1 Basic notions 1 --
1.1 Sets 1 --
1.2 Elements of mathematical logic 5 --
1.2.1 Connectives 5 --
1.2.2 Predicates 6 --
1.2.3 Quantifiers 7 --
1.3 Sets of numbers 8 --
1.3.1 The ordering of real numbers 12 --
1.3.2 Completeness of R 17 --
1.4 Factorials and binomial coefficients 18 --
1.5 Cartesian product 21 --
1.6 Relations in the plane 23 --
2 Functions 31 --
2.1 Definitions and first examples 31 --
2.2 Range and pre-image 36 --
2.3 Surjective and injective functions
inverse function 38 --
2.4 Monotone functions 41 --
2.5 Composition of functions 43 --
2.5.1 Translations, rescalings, reflections 45 --
2.6 Elementary functions and properties 47 --
2.6.1 Powers 48 --
2.6.2 Polynomial and rational functions 50 --
2.6.3 Exponential and logarithmic functions 50 --
2.6.4 Trigonometric functions and inverses 51 --
3 Limits and continuity I 65 --
3.1 Neighbourhoods 65 --
3.2 Limit of a sequence 66 --
3.3 Limits of functions
continuity 72 --
3.3.1 Limits at infinity 72 --
3.3.2 Continuity. Limits at real points 74 --
3.3.3 One-sided limits
points of discontinuity 82 --
3.3.4 Limits of monotone functions 85 --
4 Limits and continuity II 89 --
4.1 Theorems on limits 89 --
4.1.1 Uniqueness and sign of the limit 89 --
4.1.2 Comparison theorems 91 --
4.1.3 Algebra of limits. Indeterminate forms of algebraic type 96 --
4.1.4 Substitution theorem 102 --
4.2 More fundamental limits. Indeterminate forms of exponential type 105 --
4.3 Global features of continuous maps 108 --
5 Local comparison of functions. Numerical sequences and series 123 --
5.1 Landau symbols 123 --
5.2 Infinitesimal and infinite functions 130 --
5.3 Asymptotes 135 --
5.4 Further properties of sequences 137 --
5.5 Numerical series 141 --
5.5.1 Positive-term series 146 --
5.5.2 Alternating series 149 --
6 Differential calculus 167 --
6.1 The derivative 167 --
6.2 Derivatives of the elementary functions. Rules of differentiation 170 --
6.3 Where differentiability fails 175 --
6.4 Extrema and critical points 178 --
6.5 Theorems of Rolle and of the Mean Value 181 --
6.6 First and second finite increment formulas 183 --
6.7 Monotone maps 185 --
6.8 Higher-order derivatives 187 --
6.9 Convexity and inflection points 189 --
6.9.1 Extension of the notion of convexity 192 --
6.10 Qualitative study of a function 193 --
6.10.1 Hyperbolic functions 195 --
6.11 The Theorem of de l'Hopital 197 --
6.11.1 Applications of de l'Hopital's theorem 199 --
7 Taylor expansions and applications 223 --
7.1 Taylor formulas 223 --
7.2 Expanding the elementary functions 227 --
7.3 Operations on Taylor expansions 234 --
7.4 Local behaviour of a map via its Taylor expansion 242 --
8 Geometry in the plane and in space 257 --
8.1 Polar, cylindrical, and spherical coordinates 257 --
8.2 Vectors in the plane and in space 260 --
8.2.1 Position vectors 260 --
8.2.2 Norm and scalar product 263 --
8.2.3 General vectors 268 --
8.3 Complex numbers 269 --
8.3.1 Algebraic operations 270 --
8.3.2 Cartesian coordinates 271 --
8.3.3 Trigonometric and exponential form 273 --
8.3.4 Powers and nth roots 275 --
8.3.5 Algebraic equations 277 --
8.4 Curves in the plane and in space 279 --
8.5 Functions of several variables 284 --
8.5.1 Continuity 284 --
8.5.2 Partial derivatives and gradient 286 --
9 Integral calculus I 299 --
9.1 Primitive functions and indefinite integrals 300 --
9.2 Rules of indefinite integration 304 --
9.2.1 Integrating rational maps 310 --
9.3 Definite integrals 317 --
9.4 The Cauchy integral 318 --
9.5 The Riemann integral 320 --
9.6 Properties of definite integrals 326 --
9.7 Integral mean value 328 --
9.8 The Fundamental Theorem of integral calculus 331 --
9.9 Rules of definite integration 335 --
9.9.1 Application: computation of areas 337 --
10 Integral calculus II 355 --
10.1 Improper integrals 355 --
10.1.1 Unbounded domains of integration 355 --
10.1.2 Unbounded integrands 363 --
10.2 More improper integrals 367 --
10.3 Integrals along curves 368 --
10.3.1 Length of a curve and arc length 373 --
10.4 Integral vector calculus 376 --
11 Ordinary differential equations 387 --
11.1 General definitions 387 --
11.2 First order differential equations 388 --
11.2.1 Equations with separable variables 392 --
11.2.2 Linear equations 394 --
11.2.3 Homogeneous equations 397 --
11.2.4 Second order equations reducible to first order 398 --
11.3 Initial value problems for equations of the first order 399 --
11.3.1 Lipschitz functions 399 --
11.3.2 A criterion for solving initial value problems 402 --
11.4 Linear second order equations with constant coefficients 404.