm with Logarithmic Differentiation	148
	Differentiating Inverse Functions	149
	Scaling the Heights of Higher Order Derivatives	150
Chapter 11	Differentiation and the Shape of Curves	153
	Taking a Calculus Road Trip	153
	Finding Local Extrema--My Ma, She's Like, Totally Extreme	157
	Finding Absolute Extrema on a Closed Interval	163
	Finding Absolute Extrema over a Function's Entire Domain	166
	Locating Concavity and Inflection Points	168
	Looking at Graphs of Derivatives Till They Derive You Crazy	170
	The Mean Value Theorem--GRRRRR	174
Chapter 12	Your Problems Are Solved: Differentiation to the Rescue!	177
	Getting the Most (or Least) Out of Life: Optimization Problems	177
	Yo-Yo a Go-Go: Position, Velocity, and Acceleration	181
	Related Rates--They Rate, Relatively	189
	Tangents and Normals: Joined at the Hip	196
	Straight Shooting with Linear Approximations	201
	Business and Economics Problems	204
Part V	Integration and Infinite Series	209
Chapter 13	Intro to Integration and Approximating Area	211
	Integration: Just Fancy Addition	211
	Finding the Area under a Curve	214
	Dealing with Negative Area	216
	Approximating Area	216
	Getting Fancy with Summation Notation	224
	Finding Exact Area with the Definite Integral	228
	Approximating Area with the Trapezoid Rule and Simpson's Rule	231
Chapter 14	Integration: It's Backwards Differentiation	235
	Antidifferentiation--That's Differentiation in Reverse	235
	Vocabulary, Voshmabulary: What Difference Does It Make?	237
	The Annoying Area Function	237
	The Power and the Glory of the Fundamental Theorem of Calculus	240
	The Fundamental Theorem of Calculus: Take Two	244
	Finding Antiderivatives: Three Basic Techniques	251
	Finding Area with Substitution Problems	258
Chapter 15	Integration Techniques for Experts	261
	Integration by Parts: Divide and Conquer	261
	Tricky Trig Integrals	268
	Your Worst Nightmare: Trigonometric Substitution	274
	The As, Bs, and Cxs of Partial Fractions	279
Chapter 16	Forget Dr. Phil: Use the Integral to Solve Problems	285
	The Mean Value Theorem for Integrals and Average Value	286
	The Area between Two Curves--Double the Fun	289
	Finding the Volumes of Weird Solids	292
	Analyzing Arc Length	299
	Surfaces of Revolution--Pass the Bottle 'Round	301
	L'Hopital's Rule: Calculus for the Sick	304
	Improper Integrals: Just Look at the Way That Integral Is Holding Its Fork!	307
Chapter 17	Infinite Series	315
	Sequences and Series: What They're All About	316
	Convergence or Divergence? That Is the Question	321
	Alternating Series	332
	Keeping All the Tests Straight	336
Part VI	The Part of Tens	339
Chapter 18	Ten Things to Remember	341
	Your Sunglasses	341
	a[superscript 2] - b[superscript 2] = (a - b)(a + b)	341
	0/5 = 0, But 5/0 Is Undefined	341
	Anything[superscript 0] = 1	342
	SohCahToa	342
	Trigonometric Values for 30, 45, and 60 Degree Angles	342
	sin[superscript 2 theta] + cos[superscript 2 theta] = 1	343
	The Product Rule	343
	The Quotient Rule	343
	Where You Put Your Keys	343
Chapter 19	Ten Things to Forget	345
	(a + b)[superscript 2] = a[superscript 2] + b[superscript 2]--Wrong!	345
	[radical]a[superscript 2] + b[superscript 2] = a + b--Wrong!	345
	Slope = x[subscript 2] - x[subscript 1]/y[subscript 2] - y[subscript 1]--Wrong!	345
	3a + b/3a + c = b/c--Wrong!	346
	d/dx[pi superscript 3] = 3[pi superscript 2]--Wrong!	346
	If k Is a Constant, d/dx kx = k'x + kx'--Wrong!	346
	The Quotient Rule Is d/dx (u/v) = v'u - vu'/v[superscript 2]--Wrong!	346
	[function of] x[superscript 2] dx = 1/3x[superscript 3]--Wrong!	346
	[function of] (sinx) dx = cosx + C--Wrong!	347
	Green's Theorem	347
Chapter 20	Ten Things You Can't Get Away With	349
	Give Two Answers on Exam Questions	349
	Write Illegibly on Exams	349
	Don't Show Your Work on Exams	350
	Don't Do All of the Exam Problems	350
	Blame Your Study Partner for Your Low Exam Grade	350
	Tell Your Teacher That You Need an "A" in Calculus to Impress Your Significant Other	350
	Complain That Early-Morning Exams Are Unfair Because You're Not a "Morning Person"	351
	Protest the Whole Idea of Grades	351
	Pull the Fire Alarm During an Exam	351
	Use This Book as an Excuse	351
	Index	353

Read More Show Less

Customer Reviews

Average Rating 4.5

( 24 )

Your Rating:

Poor

Below Average

Good

Very Good

Exceptional

Review Guidelines

Tell the world what you think of this product.

Add Recommendations

Your Name: Create a Pen Name or Leave Anonymously

Barnes & Noble.com Review Rules

Our reader reviews allow you to share your comments on titles you liked, or didn't, with others. By submitting an online review, you are representing to Barnes & Noble.com that all information contained in your review is original and accurate in all respects, and that the submission of such content by you and the posting of such content by Barnes & Noble.com does not and will not violate the rights of any third party. Please follow the rules below to help ensure that your review can be posted.

Reviews by Our Customers Under the Age of 13

We highly value and respect everyone's opinion concerning the titles we offer. However, we cannot allow persons under the age of 13 to have accounts at BN.com or to post customer reviews. Please see our Terms of Use for more details.

What to exclude from your review:

Please do not write about reviews, commentary, or information posted on the product page. If you see any errors in the information on the product page, please send us an email.

Reviews should not contain any of the following:

- HTML tags, profanity, obscenities, vulgarities, or comments that defame anyone
- Time-sensitive information such as tour dates, signings, lectures, etc.
- Single-word reviews. Other people will read your review to discover why you liked or didn't like the title. Be descriptive.
- Comments focusing on the author or that may ruin the ending for others
- Phone numbers, addresses, URLs
- Pricing and availability information or alternative ordering information
- Advertisements or commercial solicitation

Reminder:

- By submitting a review, you grant to Barnes & Noble.com and its sublicensees the royalty-free, perpetual, irrevocable right and license to use the review in accordance with the Barnes & Noble.com Terms of Use.
- Barnes & Noble.com reserves the right not to post any review -- particularly those that do not follow the terms and conditions of these Rules. Barnes & Noble.com also reserves the right to remove any review at any time without notice.
- See Terms of Use for other conditions and disclaimers.

If you find inappropriate content, please report it to Barnes & Noble

Overview

Pick Up In Store

Customers Who Bought This Also Bought

More About This Book

Overview

Product Details

Meet the Author

Read an Excerpt

Calculus For Dummies

John Wiley & Sons

Chapter One

Table of Contents

Customer Reviews

5 Star

4 Star

3 Star

2 Star

1 Star

Your Rating:

Your Recommendations:

Barnes & Noble.com Review Rules

Reviews by Our Customers Under the Age of 13

What to exclude from your review:

Reviews should not contain any of the following:

Reminder:

Calculus for Dummies

Overview

Pick Up In Store

Customers Who Bought This Also Bought

More About This Book

Overview

Product Details

Related Subjects

Meet the Author

Read an Excerpt

Calculus For Dummies

John Wiley & Sons

Chapter One

Table of Contents

Customer Reviews

5 Star

4 Star

3 Star

2 Star

1 Star

Your Rating:

Your Recommendations:

Barnes & Noble.com Review Rules

Reviews by Our Customers Under the Age of 13

What to exclude from your review:

Reviews should not contain any of the following:

Reminder:

Recommend other products that relate to your review. Just search for them below and share!

Great Book!

A Rosetta stone for Calculus

A True Calculus Bible

Connie B

Makes ANYONE able to understand calculus!

Helps...

In Calculus for Dummies, Mark Ryan presents the fundamental concepts of calculus in a clear and concise manner.

A good introduction:

Makes a complex topic understandable and even enjoyable!

Good book

a terrific resource

Great book!

Best Introduction to Calculus out there

Excellent Introduction to Calculus Basics