This course will cover basic university-level mathematics. It is based on the book Essential Mathematics for Economic Analysis by Knut Sydsaeter, Peter Hammond, Arne Strom, and Andrés Carvajal. Below is a description of what you will learn throughout the course:
Chapter 1: Mathematical foundations
In this chapter, we look at the foundations of mathematics. In particular, we will look at logic and sets. You can go quite far in mathematics without the foundations. However, in order to develop a better understanding of mathematics, foundations are very useful as you will know the exact definitions of concepts we use in mathematics.
Chapter 2: Algebra
Algebra is one of the main and most important parts of mathematics. Chapter 2 will cover the algebra needed for this course, such as rules of algebra, fractions, powers, inequalities, and logarithms. Most of what you will see in this chapter will be known to you. However, unless you had studied mathematics at the university level, it might be a good idea to go through this chapter. First of all, it will give you a chance to refresh and review material that you have done before. Second, the structure of the material in this chapter is more formalized with a clear separation of definitions and results.
Chapter 3: Equations and sums
Most of chapter 3 are devoted to equations but we have also included a section on the summation sign. We will introduce all the terminology that we need for a fundamental understanding of what it means to solve an equation. We will focus most of our attention on quadratic equations and systems of linear equations.
Chapter 4: Functions of one variable
This chapter is devoted entirely to one of the most important concepts in mathematics namely functions. We begin the chapter by carefully looking at exactly what we mean by a function introducing the domain, codomain, and range of a function. The most important class of functions are the linear functions which we will study extensively. Next, we look at some of the most important nonlinear functions. The chapter is concluded with a few slightly more advanced topics related to functions.
Chapter 5: Derivatives and limits
This chapter is entirely devoted to the derivative of a function of one variable. The derivative of a function is defined as a limit of a specific ratio (the Newton quotient) and we begin the chapter with a brief introduction to limits. In the rest of the chapter, we will learn how to differentiate various functions. To our help, we will have a bunch of rules such as the chain rule. We will also need higher-order derivatives. For example, we can sometimes use the second-order derivative to distinguish between a maximum point and a minimum point. This chapter is concluded with a few more advanced topics such as implicit differentiation.
Chapter 6: Single variable optimization
In this chapter, we will look at how to maximize or minimize a function of one variable. There is not much theory in this chapter, the best way of learning how to optimize functions is through lots of problems.
Chapter 7: Integrals
This chapter introduces integrals and anti-derivatives, also called primitive functions or indefinite integrals. If you differentiate an anti-derivative you will come back to the function you started with. We will look at the connection between integrals, which is an area under the graph of the function, and the anti-derivative of the function.
Chapter 8: Functions of several variables
So far, we have only considered the functions of one variable. In this chapter, we will look at the functions of an arbitrary number of variables. An important concept for a function of one variable was the derivative. When we have a function of several variables, we will have several different derivatives which we will call partial derivatives. Fortunately, finding partial derivatives is generally no more difficult than finding ordinary derivatives. This chapter also introduces the Hessian, a matrix where we have collected all the second-order partial derivatives.
Chapter 9: Optimization, several variables
In the final chapter of this course, we look at the maximization and minimization of functions of several variables. Optimization can be either unconstrained or constrained. By constrained optimization, we mean that the variables must satisfy one or several constraints, we are not free to pick any values when we optimize the function. Unconstrained optimization is very similar to optimization of a function of a single variable. For constrained optimization, we will study a solution technique called the method of Lagrange.