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Book Description

This book introduces integrals, the fundamental theorem of calculus, initial value problems, and Riemann sums.

It introduces properties of polynomials, including roots and multiplicity, and uses them as a framework for introducing additional calculus concepts including Newton's method, L'Hôpital's Rule, and Rolle's theorem. Both the differential and integral calculus of parametric, polar, and vector functions are introduced. The book concludes with a survey of methods of integration, including u-substitution, integration by parts, special trigonometric integrals, trigonometric substitution, and partial fractions.

Table of Contents

  1. Preface
  2. Acknowledgments
  3. Integration, Area, and Initial Value Problems
    1. Anti-derivatives
    2. The Fundamental Theorem (1/3)
    3. The Fundamental Theorem (2/3)
    4. The Fundamental Theorem (3/3)
      1. Even and Odd Functions (1/2)
      2. Even and Odd Functions (2/2)
    5. Initial Value Problems (1/2)
    6. Initial Value Problems (2/2)
    7. Induction and Sums of Rectangles (1/2)
    8. Induction and Sums of Rectangles (2/2)
  4. Parametric, Polar, and Vector Functions
    1. Parametric Functions
      1. The Derivative of a Parametric Curve
    2. Polar Coordinates (1/2)
    3. Polar Coordinates (2/2)
      1. Polar Calculus (1/2)
      2. Polar Calculus (2/2)
    4. Vector Functions (1/2)
    5. Vector Functions (2/2)
      1. Calculus with Vector Curves (1/2)
      2. Calculus with Vector Curves (2/2)
  5. The Arithmetic, Geometry, and Calculus of Polynomials
    1. Polynomial Arithmetic
    2. Qualitative Properties of Polynomials
      1. Multiplicity of Roots
    3. L'Hôpital's Rule; Strange Polynomials (1/2)
    4. L'Hôpital's Rule; Strange Polynomials (2/2)
      1. Strange Polynomials (1/2)
      2. Strange Polynomials (2/2)
  6. Methods of Integration I
    1. u-Substitution
      1. Substitution in Definite Integrals
    2. Integration by Parts (1/2)
    3. Integration by Parts (2/2)
    4. Integrating Trig Functions (1/2)
    5. Integrating Trig Functions (2/2)
  7. Methods of Integration II
    1. Trigonometric Substitution
    2. Partial Fractions (1/3)
    3. Partial Fractions (2/3)
    4. Partial Fractions (3/3)
    5. Practicing Integration
  8. Useful Formulas
    1. Powers, Logs, and Exponentials
    2. Trigonometric Identities
    3. Speed of Function Growth
    4. Derivative Rules
    5. Vector Arithmetic
    6. Polar and Rectangular Conversion
    7. Integral Rules
  9. Author's Biography
  10. Index
  11. Blank Page (1/3)
  12. Blank Page (2/3)
  13. Blank Page (3/3)
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