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Chapter 1 Real number system and functions 1

1.1 Real number system 1

1.2 Inequalities 11

1.3 Functions 15

Chapter 2 Sequence Limit 40

2.1 Concept of sequence limit 40

2.2 Properties of convergent sequences 44

2.3 Fundamental theorems of sequenge limit 54

2.4 Upper limit and lower limit of a sequence 78

Chapter 3 Function limits and continuity 91

3.1 Concept of function limits 91

3.2 Properties of function limits 99

3.3 Two important limits 110

3.4 Infinitesimal and infinity 113

3.5 Concept of continuity 121

3.6 Properties of continuous functions 133

3.7 Continuity of primary functions 141

3.8 Uniform continuity 144

Chapter 4 Derivatives and differentials 168

4.1 Concept of derivatives 168

4.2 Computation of derivatives 183

4.3 Differentials 201

4.4 Derivatives and differentials of higher order 207

Chapter 5 Mean value theorems' and applications of derivative 226

5.1 Mean value theorems 226

5.2 Monotony and extremum of functions 256

5.3 Graph of a function 265

5.4 L'Hospital-rules 274

5.5 Newton-Raphson method 280

Chapter 6 Indefinite integrals 292

6.1 Concept of indefinite integrals and fundamental formulas 292

6.2 Techniques of integration 298

6.3 Integration of some special kinds of functions 309

Chapter 7 Definite integrals 330

7.1 Concept of definite integrals 330

7.2 Properties of definite integrals 348

7.3 The fundamental theorems of calculus 357

7.4 Integration techniques of definite integrals 367

7.5 Improper integrals 375

7.6 Numerical integration 405

Chapter 8 Applications of definite integrals 421

8.1 Applications in geometry 421

8.2 Applications in physics 439

Chapter 9 Preliminary of differential equations 465

9.1 Basic concepts of differential equations 465

9.2 Differential equations of first-order 467

9.3 Degrading method of second-order differential equations 482

9.4 Linear differential equations of second-order 487

9.5 Second-order linear equations with constant coefficients 495

9.6 Euler Equation 506