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Book Description

This book reviews the algebraic prerequisites of calculus, including solving equations, lines, quadratics, functions, logarithms, and trig functions.

It introduces the derivative using the limit-based definition and covers the standard function library and the product, quotient, and chain rules. It explores the applications of the derivative to curve sketching and optimization and concludes with the formal definition of the limit, the squeeze theorem, and the mean value theorem.

Table of Contents

  1. Preface
  2. Acknowledgments
  3. Review of Algebra
    1. Solving Equations
    2. Lines
      1. Computing Slopes, Finding Lines from Points
      2. Parallel and Orthogonal Lines (1/2)
      3. Parallel and Orthogonal Lines (2/2)
    3. Quadratic Equations
      1. Factoring a Quadratic
      2. Completing the Square
      3. The Quadratic Equation (1/2)
      4. The Quadratic Equation (2/2)
    4. Functions
      1. Domain and Range (1/2)
      2. Domain and Range (2/2)
  4. The Library of Functions
    1. Polynomials
    2. Powers, Logs, and Exponentials
      1. Powers and Roots
      2. Exponentials and Logs (1/2)
      3. Exponentials and Logs (2/2)
    3. Trigonometric Functions
      1. Graphs of the Basic Trig Functions
      2. Theorems about Triangles
      3. The Sum, Difference, and Double and Half-angle Identities
      4. Euler's Identity (1/2)
      5. Euler's Identity (2/2)
  5. Limits, Derivatives, Rules, and the Meaning of the Derivative
    1. Limits
      1. Split-rule Functions (1/2)
      2. Split-rule Functions (2/2)
    2. Derivatives
      1. Derivatives of Sums and Constant Multiples
    3. Derivatives of the Library of Functions
      1. Logs and Exponents
      2. The Trigonometric Functions
      3. Inverse Trigonometric Functions (1/2)
      4. Inverse Trigonometric Functions (2/2)
    4. The Product, Quotient, Reciprocal, and Chain Rules
      1. Functional Composition and the Chain Rule (1/2)
      2. Functional Composition and the Chain Rule (2/2)
    5. Physical Interpretation of Derivatives
      1. Implicit Derivatives
      2. Logarithmic Differentiation (1/2)
      3. Logarithmic Differentiation (2/2)
  6. Curve Sketching
    1. Limits at Infinity
    2. Information from the Derivative
      1. Increasing and Decreasing Ranges
      2. Concavity (1/2)
      3. Concavity (2/2)
    3. The Full Report for Curve Sketching (1/3)
    4. The Full Report for Curve Sketching (2/3)
    5. The Full Report for Curve Sketching (3/3)
  7. Optimization
    1. Optimization with Derivatives
      1. Optimization Story Problems (1/3)
      2. Optimization Story Problems (2/3)
      3. Optimization Story Problems (3/3)
  8. Limits and Continuity: The Details
    1. Limits and Continuity
    2. The Squeeze Theorem and the Mean Value Theorem (1/2)
    3. The Squeeze Theorem and the Mean Value Theorem (2/2)
  9. Useful Formulas
    1. Powers, Logs, and Exponentials
    2. Trigonometric Identities
    3. Speed of Function Growth
    4. Derivative Rules
  10. Author's Biography
  11. Index
  12. Blank Page (1/3)
  13. Blank Page (2/3)
  14. Blank Page (3/3)
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