Calculus with Early Transcendentals

by Paul Sisson and Tibor Szarvas

A comprehensive, mathematically rigorous exposition, Calculus with Early Transcendentals blends precision and depth with a conversational tone to include the reader in developing the ideas and intuition of calculus. A consistent focus on historical context, theoretical discovery, and extensive exercise sets provides insight into the many applications and inherent beauty of the subject.

Formats: Textbook, Software (Beta)

Product ISBN
Textbook 978-1-935782-21-6
Courseware (Beta) 978-1-941552-30-8
Textbook + Courseware (Beta) 978-1-946158-44-4

About the Authors:

Paul Sisson

Paul Sisson

Paul Sisson received his bachelor's degree in mathematics and physics from New Mexico Tech and his PhD from the University of South Carolina. Since then, he has taught a wide variety of math and computer science courses, including Intermediate Algebra, College Algebra, Calculus, Topology, Mathematical Art, History of Mathematics, Real Analysis, Mathematica Programming, and Network Operating Systems. He is Professor of Mathematics and Provost Emeritus at Louisiana State University in Shreveport.


Tibor Szarvas

Tibor Szarvas

Tibor Szarvas received his master's degree in mathematics from the University of Szeged, Hungary, and his PhD from the University of South Carolina. His teaching experience in mathematics includes courses such as Intermediate Algebra, Liberal Arts Mathematics, Mathematics for Elementary Teachers, Precalculus, Calculus, Advanced Calculus, Discrete Mathematics, Differential Equations, College Geometry, Mathematical Modeling, Linear Algebra, Abstract Algebra, Topology, Real Analysis, Complex Variables, Number Theory, and Mathematical Logic. He is currently serving as Professor and Chair of Mathematics at Louisiana State University in Shreveport.


Resources

  1. Formulas – Algebra, Geometry, Conic Sections, Limits, Derivatives, Integration, Sequences and Series
  2. Table of Integrals
  3. Appendices
    1. Fundamentals of Mathematica
    2. Properties of Exponents and Logarithms, Graphs of Exponential and Logarithmic Functions
    3. Trigonometric and Hyperbolic Functions
    4. Complex Numbers
    5. Proofs of Selected Theorems

Table of Contents

  1. A Function Primer

    1. Functions and How We Represent Them
    2. A Function Repertory
    3. Transforming and Combining Functions
    4. Inverse Functions
    5. Calculus, Calculators, and Computer Algebra Systems
  2. Limits and the Derivative

    1. Rates of Change and Tangents
    2. Limits All Around the Plane
    3. The Mathematical Definition of Limit
    4. Determining Limits of Functions
    5. Continuity
    6. Rate of Change Revisited: The Derivative
  3. Differentiation

    1. Differentiation Notation and Consequences
    2. Derivatives of Polynomials, Exponentials, Products, and Quotients
    3. Derivatives of Trigonometric Functions
    4. The Chain Rule
    5. Implicit Differentiation
    6. Derivatives of Inverse Functions
    7. Rates of Change in Use
    8. Related Rates
    9. Linearization and Differentials
  4. Applications of Differentiation

    1. Extreme Values of Functions
    2. The Mean Value Theorem
    3. The First and Second Derivative Tests
    4. L'Hôpital's Rule
    5. Calculus and Curve Sketching
    6. Optimization Problems
    7. Antiderivatives
  5. Integration

    1. Area, Distance, and Riemann Sums
    2. The Definite Integral
    3. The Fundamental Theorem of Calculus
    4. Indefinite Integrals and the Substitution Rule
    5. The Substitution Rule and Definite Integration
  6. Applications of the Definite Integral

    1. Finding Volumes Using Slices
    2. Finding Volumes Using Cylindrical Shells
    3. Arc Length and Surface Area
    4. Moments and Centers of Mass
    5. Force, Work, and Pressure
    6. Hyperbolic Functions
  7. Techniques of Integration

    1. Integration by Parts
    2. The Partial Fractions Method
    3. Trigonometric Integrals
    4. Trigonometric Substitutions
    5. Integration Summary and Integration Using Computer Algebra Systems
    6. Numerical Integration
    7. Improper Integrals
  8. Differential Equations

    1. Separable Differential Equations
    2. First-Order Linear Differential Equations
    3. Autonomous Differential Equations and Slope Fields
    4. Second-Order Differential Equations
  9. Parametric Equations and Polar Coordinates

    1. Parametric Equations
    2. Calculus and Parametric Equations
    3. Polar Coordinates
    4. Calculus in Polar Coordinates
    5. Conic Sections in Cartesian Coordinates
    6. Conic Sections in Polar Coordinates
  10. Sequences and Series

    1. Sequences
    2. Infinite Series
    3. The Integral Test
    4. Comparison Tests
    5. The Ratio and Root Tests
    6. Absolute and Conditional Convergence
    7. Power Series
    8. Taylor and Maclaurin Series
    9. Further Applications of Series
  11. Chapter 11: Vectors and the Geometry of Space

    1. Three-Dimensional Cartesian Space
    2. Vectors and Vector Algebra
    3. The Dot Product
    4. The Cross Product
    5. Describing Lines and Planes
    6. Cylinders and Quadric Surfaces
  12. Chapter 12: Vector Functions

    1. Vector-Valued Functions
    2. Arc Length and the Unit Tangent Vector
    3. The Unit Normal and Binormal Vectors, Curvature, and Torsion
    4. Planetary Motion and Kepler's Laws
  13. Chapter 13: Partial Derivatives

    1. Functions of Several Variables
    2. Limits and Continuity of Multivariable Functions
    3. Partial Derivatives
    4. The Chain Rule
    5. Directional Derivatives and Gradient Vectors
    6. Tangent Planes and Differentials
    7. Extreme Values of Multivariable Functions
    8. Lagrange Multipliers
  14. Chapter 14: Multiple Integrals

    1. Double Integrals
    2. Applications of Double Integrals
    3. Double Integrals in Polar Coordinates
    4. Triple Integrals
    5. Triple Integrals in Cylindrical and Spherical Coordinates
    6. Substitutions and Multiple Integrals
  15. Chapter 15: Vector Calculus

    1. Vector Fields
    2. Line Integrals
    3. The Fundamental Theorem for Line Integrals
    4. Green's Theorem
    5. Parametric Surfaces and Surface Area
    6. Surface Integrals
    7. Stokes' Theorem
    8. The Divergence Theorem