Pages: 408

Trade Paper: ISBN 9788885486034; $69.95

Pdf: ISBN 9788885486041; $34,99

Authors: Lorenzo Peccati, Sandro Salsa, Annamaria Squellati

### Table of Contents

** **

**Preface**

**Structure of the book**

**1. Numbers**

1.1 Natural and relative integers

1.1.1 Natural integers

1.1.2 Relative integers

1.2 Rational numbers

1.3 Real numbers

1.4 Sum of terms in a progression

1.4.1 The summation symbol

1.4.2 Sum of terms in an arithmetic progression

1.4.3 Sum of terms in a geometric progression

1.5 An outline of set theory

1.5.1 Sets

1.5.2 Relations and operations with sets

1.5.3 Cartesian product

1.6 Sets of real numbers

1.6.1 Maximum and minimum of a set

1.7 The cartesian plane

1.8 Howmany elements in a set?

1.8.1 Finite set. Combinatorics

1.8.2 Infinite sets. Countability, power of the continuum

1.9 Exercises

**2. Functions**

2.1 The concept of function

2.2 Sequences

2.2.1 Recursive sequences

2.2.2 Geometric sequences

2.3 Linear functions

2.4 Quadratic and inverse proportionality

2.4.1 Quadratic functions

2.4.2 Inverse proportionality

2.5 Composite function. Inverse function

2.5.1 Composite function

2.5.2 Inverse function

2.6 Monotonic, bounded, convex functions

2.6.1 Bounded functions

2.6.2 Monotonic functions and sequences

2.6.3 Maximum and minimum values

2.6.4 Convex and concave functions

2.6.5 Local properties

2.7 Power function

2.8 Exponential, logarithmic, trigonometric functions

2.8.1 Exponential function

2.8.2 Logarithmic functions

2.8.3 Trigonometric functions

2.9 Geometric transformations

2.10 Exercises

**3. Limits**

3.1 Limits of sequences

3.1.1 Asymptotic properties of a sequence

3.1.2 Convergent sequences

3.1.3 Divergent sequences

3.1.4 Irregular sequences

3.1.5 Uniqueness of the limit

3.1.6 Limits of elementary sequences

3.2 Limits of functions

3.2.1 Right-hand limit

3.2.2 Left-hand limit. Limit

3.2.3 Limit as x → +∞, −∞

3.3 Existence of the limit

3.3.1 Limit of a monotonic sequence

3.3.2 Limit of a monotonic function

3.3.3 Limits of elementary functions

3.4 The number *e*

3.5 Calculation of limits

3.5.1 The set `RR`^{∗}

3.5.2 Limits and algebraic operations

3.5.3 Limits and inequalities

3.5.4 Change of variable

3.6 Comparisons

3.6.1 The symbols “o” and “∼”

3.6.2 The hierarchy of infinities

3.6.3 The hierarchy of infinitesimals

3.7 Exercises

**4. Continuity**

4.1 An intuitive idea of continuity

4.2 Continuous functions

4.2.1 Continuity of elementary functions

4.2.2 Discontinuities

4.3 Properties of continuous functions

4.4 Exercises

**5. Differential Calculus and Optimization**

5.1 Derivative and tangent line

5.1.1 Derivatives and continuity. Right and left derivatives

5.1.2 Interpretations of the derivative

5.2 Elementary formulae

5.3 Algebra of derivatives

5.4 Composite functions and inverse functions

5.4.1 The derivative of a composite function

5.4.2 The derivative of the inverse function

5.5 The differential

5.6 Elasticity and semi-elasticity

5.6.1 Elasticity

5.6.2 Logarithmic derivative or semi-elasticity

5.7 Optimization and stationary points

5.8 Lagrange’s mean value theorem

5.9 Monotonicity test

5.10 De l’Hospital’s theorem

5.11 Taylor’s formula

5.12 Test for convexity (or concavity)

5.13 Taylor’s formula of order *n*

5.14 Exercises

**6. Series**

6.1 The concept of series

6.2 Geometric series

6.3 The problem of convergence

6.3.1 A necessary condition for convergence

6.4 Series with non-negative terms

6.5 Series with terms of non-constant sign

6.5.1 Series with terms of alternate sign

6.6 Exercises

**7. Integral Calculus**

7.1 Introduction

7.2 The Riemann integral

7.3 Properties of the integral

7.3.1 Additivity, linearity, monotonicity

7.3.2 Mean value theorem

7.4 The Fundamental Theorem of Calculus

7.5 The indefinite integral

7.5.1 Linearity and the decomposition method

7.5.2 Integration by parts

7.5.3 Integration by substitution

7.6 Improper integrals

7.6.1 Preliminary considerations

7.6.2 Integrals over unbounded intervals

7.6.3 Bounded intervals and unbounded functions

7.6.4 Unbounded intervals and unbounded functions

7.6.5 Properties of improper integrals

7.7 Integrability criteria

7.8 Series and integrals

7.9 Integral functions

7.10 Exercises

**8. Vectors and Matrices**

8.1 Vectors in R^{n}

8.2 Operations with vectors

8.2.1 Linear combinations

8.3 Inner product of two vectors

8.3.1 Modulus, distance

8.4 Subspaces of R^{n}

8.5 Linear dependence

8.6 Bases and dimension of a subspace of R^{n}

8.7 Matrices

8.8 Operations with matrices

8.8.1 Sum of matrices and product of a matrix by a scalar

8.8.2 Product of matrices

8.8.3 Inverse matrix

8.9 The determinant

8.9.1 Properties of the determinant

8.10 Inverse matrix

8.11 Rank of a matrix

8.12 Exercises

**9. Linear Systems and Functions**

9.1 Linear systems

9.1.1 Elimination method

9.1.2 Linear systems and matrices

9.2 Systems with n equations and n unknowns

9.3 General systems

9.3.1 Solution scheme

9.4 Structure of the solutions

9.4.1 Homogeneous systems

9.4.2 Structure of the solutions of a linear system

9.5 Economic applications

9.6 Linear functions from R* ^{n}* to R

^{m}9.6.1 Image and kernel of a linear function

9.7 Exercises

**10. Multivariable Differential Calculus**

10.1 Introduction

10.1.1 Graph of two-variable functions

10.1.2 Level curves

10.2 Domain of a function

10.3 Global and local extrema

10.3.1 Concave and convex functions

10.4 Quadratic forms

10.5 Continuity

10.6 Partial derivatives

10.7 Differentiability and tangent plane

10.7.1 The chain rule

10.8 Implicit functions

10.9 Second order Taylor’s formula

10.9.1 Second derivatives and Hessian matrix

10.9.2 Second differential and second order Taylor’s formula

10.10 Functions of n variables

10.11 Optimization. Unconstrained extrema

10.11.1 Unconstrained and constrained extrema

10.11.2 First and second order conditions

10.12 Constrained extrema

10.12.1 Explicit constraint

10.12.2 Lagrange’s multipliers

10.12.3 Economic interpretation. Saddle points

10.12.4 Saddle points and multipliers for n-variable functions

10.13 Exercises

**11. Financial Calculus**

11.1 Accumulation and discount

11.2 Standard systems of financial laws

11.2.1 Simple interest and simple discount

11.2.2 Compound interests and compound discount

11.2.3 Bank discount and anticipated simple interests

11.2.4 Force of interest

11.3 Typical applications of compound interests

11.3.1 Simple annuities with constant instalments

11.3.2 Discounted Cash Flow

11.3.3 Amortization plans

11.3.4 A theoretical issue: decomposability

11.4 Exercises

**Index**