A First Course in Calculus

1. Front Matter

   Pages N2-XV

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2. Review of Basic Material

   1. Front Matter

      Pages 1-1

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   2. Numbers and Functions

      • Serge Lang

      Pages 3-20

   3. Graphs and Curves

      • Serge Lang

      Pages 21-54

3. Differentiation and Elementary Functions

   1. Front Matter

      Pages 55-55

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   2. The Derivative

      • Serge Lang

      Pages 57-116

   3. Sine and Cosine

      • Serge Lang

      Pages 117-158

   4. The Mean Value Theorem

      • Serge Lang

      Pages 159-180

   5. Sketching Curves

      • Serge Lang

      Pages 181-215

   6. Inverse Functions

      • Serge Lang

      Pages 216-235

   7. Exponents and Logarithms

      • Serge Lang

      Pages 236-283

4. Integration

   1. Front Matter

      Pages 285-285

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   2. Integration

      • Serge Lang

      Pages 287-311

   3. Properties of the Integral

      • Serge Lang

      Pages 312-334

   4. Techniques of Integration

      • Serge Lang

      Pages 335-378

   5. Applications of Integration

      • Serge Lang

      Pages 379-423

5. Taylor’s Formula and Series

   1. Front Matter

      Pages 425-425

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   2. Taylor’s Formula

      • Serge Lang

      Pages 427-472

   3. Series

      • Serge Lang

      Pages 473-500

6. Functions of Several Variables

   1. Front Matter

      Pages 521-521

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The purpose of a first course in calculus is to teach the student the basic notions of derivative and integral, and the basic techniques and applica­ tions which accompany them. The very talented students, with an ob­ vious aptitude for mathematics, will rapidly require a course in functions of one real variable, more or less as it is understood by professional is not primarily addressed to them (although mathematicians. This book I hope they will be able to acquire from it a good introduction at an early age). I have not written this course in the style I would use for an advanced monograph, on sophisticated topics. One writes an advanced monograph for oneself, because one wants to give permanent form to one's vision of some beautiful part of mathematics, not otherwise ac­ cessible, somewhat in the manner of a composer setting down his sym­ phony in musical notation. This book is written for the students to give them an immediate, and pleasant, access to the subject. I hope that I have struck a proper com­ promise, between dwelling too much on special details and not giving enough technical exercises, necessary to acquire the desired familiarity with the subject. In any case, certain routine habits of sophisticated mathematicians are unsuitable for a first course. Rigor. This does not mean that so-called rigor has to be abandoned.

• calculus
• derivative
• differential equation
• logarithm
• mean value theorem

• Department of Mathematics, Yale University, New Haven, USA

  Serge Lang

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