Calculus: Graphical, Numerical, Algebraic

Prerequisites for Calculus 2
1.1 Linear Functions 3
Increments and Slope • Point-Slope Equation of a Linear Function • Other Linear
Equation Forms • Parallel and Perpendicular Lines • Applications of Linear Functions
• Solving Two Linear Equations Simultaneously
1.2 Functions and Graphs 13
Functions • Domains and Ranges • Viewing and Interpreting Graphs • Even Functions
and Odd Functions—Symmetry • Piecewise-Defined Functions • Absolute Value
Function • Composite Functions
1.3 Exponential Functions 23
Exponential Growth • Exponential Decay • Compound Interest • The Number e
1.4 Parametric Equations 29
Relations • Circles • Ellipses • Lines and Other Curves
1.5 Inverse Functions and Logarithms 36
One-to-One Functions • Inverses • Finding Inverses • Logarithmic Functions
• Properties of Logarithms • Applications
1.6 Trigonometric Functions 45
Radian Measure • Graphs of Trigonometric Functions • Periodicity • Even and Odd
Trigonometric Functions • Transformations of Trigonometric Graphs • Inverse
Trigonometric Functions
Key Terms 53
Review Exercises 54
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 • Squeeze Theorem
2.2 Limits Involving Infinity 70
Finite Limits as x S ±q • Squeeze Theorem Revisited • Infinite Limits as x S a
• End Behavior Models • “Seeing” Limits as x S ±q
2.3 Continuity 78
Continuity at a Point • Continuous Functions • Algebraic Combinations • Composites
• Intermediate Value Theorem for Continuous Functions
2.4 Rates of Change, Tangent Lines, and Sensitivity 87
Average Rates of Change • Tangent to a Curve • Slope of a Curve • Normal to a Curve
• Speed Revisited • Sensitivity
Key Terms 96
Review Exercises 97
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Chapter 3 Derivatives 100
3.1 Derivative of a function 101
Definition of Derivative • Notation • Relationships Between the Graphs of ƒ and ƒ′
• Graphing the Derivative from Data • One-Sided Derivatives
3.2 Differentiability 111
How ƒ′1a2 Might Fail to Exist • Differentiability Implies Local Linearity • Numerical
Derivatives on a Calculator • Differentiability Implies Continuity • Intermediate Value
Theorem for Derivatives
3.3 rules for Differentiation 118
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 129
Instantaneous Rates of Change • Motion Along a Line • Sensitivity to Change
• Derivatives in Economics
3.5 Derivatives of trigonometric functions 143
Derivative of the Sine Function • Derivative of the Cosine Function • Simple Harmonic
Motion • Jerk • Derivatives of the Other Basic Trigonometric Functions
Key terms 150
review exercises 150
Chapter 4 More Derivatives 154
4.1 Chain rule 155
Derivative of a Composite Function • “Outside-Inside” Rule • Repeated Use of the
Chain Rule • Slopes of Parametrized Curves • Power Chain Rule
4.2 implicit Differentiation 164
Implicitly Defined Functions • Lenses, Tangents, and Normal Lines • Derivatives of
Higher Order • Rational Powers of Differentiable Functions
4.3 Derivatives of inverse trigonometric functions 173
Derivatives of Inverse Functions • Derivative of the Arcsine • Derivative of the
Arctangent • Derivative of the Arcsecant • Derivatives of the Other Three
4.4 Derivatives of exponential and logarithmic functions 180
Derivative of ex • Derivative of ax • Derivative of ln x • Derivative of logax • Power
Rule for Arbitrary Real Powers
Key terms 189
review exercises 189
Chapter 5 applications of Derivatives 192
5.1 extreme values of functions 193
Absolute (Global) Extreme Values • Local (Relative) Extreme Values • Finding Extreme
Values
5.2 Mean value theorem 202
Mean Value Theorem • Physical Interpretation • Increasing and Decreasing Functions
• Other Consequences
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Contents xiii
5.3 Connecting ƒ′ and ƒ ″ with the graph of ƒ 211
First Derivative Test for Local Extrema • Concavity • Points of Inflection • Second
Derivative Test for Local Extrema • Learning About Functions from Derivatives
5.4 Modeling and optimization 224
A Strategy for Optimization • Examples from Mathematics • Examples from Business
and Industry • Examples from Economics • Modeling Discrete Phenomena with
Differentiable Functions
5.5 linearization, sensitivity, and Differentials 238
Linear Approximation • Differentials • Sensitivity Analysis • Absolute, Relative, and
Percentage Change • Sensitivity to Change • Newton’s Method • Newton’s Method
May Fail
5.6 related rates 252
Related Rate Equations • Solution Strategy • Simulating Related Motion
Key terms 261
review exercises 262
Chapter 6 the Definite integral 268
6.1 estimating with finite sums 269
Accumulation Problems as Area • Rectangular Approximation Method (RAM)
• Volume of a Sphere • Cardiac Output
6.2 Definite integrals 281
Riemann Sums • Terminology and Notation of Integration • Definite Integral and Area
• Constant Functions • Definite Integral as an Accumulator Function • Integrals on a
Calculator • Discontinuous Integrable Functions
6.3 Definite integrals and antiderivatives 293
Properties of Definite Integrals • Average Value of a Function • Mean Value Theorem
for Definite Integrals • Connecting Differential and Integral Calculus
6.4 fundamental theorem of Calculus 302
Fundamental Theorem, Antiderivative Part • Graphing the Function 1
x
a ƒ1t2 dt
• Fundamental Theorem, Evaluation Part • Area Connection • Analyzing Antiderivatives
Graphically
6.5 trapezoidal rule 314
Trapezoidal Approximations • Other Algorithms • Error Analysis
Key terms 323
review exercises 323
Differential equations and Mathematical
Chapter 7 Modeling 328
7.1 slope fields and euler’s Method 329
Differential Equations • Slope Fields • Euler’s Method
7.2 antidifferentiation by substitution 340
Indefinite Integrals • Leibniz Notation and Antiderivatives • Substitution in
Indefinite Integrals • Substitution in Definite Integrals
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7.3 antidifferentiation by Parts 349
Product Rule in Integral Form • Solving for the Unknown Integral • Tabular Integration
• Inverse Trigonometric and Logarithmic Functions
7.4 exponential growth and Decay 358
Separable Differential Equations • Law of Exponential Change • Continuously
Compounded Interest • Radioactivity • Modeling Growth with Other Bases
• Newton’s Law of Cooling
7.5 logistic growth 369
How Populations Grow • Partial Fractions • The Logistic Differential Equation
• Logistic Growth Models
Key terms 378
review exercises 379
Chapter 8 applications of Definite integrals 384
8.1 accumulation and net Change 385
Linear Motion Revisited • General Strategy • Consumption over Time • Coming and
Going • Net Change from Data • Density • Work
8.2 areas in the Plane 397
Area Between Curves • Area Enclosed by Intersecting Curves • Boundaries with
Changing Functions • Integrating with Respect to y • Saving Time with Geometry
Formulas
8.3 volumes 406
Volume as an Integral • Square Cross Sections • Circular Cross Sections • Cylindrical
Shells • Other Cross Sections
8.4 lengths of Curves 420
A Sine Wave • Length of a Smooth Curve • Vertical Tangents, Corners, and Cusps
8.5 applications from science and statistics 427
Work Revisited • Fluid Force and Fluid Pressure • Normal Probabilities
Key terms 438
review exercises 438
sequences, l’Hospital’s rule, and improper
Chapter 9 integrals 442
9.1 sequences 443
Defining a Sequence • Arithmetic and Geometric Sequences • Graphing a Sequence
• Limit of a Sequence
9.2 l’Hospital’s rule 452
Indeterminate Form 0>0 • Indeterminate Forms q>q, q # 0, and q- q
• Indeterminate Forms 1q, 00, q0
9.3 relative rates of growth 461
Comparing Rates of Growth • Using L’Hospital’s Rule to Compare Growth Rates
• Sequential Versus Binary Search
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9.4 improper integrals 467
Infinite Limits of Integration • Integrands with Infinite Discontinuities • Test for
Convergence and Divergence • Applications
Key terms 477
review exercises 478
Chapter 10 infinite series 480
10.1 Power series 481
Geometric Series • Representing Functions by Series • Differentiation and Integration
• Identifying a Series
10.2 taylor series 492
Constructing a Series • Series for sin x and cos x • Beauty Bare • Maclaurin and
Taylor Series • Combining Taylor Series • Table of Maclaurin Series
10.3 taylor’s theorem 503
Taylor Polynomials • The Remainder • Bounding the Remainder • Analyzing
Truncation Error • Euler’s Formula
10.4 radius of Convergence 513
Convergence • nth-Term Test • Comparing Nonnegative Series • Ratio Test
• Endpoint Convergence
10.5 testing Convergence at endpoints 523
Integral Test • Harmonic Series and p-series • Comparison Tests • Alternating Series
• Absolute and Conditional Convergence • Intervals of Convergence • A Word of
Caution
Key terms 537
review exercises 537
Chapter 11 Parametric, vector, and Polar functions 542
11.1 Parametric functions 543
Parametric Curves in the Plane • Slope and Concavity • Arc Length • Cycloids
11.2 vectors in the Plane 550
Two-Dimensional Vectors • Vector Operations • Modeling Planar Motion • Velocity,
Acceleration, and Speed • Displacement and Distance Traveled
11.3 Polar functions 561
Polar Coordinates • Polar Curves • Slopes of Polar Curves • Areas Enclosed by Polar
Curves • A Small Polar Gallery
Key terms 574
review exercises 575
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appe NDIC e S
a1 formulas from Precalculus Mathematics 577
a2 a formal Definition of limit 583
a3 a Proof of the Chain rule 591
a4 Hyperbolic functions 592
a5 a very Brief table of integrals 601
glossary 604
selected answers 615
applications index 672
subject index 676