# Calculus: Early Transcendentals

###### Dale Varberg, Edwin J. Purcell, and Steven E. Rigdon
Publisher:
Pearson/Prentice Hall
Publication Date:
2007
Number of Pages:
790
Format:
Hardcover
Price:
109.33
ISBN:
0132273462
Category:
Textbook
[Reviewed by
Fernando Q. Gouêa
, on
05/30/2006
]

This is the "early transcendentals" version of a text that was reviewed here earlier. Most of what was said in that review applies to this book also. The main difference is that the logarithm and exponential functions are introduced much earlier in this version of the text.

The need for an "early transcendentals" option is a consequence of the widespread adoption of the idea (first proposed, I believe, by Felix Klein in his Elementary Mathematics from an Advanced Standpoint, where he argues that it be used in the schools ) that the natural logarithm is best introduced as the integral of 1/x and that the exponential is best defined as the inverse of the natural logarithm.

Adopting Klein's suggestion was always a rather strange decision in the context of the average undergraduate calculus text. After all, no one felt the need to include a rigorous definition of the sine and cosine functions. Giving such a definition also requires doing some (rather hard) analysis, as one can see in Spivak's Calculus and in most standard real analysis texts. Furthermore, undergraduates arrive in college already "knowing" about these functions; it seems reasonable to simply recall their properties and use them, as we do for the trigonometric functions.

The approach taken in this book is less radical than that. The authors define ax for rational x in the usual way (without bothering to prove the existence of n-th roots, of course), then extend the definition to irrational exponents using a limit argument (with at least some proofs). They then define the logarithm as the inverse function, and finally use the limit of (1+r/n)n to introduce the "natural" exponential function. (At this point of the text, not a word is said to justify that adjective!) The derivative of the logarithm function is obtained first, then the inverse function theorem is used to find the derivative of the exponential function.

This approach does have the advantage of introducing a serious example of the use of limits into what is usually a dreary chapter. (Until we meet the derivative, why do we care what the limit is?) I believe it is an improvement on the "late transcendentals" approach. But in the end I still have my doubts whether students are ready for this kind of analysis so early in the course.

For what it's worth, I would simply take for granted the existence of the logarithm and exponential functions and their basic properties. Then, using the definition of the derivative, it's fairly easy to prove that the derivative of ax is some constant times ax. Choosing the right value for a makes that constant equal to 1, which explains what is "natural" about ex . An analogous argument explains why we must use radian measure if we want the derivative of sin(x) to be cos(x) rather than a constant times cos(x). This seems both simpler and more illuminating, even if it hides the inherent trickiness of defining these functions. There's time for that in the analysis course.

Fernando Q. Gouvêa is professor of mathematics at Colby College in Waterville, ME.

 0 PRELIMINARIES 0.1 Real Numbers, Logic and Estimation 0.2 Inequalities and Absolute Values 0.3 The Rectangular Coordinate System 0.4 Graphs of Equations 1 FUNCTIONS 1.1 Functions and Their Graphs 1.2 Operations on Functions 1.3 Exponential and Logarithmic Functions 1.4 The Trigonometric Functions & Their Inverses 1.5 Chapter Review 2 LIMITS 2.1 Introduction to Limits 2.2 Rigorous Study of Limits 2.3 Limit Theorems 2.4 Limits Involving Transcendental Functions 2.5 Limits at Infinity, Infinite Limits 2.6 Continuity of Functions 2.7 Chapter Review 3 THE DERIVATIVE 3.1 Two Problems with One Theme 3.2 The Derivative 3.3 Rules for Finding Derivatives 3.4 Derivatives of Trigonometric Functions 3.5 The Chain Rule 3.6 Higher-Order Derivatives 3.7 Implicit Differentiation 3.8 Related Rates 3.9 Differentials and Approximations 3.10 Chapter Review 4 APPLICATIONS OF THE DERIVATIVE 4.1 Maxima and Minima 4.2 Monotonicity and Concavity 4.3 Local Extrema and Extrema on Open Intervals 4.4 Graphing Functions Using Calculus 4.5 The Mean Value Theorem for Derivatives 4.6 Solving Equations Numerically 4.7 Antiderivatives 4.8 Introduction to Differential Equations 5 THE DEFINITE INTEGRAL 5.1 Introduction to Area 5.2 The Definite Integral 5.3 The 1st Fundamental Theorem of Calculus 5.4 The 2nd Fundamental Theorem of Calculus and the Method of Substitution 5.5 The Mean Value Theorem for Integrals & the Use of Symmetry 5.6 Numerical Integration 5.7 Chapter Review 6 APPLICATIONS OF THE INTEGRAL 6.1 The Area of a Plane Region 6.2 Volumes of Solids: Slabs, Disks, Washers 6.3 Volumes of Solids of Revolution: Shells 6.4 Length of a Plane Curve 6.5 Work and Fluid Pressure 6.6 Moments, Center of Mass 6.8 Probability and Random Variables 6.8 Chapter Review 7 TECHNIQUES OF INTEGRATION & DIFFERENTIAL EQUATIONS 7.1 Basic Integration Rules 7.2 Integration by Parts 7.3 Some Trigonometric Integrals 7.4 Rationalizing Substitutions 7.5 The Method of Partial Fractions 7.6 Strategies for Integration 7.7 Growth and Decay 7.8 First-Order Linear Differential Equations 7.9 Approximations for Differential Equations 7.10 Chapter Review 8 INDETERMINATE FORMS & IMPROPER INTEGRALS 8.1 Indeterminate Forms of Type 0/0 8.2 Other Indeterminate Forms 8.3 Improper Integrals: Infinite Limits of Integration 8.4 Improper Integrals: Infinite Integrands 8.5 Chapter Review 9 INFINITE SERIES 9.1 Infinite Sequences 9.2 Infinite Series 9.3 Positive Series: The Integral Test 9.4 Positive Series: Other Tests 9.5 Alternating Series, Absolute Convergence, and Conditional Convergence 9.6 Power Series 9.7 Operations on Power Series 9.8 Taylor and Maclaurin Series 9.9 The Taylor Approximation to a Function 9.10 Chapter Review 10 CONICS AND POLAR COORDINATES 10.1 The Parabola 10.2 Ellipses and Hyperbolas 10.3 Translation and Rotation of Axes 10.4 Parametric Representation of Curves 10.5 The Polar Coordinate System 10.6 Graphs of Polar Equations 10.7 Calculus in Polar Coordinates 10.8 Chapter Review 11 GEOMETRY IN SPACE, VECTORS 11.1 Cartesian Coordinates in Three-Space 11.2 Vectors 11.3 The Dot Product 11.4 The Cross Product 11.5 Vector Valued Functions & Curvilinear Motion 11.6 Lines in Three-Space 11.7 Curvature and Components of Acceleration 11.8 Surfaces in Three Space 11.9 Cylindrical and Spherical Coordinates 11.10 Chapter Review 12 DERIVATIVES OF FUNCTIONS OF TWO OR MORE VARIABLES 12.1 Functions of Two or More Variables 12.2 Partial Derivatives 12.3 Limits and Continuity 12.4 Differentiability 12.5 Directional Derivatives and Gradients 12.6 The Chain Rule 12.7 Tangent Planes, Approximations 12.8 Maxima and Minima 12.9 Lagrange Multipliers 12.10 Chapter Review 13 MULTIPLE INTEGRATION 13.1 Double Integrals over Rectangles 13.2 Iterated Integrals 13.3 Double Integrals over Nonrectangular Regions 13.4 Double Integrals in Polar Coordinates 13.5 Applications of Double Integrals 13.6 Surface Area 13.7 Triple Integrals (Cartesian Coordinates) 13.8 Triple Integrals (Cyl & Sph Coordinates) 13.9 Change of Variables in Multiple Integrals 13.1 Chapter Review 14 VECTOR CALCULUS 14.1 Vector Fields 14.2 Line Integrals 14.3 Independence of Path 14.4 Green's Theorem in the Plane 14.5 Surface Integrals 14.6 Gauss's Divergence Theorem 14.7 Stokes's Theorem 14.8 Chapter Review APPENDIX A.1 Mathematical Induction A.2 Proofs of Several Theorems A.3 A Backward Look
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