Calculus A Modern Approach

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ISBN: 9780486457710 | 0486457710
Cover: Paperback
Copyright: 6/5/2007

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One of the twentieth century's most original mathematicians and thinkers, Karl Menger taught students of many backgrounds. In this, his radical revision of the traditional calculus text, he presents pure and applied calculus in a unified conceptual frame, offering a thorough understanding of theory as well as of the methodology underlying the use of calculus as a tool.

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