Calculus

The following sections are included:

• Preface

• Contents

The following sections are included:

• Set Notation

• Completeness Of The Real Numbers

• A Few Algebraic Conventions And Techniques

• The Circular Arc Subtended By An Angle

• The Trigonometric Functions

• Exercises

• Parabolas, Ellipses, And Hyperbolas
  • The Parabola
  • The Ellipse
  • The Hyperbola

• Exercises

• The Complex Numbers

• Exercises

• Solving Systems Of Equations
  • Gauss Elimination
  • Row Reduced Echelon Form

• Exercises

The following sections are included:

• Functions And Sequences

• Exercises

• Continuous Functions

• Sufficient Conditions For Continuity

• Continuity Of Circular Functions

• Exercises

• Properties Of Continuous Functions

• Exercises

• Limit Of A Function

• Exercises

• The Limit Of A Sequence
  • Sequences And Completeness
  • Decimals
  • Continuity And The Limit Of A Sequence

• Exercises

• Uniform Continuity, Sequential Compactness

• Exercises

• Fundamental Theory
  • Combinations Of Functions And Sequences
  • The Intermediate Value Theorem
  • Extreme Value Theorem, Nested Intervals
  • Sequential Compactness Of Closed Intervals
  • Different Versions Of Completeness

• Exercises

The following sections are included:

• Velocity

• The Derivative

• Exercises

• Local Extrema

• Exercises

• Mean Value Theorem

• Exercises

• Curve Sketching

• Exercises

The following sections are included:

• The Circular Functions

• Exercises

• The Exponential And Log Functions
  • The Rules Of Exponents
  • The Exponential Functions, A Wild Assumption
  • The Special Number e
  • The Function ln |x|
  • Logarithm Functions

• Exercises

The following sections are included:

• The Chain Rule And Derivatives Of Inverse Functions
  • The Chain Rule
  • Implicit Differentiation

• Exercises

• The Function x^r For r A Real Number
  • Logarithmic Differentiation

• Exercises

• The Inverse Trigonometric Functions

• The Hyperbolic And Inverse Hyperbolic Functions

• Exercises

The following sections are included:

• L'Hôpital's Rule
  • Interest Compounded Continuously

• Exercises

• Related Rates

• Exercises

• The Derivative And Optimization

• Exercises

• The Newton Raphson Method

• Exercises

• Review Exercises

The following sections are included:

• Initial Value Problems

• The Method Of Substitution

• Exercises

• Integration By Parts

• Exercises

• Trig. Substitutions

• Exercises

• Partial Fractions
  • Rules For Finding Partial Fractions Expansion Of A Rational Function

• Rational Functions Of Trig. Functions

• Exercises

• Practice Problems For Antiderivatives

• Computers And Antiderivatives

The following sections are included:

• Areas

• Area Between Graphs

• Exercises

• Volumes
  • Volumes Using Cross Sections
  • Volumes Using Shells

• Exercises

• Lengths Of Curves And Areas Of Surfaces Of Revolution
  • Lengths
  • Surfaces Of Revolution

• Exercises

• Force On A Dam And Work
  • Force On A Dam
  • Work

• Exercises

The following sections are included:

• The Equation y′ + a(t) y = b(t)

• Separable Differential Equations

• Exercises

• The Equations Of Undamped And Damped Oscillation

• Exercises

• Review Exercises

The following sections are included:

• Upper And Lower Sums

• Exercises

• Functions Of Riemann Integrable Functions

• Properties Of The Integral

• Fundamental Theorem Of Calculus

• The Riemann Integral

• Exercises

• Return Of The Wild Assumption

• Exercises

• Techniques Of Integration
  • The Method Of Substitution
  • Integration By Parts

• Exercises

• Improper Integrals

• Exercises

The following sections are included:

• Approximation By Taylor Polynomials

• Exercises

• Infinite Series Of Numbers
  • Basic Considerations

• Exercises

• More Tests For Convergence
  • Convergence Because Of Cancelation
  • Ratio And Root Tests

• Double Series

• Exercises

• Power Series
  • Functions Defined In Terms Of Series
  • Operations On Power Series

• Exercises

• Some Other Theorems

The following sections are included:

• ℝⁿ

• Algebra in ℝⁿ

• Geometric Meaning Of Vector Addition In ℝ³

• Lines

• Distance in ℝⁿ

• Geometric Meaning Of Scalar Multiplication In ℝ³

• Exercises

• Physical Vectors

• Exercises

The following sections are included:

• The Dot Product

• The Geometric Significance Of The Dot Product
  • The Angle Between Two Vectors
  • Work And Projections
  • The Parabolic Mirror, An Application
  • The Dot Product And Distance In ℙⁿ

• Exercises

• The Cross Product
  • The Distributive Law For The Cross Product
  • Torque
  • Center Of Mass
  • Angular Velocity
  • The Box Product
    • An Alternate Proof Of The Distributive Law

• Vector Identities And Notation

• Exercises

• Planes

• Quadric Surfaces

• Exercises

The following sections are included:

• Polar Coordinates
  • Graphs In Polar Coordinates

• The Area In Polar Coordinates

• Exercises

• The Acceleration In Polar Coordinates

• Planetary Motion
  • The Equal Area Rule, Kepler's Second Law
  • Inverse Square Law, Kepler's First Law
  • Kepler's Third Law

• Exercises

• Spherical And Cylindrical Coordinates

• Exercises

The following sections are included:

• Appendix A Basic Plane Geometry

• Appendix B The Fundamental Theorem Of Algebra

• Appendix C Newton's Laws Of Motion

• Bibliography

• Appendix D Answers To Selected Exercises

• Index