Calculus: Graphical, Numerical, Algebraic

435 Pages·2006·35.79 MB·English

by  Ross L. Finney

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Contents
CHAPTER 1 Prerequisites for Calculus 2
1.1 Lines 3
•Increments •Slope of a Line •Parallel and Perpendicular Lines •Equations of
Lines •Applications
1.2 Functions and Graphs 12
•Functions •Domains and Ranges •Viewing and Interpreting Graphs •Even
Functions and Odd Functions—Symmetry •Functions Defined in Pieces •
Absolute Value Function •Composite Functions
1.3 Exponential Functions 22
•Exponential Growth •Exponential Decay •Applications •The Number e
1.4 Parametric Equations 30
•Relations •Circles •Ellipses •Lines and Other Curves
1.5 Functions and Logarithms 37
•One-to-One Functions •Inverses •Finding Inverses •Logarithmic Functions
•Properties of Logarithms •Applications
1.6 Trigonometric Functions 46
•Radian Measure •Graphs of Trigonometric Functions •Periodicity •Even and
Odd Trigonometric Functions •Transformations of Trigonometric Graphs •
Inverse Trigonometric Functions
Key Terms 55
Review Exercises 56
CHAPTER 2 Limits and Continuity 58
2.1 Rates of Change and Limits 59
•Average and Instantaneous Speed •Definition of Limit •Properties of Limits
•One-sided and Two-sided Limits •Sandwich Theorem
2.2 Limits Involving Infinity 70
•Finite Limits as x→!"•Sandwich Theorem Revisited •Infinite Limits as
x→a•End Behavior Models •“Seeing”Limits as x→!"
2.3 Continuity 78
•Continuity at a Point •Continuous Functions •Algebraic Combinations •
Composites •Intermediate Value Theorem for Continuous Functions
Every section throughout the book 2.4 Rates of Change and Tangent Lines 87
also includes “Exploration”and
“Extending the Ideas”features which •Average Rates of Change •Tangent to a Curve •Slope of a Curve •Normal to
follow the exercises. a Curve •Speed Revisited
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Key Terms 95
Review Exercises 95
CHAPTER 3 Derivatives 98
3.1 Derivative of a Function 99
•Definition of a Derivative •Notation •Relationship Between the Graphs of
ƒand ƒ!•Graphing the Derivative from Data •One-sided Derivatives
3.2 Differentiability 109
•How ƒ!(a) Might Fail to Exist •Differentiability Implies Local Linearity •
Derivatives on a Calculator •Differentiability Implies Continuity •Intermediate
Value Theorem for Derivatives
3.3 Rules for Differentiation 116
•Positive Integer Powers,Multiples,Sums,and Differences •Products and
Quotients •Negative Integer Powers of x•Second and Higher Order
Derivatives
3.4 Velocity and Other Rates of Change 127
•Instantaneous Rates of Change •Motion along a Line •Sensitivity to Change
•Derivatives in Economics
3.5 Derivatives of Trigonometric Functions 141
•Derivative of the Sine Function •Derivative of the Cosine Function •Simple
Harmonic Motion •Jerk •Derivatives of Other Basic Trigonometric Functions
3.6 Chain Rule 148
•Derivative of a Composite Function •“Outside-Inside”Rule •Repeated Use
of the Chain Rule •Slopes of Parametrized Curves •Power Chain Rule
3.7 Implicit Differentiation 157
•Implicitly Defined Functions •Lenses,Tangents,and Normal Lines •
Derivatives of Higher Order •Rational Powers of Differentiable Functions
3.8 Derivatives of Inverse Trigonometric Functions 165
•Derivatives of Inverse Functions •Derivative of the Arcsine •Derivative of the
Arctangent •Derivative of the Arcsecant •Derivatives of the Other Three
3.9 Derivatives of Exponential and Logarithmic Functions 172
•Derivative of ex•Derivative of ax•Derivative of ln x•Derivative of log x•
a
Power Rule for Arbitrary Real Powers
Calculus at Work 181
Key Terms 181
Review Exercises 181
Contents v
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CHAPTER 4 Applications of Derivatives 186
4.1 Extreme Values of Functions 187
•Absolute (Global) Extreme Values •Local (Relative) Extreme Values •Finding
Extreme Values
4.2 Mean Value Theorem 196
•Mean Value Theorem •Physical Interpretation •Increasing and Decreasing
Functions •Other Consequences
4.3 Connecting ƒ! and ƒ" with the Graph of ƒ 205
•First Derivative Test for Local Extrema •Concavity •Points of Inflection •
Second Derivative Test for Local Extrema •Learning about Functions from
Derivatives
4.4 Modeling and Optimization 219
•Examples from Mathematics •Examples from Business and Industry •
Examples from Economics •Modeling Discrete Phenomena with Differentiable
Functions
4.5 Linearization and Newton’s Method 233
•Linear Approximation •Newton’s Method •Differentials •Estimating Change
with Differentials •Absolute,Relative,and Percentage Change •Sensitivity to
Change
4.6 Related Rates 246
•Related Rate Equations •Solution Strategy •Simulating Related Motion
Key Terms 255
Review Exercises 256
CHAPTER 5 The Definite Integral 262
5.1 Estimating with Finite Sums 263
•Distance Traveled •Rectangular Approximation Method (RAM) •Volume of a
Sphere •Cardiac Output
5.2 Definite Integrals 274
•Riemann Sums •Terminology and Notation of Integration •Definite Integral
and Area •Constant Functions •Integrals on a Calculator •Discontinuous
Integrable Functions
5.3 Definite Integrals and Antiderivatives 285
•Properties of Definite Integrals •Average Value of a Function •Mean Value
Theorem for Definite Integrals •Connecting Differential and Integral Calculus
5.4 Fundamental Theorem of Calculus 294
•Fundamental Theorem,Part 1 •Graphing the Function ∫axƒ(t)dt•Fundamental
Theorem,Part 2 •Area Connection •Analyzing Antiderivatives Graphically
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5.5 Trapezoidal Rule 306
•Trapezoidal Approximations •Other Algorithms •Error Analysis
Key Terms 315
Review Exercises 315
Calculus at Work 319
CHAPTER 6 Differential Equations and
Mathematical Modeling 320
6.1 Slope Fields and Euler’s Method 321
•Differential Equations •Slope Fields •Euler’s Method
6.2 Antidifferentiation by Substitution 331
•Indefinite Integrals •Leibniz Notation and Antiderivatives •Substitution in
Indefinite Integrals •Substitution in Definite Integrals
6.3 Antidifferentiation by Parts 341
•Product Rule in Integral Form •Solving for the Unknown Integral •Tabular
Integration •Inverse Trigonometric and Logarithmic Functions
6.4 Exponential Growth and Decay 350
•Separable Differential Equations •Law of Exponential Change •Continuously
Compounded Interest •Radioactivity •Modeling Growth with Other Bases •
Newton’s Law of Cooling
6.5 Logistic Growth 362
•How Populations Grow •Partial Fractions •The Logistic Differential Equation
•Logistic Growth Models
Key Terms 372
Review Exercises 372
Calculus at Work 376
CHAPTER 7 Applications of Definite Integrals 378
7.1 Integral As Net Change 379
•Linear Motion Revisited •General Strategy •Consumption Over Time •Net
Change from Data •Work
7.2 Areas in the Plane 390
•Area Between Curves •Area Enclosed by Intersecting Curves •Boundaries
with Changing Functions •Integrating with Respect to y•Saving Time with
Geometry Formulas
7.3 Volumes 399
•Volume As an Integral •Square Cross Sections •Circular Cross Sections •
Cylindrical Shells •Other Cross Sections
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7.4 Lengths of Curves 412
•A Sine Wave •Length of Smooth Curve •Vertical Tangents,Corners,and
Cusps
7.5 Applications from Science and Statistics 419
•Work Revisited •Fluid Force and Fluid Pressure •Normal Probabilities
Calculus at Work 430
Key Terms 430
Review Exercises 430
CHAPTER 8 Sequences, L’Hôpital’s Rule,
and Improper Integrals 434
8.1 Sequences 435
•Defining a Sequence •Arithmetic and Geometric Sequences •Graphing a
Sequence •Limit of a Sequence
8.2 L’Hôpital’s Rule 444
• Indeterminate Form 0/0 •Indeterminate Forms #/#,#$0,and #%#•
Indeterminate Forms 1#,00,#0
8.3 Relative Rates of Growth 453
•Comparing Rates of Growth •Using L’Hôpital’s Rule to Compare Growth
Rates •Sequential versus Binary Search
8.4 Improper Integrals 459
•Infinite Limits of Integration •Integrands with Infinite Discontinuities •Test
for Convergence and Divergence •Applications
Key Terms 470
Review Exercises 470
CHAPTER 9 Infinite Series 472
9.1 Power Series 473
•Geometric Series •Representing Functions by Series •Differentiation and
Integration •Identifying a Series
9.2 Taylor Series 484
•Constructing a Series •Series for sin xand cos x•Beauty Bare •Maclaurin
and Taylor Series •Combining Taylor Series •Table of Maclaurin Series
9.3 Taylor’s Theorem 495
•Taylor Polynomials •The Remainder •Remainder Estimation Theorem •
Euler’s Formula
9.4 Radius of Convergence 503
•Convergence •nth-Term Test •Comparing Nonnegative Series •Ratio Test •
Endpoint Convergence
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9.5 Testing Convergence at Endpoints 513
•Integral Test •Harmonic Series and p-series •Comparison Tests •Alternating
Series •Absolute and Conditional Convergence •Intervals of Convergence •
A Word of Caution
Key Terms 526
Review Exercises 526
Calculus at Work 529
CHAPTER 10 Parametric, Vector, and Polar Functions 530
10.1 Parametric Functions 531
•Parametric Curves in the Plane •Slope and Concavity •Arc Length •Cycloids
10.2 Vectors in the Plane 538
•Two-Dimensional Vectors •Vector Operations •Modeling Planar Motion •
Velocity,Acceleration,and Speed •Displacement and Distance Traveled
10.3 Polar Functions 548
•Polar Coordinates •Polar Curves •Slopes of Polar Curves •Areas Enclosed
by Polar Curves •A Small Polar Gallery
Key Terms 559
Review Exercises 560
APPENDIX
A1 Formulas from Precalculus Mathematics 562
A2 Mathematical Induction 566
A3 Using the Limit Definition 569
A4 Proof of the Chain Rule 577
A5 Conic Sections 578
A6 Hyperbolic Functions 603
A7 A Brief Table of Integrals 612
Glossary 618
Additional Answers 629
Applications Index 691
Index 695
Contents ix
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1
Chapter Prerequisites
for Calculus
Exponential functions are used to model
situations in which growth or decay change
dramatically. Such situations are found in
nuclear power plants, which contain rods of
plutonium-239; an extremely toxic radioactive
isotope.
Operating at full capacity for one year, a 1,000
megawatt power plant discharges about 435 lb of
plutonium-239. With a half-life of 24,400 years,
how much of the isotope will remain after 1,000
years? This question can be answered with the
mathematics covered in Section 1.3.
2
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Section 1.1 Lines 3
Chapter 1 Overview
This chapter reviews the most important things you need to know to start learning calcu-
lus. It also introduces the use of a graphing utility as a tool to investigate mathematical
ideas,to support analytic work,and to solve problems with numerical and graphical meth-
ods. The emphasis is on functions and graphs,the main building blocks of calculus.
Functions and parametric equations are the major tools for describing the real world in
mathematical terms,from temperature variations to planetary motions,from brain waves
to business cycles, and from heartbeat patterns to population growth. Many functions
have particular importance because of the behavior they describe. Trigonometric func-
tions describe cyclic, repetitive activity; exponential, logarithmic, and logistic functions
describe growth and decay; and polynomial functions can approximate these and most
other functions.
1.1 Lines
What you’ll learn about
Increments
• Increments
One reason calculus has proved to be so useful is that it is the right mathematics for relat-
• Slope of a Line ing the rate of change of a quantity to the graph of the quantity. Explaining that relation-
ship is one goal of this book. It all begins with the slopes of lines.
• Parallel and Perpendicular Lines
When a particle in the plane moves from one point to another, the net changes or
• Equations of Lines incrementsin its coordinates are found by subtracting the coordinates of its starting point
from the coordinates of its stopping point.
• Applications
. . . and why
Linear equations are used exten- DEFINITION Increments
sively in business and economic
If a particle moves from the point (x ,y ) to the point (x ,y ),the incrementsin its
applications. 1 1 2 2
coordinates are
!x"x !x and !y"y !y .
2 1 2 1
The symbols !xand !yare read “delta x”and “delta y.”The letter !is a Greek capital
dfor “difference.”Neither !xnor !ydenotes multiplication; !xis not “delta times x”nor
is !y“delta times y.”
Increments can be positive,negative,or zero,as shown in Example 1.
L EXAMPLE 1 Finding Increments
y
The coordinate increments from !4,!3"to (2,5) are
P(x , y)
2 2 2
!x"2!4"!2, !y"5!!!3""8.
!y " rise
From (5,6) to (5,1),the increments are
P1(x1, y1)!x " run Q(x2, y1) !x"5!5"0, !y"1!6"!5. Now try Exercise 1.
x
O Slope of a Line
Figure 1.1 The slope of line Lis Each nonvertical line has a slope,which we can calculate from increments in coordinates.
rise !y Let L be a nonvertical line in the plane and P (x , y ) and P (x , y ) two points on L
m"##"##. 1 1 1 2 2 2
run !x (Figure 1.1). We call !y"y !y the risefrom P to P and !x"x !x the runfrom
2 1 1 2 2 1
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4 Chapter1 Prerequisites for Calculus
P to P . Since Lis not vertical,!x!0 and we define the slope of Lto be the amount of
1 2
rise per unit of run. It is conventional to denote the slope by the letter m.
L L
1 2
DEFINITION Slope
Slope m Slope m
1 2
Let P (x ,y ) and P (x ,y ) be points on a nonvertical line,L.The slopeof Lis
1 1 1 2 2 2
m m
1 2
θ θ rise !y y %y
1 2 x m"&&"&&"&2 &1.
1 1 run !x x %x
2 1
Figure 1.2 If L !L ,then u "u and
1 2 1 2
A line that goes uphill as xincreases has a positive slope. A line that goes downhill as x
m "m . Conversely,if m "m ,then
1 2 1 2
increases has a negative slope. A horizontal line has slope zero since all of its points have
u "u and L !L .
1 2 1 2 the same y-coordinate,making !y"0. For vertical lines,!x"0 and the ratio !y#$xis
undefined. We express this by saying that vertical lines have no slope.
y
Parallel and Perpendicular Lines
L
L 1
2
Parallel lines form equal angles with the x-axis (Figure 1.2). Hence, nonvertical parallel
C
lines have the same slope. Conversely,lines with equal slopes form equal angles with the
Slope m1 !1 Slope m2 x-axis and are therefore parallel.
h If two nonvertical lines L and L are perpendicular, their slopes m and m satisfy
!1 !2 m m "%1,so each slope is 1the nega2tive reciprocalof the other: 1 2
x 1 2
O A D a B
1 1
m "%&&, m "%&&.
1 m 2 m
Figure 1.3 !ADCis similar to !CDB. 2 1
Hence f1is also the upper angle in !CDB, The argument goes like this: In the notation of Figure 1.3, m1 " tanf1 " a#h, while
where tanf1"a#h. m "tanf "%h#a.Hence,m m "(a#h)(%h#a) "%1.
2 2 1 2
Equations of Lines
The vertical line through the point (a,b) has equation x"asince every x-coordinate on
the line has the value a.Similarly,the horizontal line through (a,b) has equation y"b.
y
EXAMPLE 2 Finding Equations of Vertical and Horizontal Lines
Along this line, The vertical and horizontal lines through the point (2,3) have equations x"2 and
6 x " 2.
y"3,respectively (Figure 1.4). Now try Exercise 9.
5
We can write an equation for any nonvertical line L if we know its slope m and the
4
Along this line, coordinates of one point P (x ,y ) on it. If P(x,y) is anyother point on L,then
1 1 1
y " 3.
3 y%y
(2, 3) &&1 "m,
x%x
2 1
so that
1
y%y "m(x%x ) or y"m(x%x ) ’y .
1 1 1 1
x
0 1 2 3 4
DEFINITION Point-Slope Equation
Figure 1.4 The standard equations for
the vertical and horizontal lines through The equation
the point (2,3) are x"2 and y"3.
y"m(x%x ) ’y
(Example 2) 1 1
is the point-slope equationof the line through the point (x ,y ) with slope m.
1 1
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Section 1.1 Lines 5
EXAMPLE 3 Using the Point-Slope Equation
Write the point-slope equation for the line through the point (2,3) with slope !3"2.
SOLUTION
We substitute x #2,y #3,and m#!3"2 into the point-slope equation and obtain
1 1
3 3
y#!$$!x!2"%3 or y#!$$x%6.
2 2
Now try Exercise 13.
y The y-coordinate of the point where a nonvertical line intersects the y-axis is the
y-interceptof the line. Similarly,the x-coordinate of the point where a nonhorizontal line
Slope m
(x, y) intersects the x-axis is the x-intercept of the line. A line with slope m and y-intercept
bpasses through (0,b) (Figure 1.5),so
y = mx + b
(0, b) y#m!x!0"%b, or,more simply, y#mx%b.
b
x DEFINITION Slope-Intercept Equation
0
The equation
Figure 1.5 A line with slope mand
y-intercept b. y#mx%b
is the slope-intercept equationof the line with slope mand y-intercept b.
EXAMPLE 4 Writing the Slope-Intercept Equation
Write the slope-intercept equation for the line through (!2,!1) and (3,4).
SOLUTION
The line’s slope is
4!!!1" 5
m#$$#$$#1.
3!!!2" 5
We can use this slope with either of the two given points in the point-slope equation. For
(x ,y ) #(!2,!1),we obtain
1 1
y#1•!x!!!2""%!!1"
y#x%2%!!1"
y#x%1. Now try Exercise 17.
If Aand Bare not both zero,the graph of the equation Ax%By#Cis a line. Every line
has an equation in this form,even lines with undefined slopes.
DEFINITION General Linear Equation
The equation
Ax%By#C (Aand Bnot both 0)
is a general linear equationin xand y.