Now available in a low-priced paperback edition! Written by one of the foremost mathematicians of the 20th century, this text remains the only modern treatment to successfully integrate principles of analysis into first-year calculus. Further, Courant's treatment introduces the differential and integral calculus simultaneously, emphasizing the central point of the calculus, namely, the connection between definite integral, indefinite integral, and derivative. Exposition exhibits the close connection between analysis and its applications, making this text appropriate for students of mathematics, or of science and engineering. Courant makes the subject easier to grasp by giving proofs step-by-step, and by developing the intuition that gave rise to the calculus and guides its use today.
Partial table of contents:
The Continuum of Numbers, The Concept of Function, The Concept of the Limit of a Sequence, The Concept of Continuity.
The Fundamental Ideas of the Integral and Differential Calculus: The Definite Integral, The Derivative, The Estimation of Integrals and the Mean Value Theorem of the Integral Calculus.
Differentiation and Integration of the Elementary Functions: Maxima and Minima, The Logarithm and the Exponential Function, The Hyperbolic Functions.
Further Development of the Integral Calculus: The Method of Substitution, Integration by Parts, Integration of Rational Functions, Improper Integrals.
Taylor's Theorem and the Approximate Expression of Functions by Polynomials.
Infinite Series and Other Limiting Processes.
A Sketch of the Theory of Functions of Several Variables.
The Differential Equations for the Simplest Types of Vibration.
Answers and Hints.