CALCULUS
Instructional goals
The course aims to provide first-year economics students with the main mathematical tools that are used to analyse and solve a wide range of problems in economics, business, finance and insurance. This will be their first step towards achieving a broad training in quantitative disciplines, as well as a good methodological training for the analysis and critical interpretation of economic and business dynamics. This will allow them to acquire the useful tools for the formulation, implementation and control of decisions in the work contexts in which they will operate.
Intended learning outcomes
Real numbers. Powers. Basic Algebra. Fractions. Solving equations and inequalities. Intervals and absolute values. Cartesian coordinates. Second degree equations. Logic. Set theory. Trigonometry. Exponential and logarithmic functions.
Videos and additional materials available on the website learn.luiss.it.
Course Contents
Knowledge and understanding: By the end of the course, students will have acquired a wide range of mathematical tools and will be able to understand and solve numerous theoretical and applied problems in economics, business and finance. Ability to apply knowledge and understanding: By the end of the course, students will be able to apply quantitative tools to business decisions, to the analysis of economic phenomena and to the resolution of economic and business problems in the work contexts in which they will operate. Autonomy of judgment: By the end of the course, students will be able to develop a critical capacity in identifying the most suitable solution to the proposed problem, even in highly innovative contexts. Communication skills: By the end of the course, students will be able to develop both written and oral communication skills and to highlight the relationships between the theoretical concepts and the most important economic and managerial applications. Learning ability: By the end of the course, students will be able to stay up to date and improve their skills in examining the quantitative aspects of economics and management.
Reference Books
Essential Mathematics for Economic Analysis 6th Edition
by Knut Sydsaeter, Peter Hammond, Arne Strom, Andrés Carvajal, Pearson
Teaching Methods
The teaching activity will consist of:
- frontal teaching and tutoring
- material for in-depth study of the topics available on the learn.luiss.it page of the course.
The teacher and the teaching assistants will set up weekly office hours to help students.
Assessment Method
Student evaluation will be based on individual performance, class participation, and overall involvement. For students with consistent attendance, grading will follow these criteria:
• Class Participation and Assignments.
• Midterm Exam. The midterm exam will take place during the course break (between the 6th and 7th week). Students may choose to reject their midterm grade, in accordance with the rules and deadlines announced during the course.
• Final Exam. The final exam mirrors the format of the midterm, including multiple-choice questions and written exercises.
Scoring: graded on a 0–30 scale, with a minimum passing score of 18.
• Review Process Each written exam will be followed by a compulsory review session.
Further details will be provided before the course begins.
Thesis assignment criteria
Interview with the teacher after the exam.
Week 1
Introducing functions of one variable.
Graphs of functions. Linear and quadratic functions. Polynomials. Power, exponential and logarithmic functions.
Economic applications.
TA sessions on the lectures of the present week.
Week 2
Functions and their properties
Shifting grafts. New functions from old (operations on functions, compound and piecewise defined functions). Inverse functions. Graphs of equations and graphs of functions.
Economic applications.
TA sessions on the lectures of the present week.
Week 3
Limits and Their Properties
Informal definition of limit. Rules for calculating limits. One-sided limits. Infinite limits. Limits at infinity. Continuity and limits. Properties of continuous functions.
Economic applications.
TA sessions on the lectures of the present week
Week 4
Differentiation
The Derivative and the Tangent Line Problem. Continuity and differentiability. Basic Differentiation Rules. Product and Quotient Rules. Higher-Order Derivatives
The Chain Rule. The derivative of exponential and logarithmic functions.
Economic applications.
TA sessions on the lectures of the present week
Week 5
Using the derivative.
Increasing and decreasing functions and the first derivative test. Concavity and the second derivative test. Derivative for compound functions. Linear approximation. Quadratic approximation. Elasticity. The Intermediate Value Theorem. L’Hopital’s rule.
Economic applications.
TA sessions on the lectures of the present week.
Week 6
Optimization of one-variable functions. Global and local extrema of a function. Fermat’s Theorem. The Extreme and Mean Value Theorems. First and second derivative tests for local extrema. Tests for global extrema. Curve sketching. Economic applications. TA sessions on the lectures of the present week.
Week 7
Matrix Algebra. Matrices and vectors. Systems of linear equations. Matrix addition and multiplication. The transpose. Gaussian elimination. Economic applications. TA sessions on the lectures of the present week.
Week 8
Determinants and Inverses
The inverse of a matrix. Determinant and inverse of a 2x2 square matrix.
Geometric interpretation of vectors.
Lines and planes. Linear independence and matrix rank.
Economic applications
TA sessions on the lectures of the present week.
Week 9
Integration
Indefinite integrals. Area and definite integrals. Properties of definite integrals
Economic applications.
TA sessions on the lectures of the present week.
Week 10
Integration by parts and by substitution. Improper integrals. Economic Applications. TA sessions on the lectures of the present week
Week 11
Functions of Several Variables Introduction. Graphs and level curves. Partial Derivatives. TA sessions on the lectures of the present week.
Week 12
Functions of several variables Local extreme points. Finding extrema. Economic Applications. TA sessions on the lectures of the present week.