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 Taylor’s series(or Maclaurin series) for function f at 0 （32） 
7.6.3 The Taylor’s 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 First�睴rder Linear Differential Equations（114） 
9.2.1 Separable Equations（115） 
9.2.2 Homogeneous Differential Equations（117） 
9.2.3 First�睴rder Linear Differential Equations（118） 
9.2.4 Total (or Exact) Differential Equations（121） 
9.2.5 Bernoulli Equations(Equations reducible to a linear one)（123） 
9.2.6 Euler Equations（124） 
Exercises 9.2（126） 
9.3 Second�瞣rder Differential Equations（127） 
9.3.1 Reducible Second�睴rder 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） 


Chapter 10 Difference Equations（143） 
10.1 Introduction （143） 
10.1.1 Definition（143） 
10.1.2 Properties（144） 
Exercises 10.1（147） 
10.2 Linear Difference Equations（147） 
10.2.1 nth�睴rder Difference Equations（147） 
10.2.2 First�睴rder Difference Equations（149） 
10.2.3 Second�睴rder Difference Equations（156） 
Exercises 10.2（161）