Chapter 7 Infinite Series（1）
7.1 Series（1）
Exercises 7.1（5）
7.2 Series with Positive Terms（7）
7.2.1 The Comparison Tests（7）
7.2.2 The Root and Ratio Tests（11）
Exercises 7.2（14）
7.3 Alternating Series and Absolute Convergence（15）
7.3.1 Alternating Series （15）
7.3.2 Absolute Convergence（18）
Exercises 7.3（19）
7.4 Power Series（20）
Exercises 7.4（26）
7.5 Differentiation and Integration of Power Series（27）
Exercises 7.5（30）
7.6 Taylor Series（31）
7.6.1 The Taylor Polynomials at x=0 or Maclaurin Polynomials（31）
7.6.2 The Taylors seriesor Maclaurin series for function f at 0 （32）
7.6.3 The Taylors series for function f at a an arbitrary real number（33）
Exercises 7.6（38）
Chapter 8 Partial Derivatives and Double Integrals（39）
8.1 Functions of Two Variables（39）
Exercises 8.1（45）
8.2 Limits and Continuity（45）
8.2.1 Limits（45）
8.2.2 Continuity（48）
Exercises 8.2（50）
8.3 Partial Derivatives（51）
8.3.1 Definition（51）
8.3.2 Economical Interpretations of Partial Derivatives（55）
8.3.3 Geometric Interpretations of Partial Derivatives（56）
Exercises 8.3（57）
8.4 Strategy for Finding Partial Derivatives（58）
8.4.1 The Chain Rule（58）
8.4.2 Implicit Differentiation（62）
8.4.3 Higher Derivatives（64）
Exercises 8.4（66）
8.5 Total Differentials（68）
8.5.1 Definition（68）
8.5.2 Relations between Continuity, Partial Derivatives, and Differentiability（69）
8.5.3 Rules for Finding Total Differentials（70）
8.5.4 The Invariance of First Order Total Differential Form（71）
Exercises 8.5（73）
8.6 Extremum of Functions of Two Variables（74）
8.6.1 Locating Maxima and Minima（74）
8.6.2 Methods of Finding Absolute Maxima and Minima（78）
8.6.3 Methods of Finding Conditional Extremum（79）
Exercises 8.6（82）
8.7 Directional Derivatives and The Gradient Vector（83）
8.7.1 Vectors and Vector Operations（83）
8.7.2 Directional Derivatives and The Gradient Vector（85）
8.7.3 The Relation between Directional Derivatives and The Gradient Vector（88）
Exercises 8.7（90）
8.8 Double Integrals（91）
8.8.1 Definition and Properties（91）
8.8.2 Double Integrals in Rectangular Coordinates（94）
8.8.3 Polar Coordinates（102）
8.8.4 Double Integrals in Polar Coordinates（106）
8.8.5 Application of Double Integrals（108）
Exercises 8.8（109）
Chapter 9 Differential Equations（112）
9.1 Introduction（112）
Exercises 9.1（114）
9.2 FirstOrder Linear Differential Equations（114）
9.2.1 Separable Equations（115）
9.2.2 Homogeneous Differential Equations（117）
9.2.3 FirstOrder Linear Differential Equations（118）
9.2.4 Total or Exact Differential Equations（121）
9.2.5 Bernoulli EquationsEquations reducible to a linear one（123）
9.2.6 Euler Equations（124）
Exercises 9.2（126）
9.3 Secondorder Differential Equations（127）
9.3.1 Reducible SecondOrder Differential Equations（127）
9.3.2 Complex Numbers （129）
9.3.3 Homogeneous Linear Equations（133）
9.3.4 Nonhomogeneous Linear Equations（137）
Exercises 9.3（142）
This book is intended to cover infinite series（Chapter 7）, as well as
partial derivatives and double integrals（Chapter 8）, differential equations（Chapter 9）
and difference equations（Chapter 10）. The position of Chapter 7 is rather arbitrary.
Chapter 8 contains necessary background on vectors and geometry in 3space as well as
a bit of linear algebra that is useful, though not absolutely essential, for the
understanding of subsequent multivariable material.
The author has tried to write a textbook that is thoroughly modern and teachable,
and the capacity and needs of the student pursuing a first course in the Calculus have
been kept constantly in mind.
The text is designed for general calculus courses, especially those for business,
economics and science students.
Most of the material requires only a reasonable background in high school algebra and
analytic geometry.This book contains topics from which a selection naturally would be made
in preparing students for elementary work in applied science.We choose such subjects as
best suit the needs of our classes.
This book offers simple practical problems that illustrate the theory and at the same
time are of interest to the student.These problems do not presuppose an extended knowledge
in any particular branch of science,but are based on knowledge that all students of the
Calculus are supposed to have in common.
The expunging of errors and obscurities in a text is an ongoing and asymptotic process.
We will be grateful to any readers who call our attention, or give us any other
suggestions for future improvements.
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