Session1-slides

Course Overview

Session Information

  • Course Title: Mathematics 1 for Economics and Finance

  • Instructor: Emiliano Traversi

  • Program: ESSEC BBA Program - Fall 2024

Attendance and Grading

Attendance Policy

  • Mandatory attendance with a maximum of two unexcused absences allowed.

Grading Breakdown

  • Class Attendance and Quizzes: 40% of final grade

  • Midterm Exam: 30% of final grade

  • Final Exam: 50% of final grade

Course Objectives

  • Introduction to essential mathematical concepts:

    • Functions

    • Limits

    • Derivatives

    • Integrals

  • Understanding optimization as a core concept in decision-making.

  • Mastering financial calculus, crucial for corporate finance decision-making.

Teaching Materials

  • Resources:

    1. K. Sydsaeter et al., "Essential Mathematics for Economic Analysis, 5th Edition"

    2. A. Chiang and K. Wainwright, "Fundamental Methods of Mathematical Economics, 4th Edition"

    3. C.P. Simon and L. Blume, "Mathematics for Economists"

    4. T. Bradley, "Essential Mathematics for Economics and Business, 3rd Edition"

    5. E. Haeussler et al., "Introductory Mathematical Analysis for Business, Economics and the Life and Social Sciences"

    6. R.A. Brealey and S.G. Myers, "Principles of Corporate Finance"

    7. I. Jacques, "Mathematics for Economics and Business, 5th Edition"

Course Agenda

Summary of Sessions

  • Session 1: Functions of one variable and graphs

  • Sessions 2-3: Differentiation

  • Session 4: Optimization (univariate functions)

  • Session 5: Integration

  • Session 6: Counting and sum calculus

  • Session 7: Matrix algebra

  • Session 8: Determinants

  • Session 9: Inverse matrices

  • Session 10: Systems of linear inequalities

Sets and Functions

Basic Definitions

  • Set: A collection of objects, with elements denoted as a ∈ S.

    • Example Sets:

      • Positive integers: {1, 2, 3, ...}

      • Prime numbers: {2, 3, 5, 7, 11, ...}

  • Common Sets in Course:

    • Natural numbers (N)

    • Integer numbers (Z)

    • Rational numbers (Q)

    • Real numbers (R)

    • Irrational numbers (R \ Q)

    • Complex numbers (C)

Properties of Real Numbers

  • Transitive Property: If a = b and b = c, then a = c.

  • Commutative Properties: a + b = b + a and a·b = b·a.

  • Associative Properties: a + (b + c) = (a + b) + c and a·(b·c) = (a·b)·c.

  • Inverse Properties: Additive inverse a + (−a) = 0; Multiplicative inverse for a ≠ 0.

  • Distributive Property: a·(b + c) = (a·b) + (a·c).

Functions

Definition and Properties

  • A function is a relationship that assigns each input exactly one output: y = f(x).

  • Domain: Set of all inputs.

  • Range: Set of all outputs.

  • Real functions in this course will involve real numbers.

Special Functions

  • Polynomial Functions: Defined by degrees, where the general form includes coefficients.

  • Quadratic Function: f(x) = ax² + bx + c.

  • Exponential Function: y = ax with special case y = e^x.

  • Logarithmic Function: The inverse of exponentiation, defined by y = log_b(x).

Practical Examples

Production Functions

  • General Model: Q = f(K, L, M, P, C...) where Q represents quantity produced.

  • Cobb-Douglas Production Function: Q = A * L^α * K^β with A > 0, 0 < α < 1, 0 < β < 1.

Linear Demand Function

  • Depicted as Q = f(P) = a - bP; the relationship is linear with respect to price.

  • Inverse function: P = f⁻¹(Q) = (a - Q) / b.

Exercises

Examples and Practice Questions

  1. Evaluate functions like f(x) = 3x² - x + 5 at given points.

  2. Apply Vieta’s Formulas to find solutions of quadratic equations.

  3. Calculate exponential growth as seen in population predictions.

This summary encapsulates the key concepts, attendance policies, and course objectives important for the successful navigation of the Mathematics 1 for Economics and Finance course.