Calculus: A Modern Approach Book

Preface to the Dover edition     iii
To the Instructor and General Reader     xxiii
To the Student     xxx
The Two Basic Problems of Calculus and Their Solutions for Straight Lines     1
Numerical Solutions     2
Graphical Solutions     6
The Idea of Newton and Leibniz     13
Graphical Solutions of the Two Basic Problems     19
The Derivative of a Polygon and the Integrals of a Step Line     19
The Approximate Area under a Curve and Approximate Integrals     25
Power, Exponential, and Sine Curves     28
Approximate Derivatives of a Curve     33
Comparison of Integrals and Derivatives     37
The Fundamental Idea of Calculus Suggested by the Graphical Solution     38
Numerical Solutions of the Two Basic Problems     39
The Area under a Step Line     39
The Approximate Area under a Simple Curve     40
How Accurate Is The Approximate Equality? A Fundamental Inequality     44
The Approximate Slope of a Curve at a Point     52
The Fundamental Idea of Calculus Suggested by the Numerical Method     54
The Idea and the Use of Functions     57
Remarks on Numbers     57
Numerical Variables     60
The Concept of Function     66
Names and Symbols for Functions     73
Function Variables     77
Restrictions and Extensions. Classes of Numbers     80
Addition and Multiplication of Functions     81
Substitution     88
Polynomials, Rational Functions, Elementary Functions     97
Remarks concerning the Traditional Function Notation in Pure Mathematics     99
On Limits     107
Approximate Equalities     107
The Limit of f at a     109
The Limits of Some Compound Functions     115
The Limits of Quotients of Functions     118
The Exponential and Logarithmic Functions     123
"Infinite" Limits and Limits at "Infinity"     125
The Limit of Sequences     127
The Basic Concepts of Calculus     130
The Derivative     130
Antiderivatives     134
The Integral     139
The Fundamental Reciprocity Laws of Calculus     149
Remarks concerning the Classical Notations in Calculus     152
The Approximation of Product Sums by Integrals and of Difference Quotients by Derivatives     158
Relative Maxima and Minima     163
The Application of Calculus to Science     167
Quantities     167
Consistent Classes of Quantities     169
Fluent Variables     173
Sums, Products, and Functions of Consistent Classes of Quantities     175
Functional Connections between Consistent Classes of Quantities     177
Is w a Function of u ?     180
Variable Quantities in Geometry and Kinematics     185
Remarks concerning the Traditional Notation in Applied Mathematics and in Science     194
Approximate Functional Connections     200
Integrals and the Cumulation of One Variable Quantity with regard to Another     202
Derivatives and the Rate of Change of One Variable Quantity with Respect to Another     212
What Is the Significance of Calculus in Science?     220
Optimal Differences for the Approximate Determination of Slopes     222
The Calculus of Derivatives     225
Conventions concerning the Reach of the Symbol D     225
The Sum Rule     226
The Product Rule and Its Consequences     228
The Substitution Rule     233
The Inversion Rule     239
Logarithmic Derivation     241
Elementary Functions     244
The Calculus of Antiderivatives     246
General Remarks. Standard Formulas     246
The Addition and Constant Factor Rules     249
The Transformation Rule     251
Antiderivation by Substitution     257
Antiderivation by Parts     262
The Antiderivatives of Rational Functions     266
Resume of the Calculus of Antiderivatives     269
Applications to Integrals     270
The Mean Value Theorem and Its Consequences     274
The Mean Value Theorem     274
Indeterminate Equalities and Determinate Inequalities     277
Taylor's Expansion     280
Remarks concerning Taylor's Formula     284
Maxima and Minima     288
The Approximate Computation of Areas by Expansions     290
Two-Place Functions     292
Simple Surfaces. Volumes and Tangential Planes     292
Two-Place Functions     295
Operations on Two-Place Functions     299
Pure Analytic Geometry and Pure Kinematics     304
Partial Derivatives     306
Implicit Functions     316
Partial Integrals and the Volume Problem     320
The Mean Value Theorem and the Taylor Expansion     328
Partial Rates of Change     332
Remarks concerning Partial Derivatives and Rates of Change in the Literature     339
What Are x and y?     342
Bibliography     346
Topical Index     349
Index of Symbols     354