Calc 6E revision diary
revised 07/29/17
6E CALCULUS
Here are some of the changes for the 6th edition of Calculus.
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At the beginning of the book there are four diagnostic tests, in Basic Algebra, Analytic Geometry, Functions, and
Trigonometry. Answers are given and students who don’t do well are referred to where they should seek help (appendixes,
review sections of Chapter 1, and the web site).
In response to requests of several users, the material motivating the derivative is briefer: Sections 2.6 and 3.1 are combined
into a single section called Derivatives and Rates of Change.
The section on Higher Derivatives in Chapter 3 has disappeared and that material is integrated into various sections in
Chapter 3.
Instructors who do not cover the chapter on differential equations have commented that the section on Exponential Growth
and Decay was inconveniently located there. Accordingly, it is moved earlier in the book, to Chapter 7. This move
precipitates a reorganization of Chapter 10.
Sections 4.7 and 4.8 are merged into a single section, with a briefer treatment of optimization problems in business and
economics.
Sections 12.10 and 12.11 are merged into a single section. I had previously featured the binomial series in its own section to
emphasize its importance. But I learned that some instructors were omitting that section, so I have decided to incorporate
binomial series into 12.10.
The material on cylindrical and spherical coordinates (formerly Section 13.7) is moved to Chapter 16, where it is introduced
in the context of evaluating triple integrals.
In addition, there are hundreds of small improvements: new examples, additional steps in existing examples, updating of data in
existing examples and exercises, new phrases and margin notes to clarify the exposition, references to other sources and web sites,
redrawn art, and references to the TEC CD (Tools for Enriching Calculus). At least 25% of the exercises in each chapter are new.
The following annotated table of contents indicates some, though by no means all, of these changes.
DIAGNOSTIC TESTS
A PREVIEW OF CALCULUS
1
FUNCTIONS AND MODELS
1.1
Four Ways to Represent a Function
p. 13: Example 3 is new
pp. 14, 18: Update postage rate example
21 new exercises
1.2
Mathematical Models: A Catalog of Essential Functions
pp. 26-27: Example 2: Update CO2 data for year 2002, associated equations, figures 4, 5, 6
4 new exercises
1.3
New Functions from Old Functions
p. 41: Briefer exposition of combinations of functions
p. 41: Figure 11 is new
17 new exercises
1.4
Graphing Calculators and Computers
p. 50: TEC note
10 new exercises
Review
5 new exercises
Principles of Problem Solving
2
LIMITS
2.1
2.2
The Tangent and Velocity Problems
4 new exercises
The Limit of a Function
Calc 6E revision diary
p. 73: Margin note re “large negative”
12 new exercises
2.3
Calculating Limits Using the Limit Laws
p. 78: Solution to Example 1(b) rewritten
p. 81: Fact in Note made more prominent with box
p. 84: Figure 8 redrawn
16 new exercises
2.4
The Precise Definition of a Limit
pp. 88-94: Definitions and examples reformulated in “if, then” format
8 new exercises
2.5
Continuity
p. 102: Example 7 is new
15 new exercises
Review
10 new exercises
Problems Plus
3
DIFFERENTIATION RULES
3.1
Derivatives and Rates of Change
p. 112: Two sections combined into one, with briefer exposition
14 new exercises
3.2
The Derivative as a Function
p. 125: Example 3 is new (simpler)
p. 129: Higher derivatives introduced earlier
17 new exercises
3.3 Differentiation Formulas
p. 139: Example 5 is new
p. 143: Margin notes re constants
p. 143: Example 11 changed to incorporate arbitrary constants
p. 143: Normal lines now defined here
p. 143-144: Example 12 and Figure 5 revised to include normal lines
38 new exercises
Writing Project: Early Methods for Finding Tangents
Applied Project: Building a Better Roller Coaster
3.4 Derivatives of Trigonometric Functions
p. 149: TEC note
p. 149: Box for Equation 2
p. 152: Example 3 now incorporates acceleration
p. 153: Solution to Example 5 rewritten
18 new exercises
3.5 The Chain Rule
p. 156: Hypotheses in Chain Rule reformulated
27 new exercises
Applied Project: Where Should a Pilot Start Descent?
3.6 Implicit Differentiation
p. 168: Example 4 moved here
18 new exercises
3.7 Rates of Change in the Natural and Social Sciences
p. 170: This section moved here so all differentiation rules are available
p. 171: Parts g, h, i added to Example 1
p. 172: TEC note
10 new exercises
3.8 Related Rates
7 new exercises
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Calc 6E revision diary
3.9
Linear Approximations and Differentials
13 new exercises
Laboratory Project: Taylor Polynomials
Review
10 new exercises
Problems Plus
2 new problems
4
APPLICATIONS OF DIFFERENTIATION
4.1
Maximum and Minimum Values
24 new exercises
Applied Project: The Calculus of Rainbows
4.2
The Mean Value Theorem
7 new exercises
4.3 How Derivatives Affect the Shape of a Graph
23 new exercises
4.4 Limits at Infinity; Horizontal Asymptotes
p. 236: Warning in Example 9 made more emphatic
pp. 55-56: Definitions 5-6 and Examples 12-13 recast in “if, then” form
19 new exercises
4.5
Summary of Curve Sketching
p. 243: Briefer introduction
p. 246: Example 3 is new
19 new exercises
4.6
Graphing with Calculus and Calculators
p. 252: Extra step in solution to Example 3
11 new exercises
4.7
Optimization Problems
p. 256: Two sections combined
p. 261: Briefer treatment of business applications
8 new exercises
Applied Project: The Shape of a Can
4.8
Newton’s Method
p. 270: TEC note
14 new exercises
4.9
Antiderivatives
21 new exercises
Review
20 new exercises
Problems Plus
2 new problems
5
INTEGRALS
5.1
5.2
Areas and Distances
p. 292: TEC note
8 new exercises
The Definite Integral
p. 300: Definition of an integral reformulated for integrable functions
p. 302: Theorem on integrability of piecewise continuous functions
p. 306: TEC note
18 new exercises
Discovery Project: Area Functions
5.3
The Fundamental Theorem of Calculus
23 new exercises
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Calc 6E revision diary
5.4
Indefinite Integrals and the Net Change Theorem
18 new exercises
Writing Project: Newton, Leibniz, and the Invention of Calculus
5.5
The Substitution Rule
25 new exercises
Review
11 new exercises
Problems Plus
2 new problems
6
APPLICATIONS OF INTEGRATION
6.1
6.2
6.3
6.4
6.5
Areas between Curves
14 new exercises
Volume
pp. 356, 358, 360: TEC notes
18 new exercises
Volumes by Cylindrical Shells
p. 23: TEC note
12 new exercises
Work
7 new exercises
Average Value of a Function
6 new exercises
Review
8 new exercises
Problems Plus
1 new problem
7
INVERSE FUNCTIONS
7.1 Inverse Functions
pp. 389-390: g replaced by f^(-1)
11 new exercises
7.2 Exponential Functions and Their Derivatives
p. 398: TEC Note
24 new exercises
7.3 Logarithmic Functions
19 new exercises
7.4 Derivatives of Logarithmic Functions
22 new exercises
7.2* The Natural Logarithmic Function
20 new exercises
7.3* The Natural Exponential Function
37 new exercises
7.4* General Logarithmic and Exponential Functions
10 new exercises
7.5 Exponential Growth and Decay
p. 447: This section moved here from Chapter 10
p. 447: The treatment of population growth is briefer
4 new exercises
7.6 Inverse Trigonometric Functions
16 new exercises
Applied Project: Where to Sit at the Movies
7.7 Hyperbolic Functions
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Calc 6E revision diary
p. 464: Application to Water Waves
p. 465: Margin note on the Gateway Arch
13 new exercises
7.8 Indeterminate Forms and l'Hospital's Rule
24 new exercises
Writing Project: The Origins of l'Hospital's Rule
Review
12 new exercises
Problems Plus
1 new problem
8
TECHNIQUES OF INTEGRATION
8.1 Integration by Parts
17 new exercises
8.2 Trigonometric Integrals
18 new exercises
8.3 Trigonometric Substitution
11 new exercises
8.4 Integration of Rational Functions by Partial Fractions
19 new exercises
8.5 Strategy for Integration
21 new exercises
8.6 Integration Using Tables and Computer Algebra Systems
11 new exercises
Discovery Project: Patterns in Integrals
8.7 Approximate Integration
p. 539: New table in margin
13 new exercises
8.8 Improper Integrals
p. 546: TEC note
20 new exercises
Review
17 new exercises
Problems Plus
3 new problems
9
FURTHER APPLICATIONS OF INTEGRATION
9.1 Arc Length
12 new exercises
Discovery Project: Arc Length Contest
9.2 Area of a Surface of Revolution
10 new exercises
Discovery Project: Rotating on a Slant
9.3 Applications to Physics and Engineering
15 new exercises
Discovery Project: Complementary Coffee Cups (NEW)
9.4 Applications to Economics and Biology
5 new exercises
9.5 Probability
4 new exercises
Review
revised 07/29/17
Calc 6E revision diary
1 new exercise
Problems Plus
10
DIFFERENTIAL EQUATIONS
10.1 Modeling with Differential Equations
4 new exercises
10.2 Direction Fields and Euler’s Method
8 new exercises
10.3 Separable Equations
14 new exercises
Applied Project: How Fast Does a Tank Drain?
Applied Project: Which is Faster, Going Up or Coming Down?
10.4 Models for Population Growth
p. 627: Section rewritten to combine exponential, logistic, and other models
p. 633: New material, including Figure 6
2 new exercises
Applied Project: Calculus and Baseball
10.5 Linear Equations
9 new exercises
10.6 Predator-Prey Systems
Review
8 new exercises
Problems Plus
2 new problems
11
PARAMETRIC EQUATIONS AND POLAR COORDINATES
11.1 Curves Defined by Parametric Equations
p. 657: Margin note is new
p. 659: Example 4 is new
p. 660: Figures 11 and 12 are new
15 new exercises
Laboratory Project: Running Circles Around Circles
11.2 Calculus with Parametric Curves
p. 668: Paragraph on areas rephrased
19 new exercises
Laboratory Project: Bézier Curves
11.3 Polar Coordinates
p. 682: Figures 16 and 17 are new
22 new exercises
11.4 Areas and Lengths in Polar Coordinates
15 new exercises
11.5 Conic Sections
16 new exercises
11.6 Conic Sections in Polar Coordinates
pp. 702-703: New subsection on Kepler’s Laws
8 new exercises
Review
10 new exercises
Problems Plus
revised 07/29/17
Calc 6E revision diary
12
INFINITE SEQUENCES AND SERIES
12.1 Sequences
pp. 713-714: Definitions 2 and 5 reworded in “if, then” format
p. 716: Theorem 7 is new
p. 716: Example 8 is new
p. 717: Example 11 reworded
22 new exercises
Laboratory Project: Logistic Sequences
12.2 Series
p. 726: Extra step in Example 3
p. 727: Introductory sentence in Example 7
23 new exercises
12.3 The Integral Test and Estimates of Sums
p. 735: Margin note added
13 new exercises
12.4 The Comparison Tests
12 new exercises
12.5 Alternating Series
9 new exercises
12.6 Absolute Convergence and the Ratio and Root Tests
p. 754: Sentence added before Example 6
10 new exercises
12.7 Strategy for Testing Series
10 new exercises
12.8 Power Series
p. 759: Margin note for Example 1
p. 760: Margin note and photo
11 new exercises
12.9 Representations of Functions as Power Series
10 new exercises
12.10
Taylor and Maclaurin Series
pp. 777-779: Material on binomial series incorporated here instead of in its own section
20 new exercises
Laboratory Project: An Elusive Limit
Writing Project: How Newton Discovered the Binomial Series
12.11
Applications of Taylor Polynomials
13 new exercises
Applied Project: Radiation from the Stars
Review
1 new exercise
Problems Plus
2 new problems
13
VECTORS AND THE GEOMETRY OF SPACE
13.1 Three-Dimensional Coordinate Systems
12 new exercises
13.2 Vectors
14 new exercises
13.3 The Dot Product
p. 818: First two paragraphs under Projections reworded
pp. 818-819: Figures 4 and 5 redrawn
p. 820: Example 7 is new
18 new exercises
13.4 The Cross Product
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Calc 6E revision diary
p. 826: Two new danger symbols
p. 827: New paragraph after Example 5
13 new exercises
Discovery Project: The Geometry of a Tetrahedron
13.5 Equations of Lines and Planes
21 new exercises
Laboratory Project: Putting 3D in Perspective
13.6 Cylinders and Quadric Surfaces
p. 846: New subsection on applications of quadric surfaces
14 new exercises
Review
5 new exercises
Problems Plus
14
VECTOR FUNCTIONS
14.1 Vector Functions and Space Curves
12 new exercises
14.2 Derivatives and Integrals of Vector Functions
p. 862: New margin note after Example 3
16 new exercises
14.3 Arc Length and Curvature
p. 867: Definition of smooth curve changed and moved to this section
14 new exercises
14.4 Motion in Space: Velocity and Acceleration
11 new exercises
Applied Project: Kepler’s Laws
Review
6 new exercises
Problems Plus
1 new problem
15
PARTIAL DERIVATIVES
15.1 Functions of Several Variables
p. 895: New note after Example 6
p. 901: Figure 20 revised
20 new exercises
15.2 Limits and Continuity
pp. 907, 912: Limit definitions reformulated in if-then format
pp. 909-910: Example 4 in if-then format
12 new exercises
15.3 Partial Derivatives
25 new exercises
15.4 Tangent Planes and Differentials
14 new exercises
15.5 The Chain Rule
15 new exercises
15.6 Directional Derivatives and the Gradient Vector
21 new exercises
15.7 Maximum and Minimum Values
p. 959: New first paragraph
14 new exercises
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Calc 6E revision diary
Applied Project: Designing a Dumpster
Discovery Project: Quadratic Approximations and Critical Points
15.8 Lagrange Multipliers
p. 971: Margin note expanded to warn of the dangers of zero gradients
9 new exercises
Applied Project: Rocket Science
Applied Project: Hydro-Turbine Optimization
Review
13 new exercises
Problems Plus
16
MULTIPLE INTEGRALS
16.1 Double Integrals over Rectangles
p. 989: Margin note on unequal rectangles
p. 989: New paragraph on integrable functions
p. 956: Example 4 updated to 2006 snowfall
5 new exercises
16.2 Iterated Integrals
p. 999: Figure 5 redrawn
13 new exercises
16.3 Double Integrals over General Regions
20 new exercises
16.4 Double Integrals in Polar Coordinates
12 new exercises
16.5 Applications of Double Integrals
p. 1020: Steps added in second display after Example 4
9 new exercises
16.6 Triple Integrals
13 new exercises
Discovery Project: Volumes of Hyperspheres
16.7 Triple Integrals in Cylindrical Coordinates
p. 1036: Cylindrical coordinates moved here in time to evaluate triple integrals
9 new exercises
Discovery Project: The Intersection of Three Cylinders
16.8 Triple Integrals in Spherical Coordinates
p. 1041: Spherical coordinates moved here
13 new exercises
Applied Project: Roller Derby
16.9 Change of Variables in Multiple Integrals
7 new exercises
Review
Problems Plus
17
VECTOR CALCULUS
17.1 Vector Fields
p. 1063: Figure 1 updated
12 new exercises
17.2 Line Integrals
14 new exercises
17.3 The Fundamental Theorem for Line Integrals
12 new exercises
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Calc 6E revision diary
17.4 Green’s Theorem
8 new exercises
17.5 Curl and Divergence
9 new exercises
17.6 Parametric Surfaces and Their Areas
p. 1106: Introductory paragraph rewritten
19 new exercises
17.7 Surface Integrals
p. 1117: Section reorganized so integrals over parametric surfaces are covered first
10 new exercises
17.8 Stokes’ Theorem
5 new exercises
Writing Project: Three Men and Two Theorems
17.9 The Divergence Theorem
3 new exercises
17.10
Summary
Review
5 new exercises
Problems Plus
2 new problems
18
SECOND-ORDER DIFFERENTIAL EQUATIONS
18.1 Second-Order Linear Equations
pp. 1113-1115: Figures 1-3 redrawn to emphasize the basic solutions
16 new exercises
18.2 Nonhomogeneous Linear Equations
7 new exercises
18.3 Applications of Second-Order Differential Equations
4 new exercises
18.4 Series Solutions
1 new exercise
Review
revised 07/29/17