Single Variable Calculus

Yinzhu Zou, Sichuan University, Chengdu, China

• Chapter 1 Prerequisites for Calculus1.1 Overview of Calculus1.2 Sets and Numbers1.3 Functions1.4 ExercisesChapter 2 Limits and Continuity2.1 Rates of Change and Derivatives2.2 Limits of a Function2.3 Limits of Sequences2.4 Squeeze Theorem and Cauchy's Theorem2.5 Infinitesimal Functions and Asymptotic Functions2.6 Continuous and Discontinuous Functions2.7 Some Proofs in Chapter 22.8 ExercisesChapter 3 The Derivative3.1 Derivative of a Function at a Point3.2 Derivative as a Function3.3 Derivative Laws3.4 Derivative of an Inverse Function3.5 Differentiating a Composite Function - The Chain Rule3.6 Derivatives of Higher Orders3.7 Implicit Differentiation3.8 Functions Defined by Parametric and Polar Equations3.9 Related Rates of Change3.10 The Tangent Line Approximation and the Differential3.11 Derivative Rules-Summar3.12 ExercisesChapter 4 Applications of the Derivative4.1 Extreme Values and The Candidate Theorem4.2 The Mean Value Theorem4.3 Monotonic Functions and The First Derivative Test4.4 Extended Mean Value Theorem and the L'opital's Rules4.5 Taylor's Theorem4.6 Concave Functions and The Second Derivative Test4.7 Extreme Values of Functions Revisited4.8 Curve Sketching4.9 Solving Equations Numerically4.10 Curvatures and the Differential of the Arc LengthChapter 5 The Definite Integral5.1 Definite Integrals and Properties5.2 The Fundamental Theorem of Calculus5.3 Numerical lntegration5.4 ExercisesChapter 6 Techniques for Integration and Improper Integrals6.1 Indefinite Integrals6.2 Substitution in Definite Integrals6.3 Integration by Parts in Definite Integrals6.4 lmproper Integrals6.5 ExercisesChapter 7 Applications of the Definite Integral7.1 Areas Volumes and Arc Lengths7.2 Applications in Other Disciplines7.3 ExercisesChapter 8 Infinite Series, Sequences, and Approximations8.1 Infinite Sequences8.2 Infinite Series8.3 Tests for Convergence8.4 Power Series and Taylor Series8.5 Fourier Series8.6 Exercises