## The functions and their derivatives

### Teaching unit

### Description

One of the most interesting objectives in the experimental sciences is to be able to adequately describe a variable y (dependent variable or response variable) as a function of another variable x (independent variable or explanatory variable). Mathematical functions are a basic tool to describe the dependence of some variables with respect to others. Many of the functions that are used in science can be described analytically from what are called elementary functions. Therefore, it is very important to know the fundamental properties and the graphic representation of the most common elementary functions, such as linear, polynomial, exponential, logarithmic and trigonometric functions.

The main characteristic of a natural process is its evolution over time, and the derivative is a measure that quantifies the change of one variable with respect to another. So the derivatives with respect to time describe the way in which a changing system evolves. In the environmental sciences systems are changing, so the derivative of a function has a wide variety of utilities, the most important in this context is its use in the search for models that describe environmental phenomena. It also includes a topic that contains a small introduction of real functions of several variables, the definition of partial derivatives and their application in non-linear discrete models.

### Objectives

Possessing the basic mathematical knowledge that is necessary for theunderstanding of problems applied to experimental sciences and, especially, to environmental sciences.

Recognizing and distinguishing between discrete models and continuous models.

Formulate mathematically some applied problems, analyzing and correctly interpreting the results obtained.

Recognizing and using the derivative and the integral for the treatment and resolution of basic continuous models.

Using mathematical software in solving applied problems in environmental sciences.

### Subjects

#### Lecture topics

- Real functions.
- Diferential calculus. Partial derivatives.
- Applications of the derivative.

#### Laboratory topics

- The functions and their derivatives. Applications. (Computer lab)
- Exercises and problems. (Classroom)