Calculus Early Transcendentals – Howard Anton, Irl Bivens, Stephen Davis – 8th Edition

Description

There is a mind-numbing sameness to introductory calculus texts. Some authors appear to be engaged in a form of “arms race”, trying to stand out by adding even more exercises to the ends of the chapters and sections. However, there are some significant differences between the texts and my goal here is to point out the major ones between these three books.

The first is in the introductory material, where the Sullivan book is clearly the weakest. Chapter zero contains:

  • Algebra review
  • Solving basic equations.
  • Lines
  • Rectangular coordinates.
  • Intervals, solving inequalities.
  • Lines.

I find this a waste of ink and trees. If we are letting students into our calculus classes that need a review of this basic material, then we have failed in our advising and those students will fail the course. In my opinion, students are there to learn calculus and should already know what a function is. Chapter zero of the Himonas book is devoted to a recapitulation of the categories of functions and chapter one begins the study of limits and continuity. The Anton book follows largely the same path, chapter one contains a review of functions and chapter two begins the study of limits and continuity.

Chapter 2 of the Himonas book describes exponential and logarithmic functions. While I agree that some review is probably necessary, once again, I don’t believe that such an extensive review is required. With forty pages devoted to these functions, you are spending about as much time on them as is done in a precalculus course and the limit is never mentioned in the chapter. I really don’t see any justification for this. When the exponential and logarithmic functions are mentioned in the Anton book, they are being differentiated. Chapter 2 of the Sullivan book lists the classes of functions: quadratic, power functions, exponential and logarithmic. I apply the same criticism here that I assigned to the Himonas book. I just don’t see the need.

The remainder of the Sullivan book, 320 pages, provides an overview of differential and integral calculus up through partial differentiation. It is all at the level of “plug-and-chug”, there is little to no theoretical justification of what is being done. The explanations are adequate for the purpose, but the book is suitable only for the most basic overview of calculus.

The Himonas book is also an overview of differential and integral calculus, with very little in the way of theory. There are a few theorems, although very little in the way of proof. The final chapters cover differential equations, higher-order approximations, and probability and statistics. This is the book that I would use for an overview, I much prefer having the advanced topics included in the text so that I can selectively exclude them as I wish.

The Anton book is a traditional calculus book with formal theorems and explanations of the theoretical background of how calculus works. The coverage is up through vector calculus, partial derivatives and multiple integrals. Since there are so many calculus books and the coverage in them is so similar, I cannot say that this book is outstanding. However, it is certainly better than average and I can recommend it for use in the rigorous first semester course in calculus.

Table of Contents

1. FUNCTIONS
1.1 Functions
1.2 Graphing Functions Using Calculators and Computer Algebra Systems
1.3 New Functions from Old
1.4 Families of Functions4
1.5 Inverse Functions; Inverse Trigonometric Functions
1.6 Exponential and Logarithmic Functions
1.7 Mathematical Models
1.8 Parametric Equations

2. LIMITS AND CONTINUITY
2.1 Limits (An Intuitive Approach)
2.2 Computing Limits
2.3 Limits at Infinity; End Behavior of a Function
2.4 Limits (Discussed More Rigorously)
2.5 Continuity
2.6 Continuity of Trigonometric and Inverse Functions

3. THE DERIVATIVE
3.1 Tangent Lines, Velocity, and General Rates of Change
3.2 The Derivative Function
3.3 Techniques of Differentiation9
3.4 The Product and Quotient Rules
3.5 Derivatives of Trigonometric Functions
3.6 The Chain Rule
3.7 Related Rates
3.8 Local Linear Approximation; Differentials

4. EXPONENTIAL, LOGARITHMIC, AND INVERSE TRIGONOMETRIC FUNCTIONS
4.1 Implicit Differentiation
4.2 Derivatives of Logarithmic Functions
4.3 Derivatives of Exponential and Inverse Trigonometric Functions
4.4 L’Hôpital’s Rule; Indeterminate Forms

5. THE DERIVATIVE IN GRAPHING AND APPLICATIONS
5.1 Analysis of Functions I:Increase, Decrease, and Concavity
5.2 Analysis of Functions II: Relative Extrema; Graphing Polynomials
5.3 More on Curve Sketching: Rational Functions; Curves with Cusps and Vertical Tangent Lines; Using Technology
5.4 Absolute Maxima and Minima
5.5 Applied Maximum and Minimum Problems
5.6 Newton’s Method
5.7 Rolle’s Theorem; Mean-Value Theorem
5.8 Rectilinear Motion

6. INTEGRATION
6.1 An Overview of the Area Problem
6.2 The Indefinite Integral
6.3 Integration by Substitution
6.4 The Definition of Area as a Limit; Sigma Notation
6.5 The Definite Integral
6.6 The Fundamental Theorem of Calculus
6.7 Rectilinear Motion Revisited Using Integration
6.8 Evaluating Definite Integrals by Substitution
6.9 Logarithmic Functions from the Integral Point of View

7. APPLICATIONS OF THE DEFINITE INTEGRAL IN GEOMETRY, SCIENCE, AND ENGINEERING
7.1 Area Between Two Curves
7.2 Volumes by Slicing; Disks and Washers5
7.3 Volumes by Cylindrical Shells
7.4 Length of a Plane Curve
7.5 Area of a Surface of Revolution
7.6 Average Value of a Function and its Applications
7.7 Work
7.8 Fluid Pressure and Force9
7.9 Hyperbolic Functions and Hanging Cables

8. PRINCIPLES OF INTEGRAL EVALUATION
8.1 An Overview of Integration Methods
8.2 Integration by Parts
8.3 Trigonometric Integrals
8.4 Trigonometric Substitutions3
8.5 Integrating Rational Functions by Partial Fractions
8.6 Using Computer Algebra Systems and Tables of Integrals
8.7 Numerical Integration; Simpson’s Rule
8.8 Improper Integrals

9. MATHEMATICAL MODELING WITH DIFFERENTIAL EQUATIONS
9.1 First-Order Differential Equations and Applications
9.2 Slope Fields; Euler’s Method
9.3 Modeling with First-Order Differential Equations
9.4 Second-Order Linear Homogeneous Differential Equations; The Vibrating Spring

10. INFINITE SERIES
10.1 Sequences
10.2 Monotone Sequences
10.3 Infinite Series
10.4 Convergence Tests
10.5 The Comparison, Ratio, and Root Tests
10.6 Alternating Series; Conditional Convergence
10.7 Maclaurin and Taylor Polynomials
10.8 Maclaurin and Taylor Series; PowerSeries
10.9 Convergence of Taylor Series
10.10 Differentiating and Integrating Power Series; Modeling with Taylor Series

11. ANALYTIC GEOMETRY IN CALCULUS
11.1 Polar Coordinates
11.2 Tangent Lines and Arc Length for Parametric and Polar Curves
11.3 Area in Polar Coordinates4
11.4 Conic Sections in Calculus
11.5 Rotation of Axes; Second-Degree Equations
11.6 Conic Sections in Polar Coordinates
Horizon Module: Comet Collision

12. THREE-DIMENSIONAL SPACE; VECTORS
12.1 Rectangular Coordinates in-Space; Spheres; Cylindrical Surfaces
12.2 Vectors
12.3 Dot Product; Projections
12.4 Cross Product
12.5 Parametric Equations of Lines
12.6 Planes in-Space
12.7 Quadric Surfaces
12.8 Cylindrical and Spherical Coordinates5

13. VECTOR-VALUED FUNCTIONS
13.1 Introduction to Vector-Valued Functions
13.2 Calculus of Vector-Valued Functions
13.3 Change of Parameter; Arc Length
13.4 Unit Tangent, Normal, and Binormal Vectors
13.5 Curvature
13.6 Motion Along a Curve
13.7 Kepler’s Laws of Planetary Motion

14. PARTIAL DERIVATIVES
14.1 Functions of Two or More Variables
14.2 Limits and Continuity
14.3 Partial Derivatives
14.4 Differentiability, Differentials, and Local Linearity
14.5 The Chain Rule
14.6 Directional Derivatives and Gradients
14.7 Tangent Planes and Normal Vectors
14.8 Maxima and Minima of Functions of Two Variables
14.9 Lagrange Multipliers

15. MULTIPLE INTEGRALS
15.1 Double Integrals
15.2 Double Integrals over Nonrectangular Regions
15.3 Double Integrals in Polar Coordinates
15.4 Parametric Surfaces; Surface Area
15.5 Triple Integrals
15.6 Centroid, Center of Gravity, Theorem of Pappus
15.7 Triple Integrals in Cylindrical and Spherical Coordinates
15.8 Change of Variables in Multiple Integrals; Jacobians

16. TOPICS IN VECTOR CALCULUS
16.1 Vector Fields
16.2 Line Integrals
16.3 Independence of Path; Conservative Vector Fields
16.4 Green’s Theorem
16.5 Surface Integrals
16.6 Applications of Surface Integrals; Flux
16.7 The Divergence Theorem
16.8 Stokes' Theorem

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Rashid Ali
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Good book