This textbook strengthens students' computational skills, teaches them how to prove theorems, and helps them understand proofs. Designed for junior and senior undergraduates, the book includes historical background for many concepts and theorems. It begins with the calculus of one variable and progresses to multivariable calculus in the second half of the book. An introductory section in each chapter reviews the material covered in earlier calculus courses, easing students into the more serious material. A solutions manual is available with qualifying course adoption.

Sequences and Their Limits Computing the Limits Definition of the Limit Properties of Limits Monotone Sequences The Number e Cauchy Sequences Limit Superior and Limit Inferior Computing the Limits-Part II Real Numbers The Axioms of the Set R Consequences of the Completeness Axiom Bolzano-Weierstrass Theorem Some Thoughts about R Continuity Computing Limits of Functions A Review of Functions Continuous Functions: A Geometric Viewpoint Limits of Functions Other Limits Properties of Continuous Functions The Continuity of Elementary Functions Uniform Continuity Two Properties of Continuous Functions The Derivative Computing the Derivatives The Derivative Rules of Differentiation Monotonicity. Local Extrema Taylor's Formula L'Hopital's Rule The Indefinite Integral Computing Indefinite Integrals The Antiderivative The Definite Integral Computing Definite Integrals The Definite Integral Integrable Functions Riemann Sums Properties of Definite Integrals The Fundamental Theorem of Calculus Infinite and Improper Integrals Infinite Series A Review of Infinite Series Definition of a Series Series with Positive Terms The Root and Ratio Tests Series with Arbitrary Terms Sequences and Series of Functions Convergence of a Sequence of Functions Uniformly Convergent Sequences of Functions Function Series Power Series Power Series Expansions of Elementary Functions Fourier Series Introduction Pointwise Convergence of Fourier Series The Uniform Convergence of Fourier Series Cesaro Summability Mean Square Convergence of Fourier Series The Influence of Fourier Series Functions of Several Variables Subsets of Rn Functions and Their Limits Continuous Functions Boundedness of Continuous Functions Open Sets in Rn The Intermediate Value Theorem Compact Sets Derivatives Computing Derivatives Derivatives and Differentiability Properties of the Derivative Functions from Rn to Rm Taylor's Formula Extreme Values Implicit Functions and Optimization Implicit Functions Derivative as a Linear Map Open Mapping Theorem Implicit Function Theorem Constrained Optimization The Second Derivative Test Integrals Depending on a Parameter Uniform Convergence The Integral as a Function Uniform Convergence of Improper Integrals Integral as a Function Some Important Integrals Integration in Rn Double Integrals over Rectangles Double Integrals over Jordan Sets Double Integrals as Iterated Integrals Transformations of Jordan Sets in R2 Change of Variables in Double Integrals Improper Integrals Multiple Integrals Fundamental Theorems Curves in Rn Line Integrals Green's Theorem Surface Integrals The Divergence Theorem Stokes' Theorem Differential Forms on Rn Exact Differential Forms on Rn Solutions and Answers to Selected Problems Bibliography Subject Index Author Index