Chapter 6 Elementary Functions

Section 6.1 Basics of Functions

6.1.3 Functions in Mathematics and Applications

Mathematical functions often describe relations between quantities that arise from other sciences or everyday life. For example, the volume

V

of a cube depends on the side length

a

of the cube. The volume can be considered as a mathematical function that assigns the corresponding volume

V (a) = a^{3}

to every possible side length

a > 0

V : {\begin{matrix} (0; \infty) & \to & ℝ \\ a & ⟼ & V (a) = a^{3} . \end{matrix}

The result is the cubic standard parabola (see Section 6.2.6) for the relation between side length and volume. In this way, many more examples can be found arising from sciences and everyday life: the position as a function of time in physics, the reaction rate as a function of concentration in chemistry, the amount of flour needed as a function of the desired amount of dough in a cake recipe, etc.
To this end, let us consider an example.

Example 6.1.12

The intensity of nuclear radiation is inversely proportional to the square of the distance to the source. This is called the inverse square law. Using a physical proportionality factor

c > 0

, the relation between intensity

I

of the radiation and distance

r > 0

from the source can be formulated mathematically as follows:

I : {\begin{matrix} (0; \infty) & \to & ℝ \\ r & ⟼ & \frac{c}{r^{2}} . \end{matrix}

Hence, for the intensity we have the mapping rule

I (r) = \frac{c}{r^{2}}

that describes the relation between the quantities

I

and

r

Exercise 6.1.13

In the construction of wind turbines it is known that the wind turbine power is proportional to the cube of the wind velocity. Under the condition that the proportionality factor satisfies the relation

ρ > 0

, which of the following mathematical functions describes this relation of physical quantities correctly?
a)

P : {\begin{matrix} (0; \infty) & \to & ℝ \\ v & ⟼ & P (v) = \frac{ρ}{v^{3}} \end{matrix}

P : {\begin{matrix} ℝ & \to & ℝ \\ v & ⟼ & P (v) = ρ v^{3} \end{matrix}

P : {\begin{matrix} [0; \infty) & \to & ℝ \\ v & ⟼ & P (v) = ρ v^{3} \end{matrix}

x : {\begin{matrix} [0; \infty) & \to & ℝ \\ f & ⟼ & x (f) = ρ f^{3} \end{matrix}

c) and d) are correct. a) is wrong since it describes an inverse proportionality. b) is not adequate to the problem. Only non-negative wind velocities make sense in this context and hence, negative values should be excluded from the domain. c) is correct; in this case we have the relation

P (v) = ρ v^{3}

between the power

P

and the wind velocity

v

. We see that d) is equally correct. In this case we have a function that is completely identical to case c), except for the fact that in this case the power is denoted by

x

and the wind velocity is denoted by

f

. This again clarifies that the letters used to denote the function and the variables are mathematically completely arbitrary. However, in the sciences there are conventions, which letters are generally used to denote certain quantities. So, here it is more common to denote the velocity by

v

and the power by

P

Onlinebrückenkurs Mathematik

1. Elementary Arithmetic

2. Equations in one Variable

3. Inequalities in one Variable

4. System of Linear Equations

5. Geometry

6. Elementary Functions

7. Differential Calculus

8. Integral Calculus

9. Objects in the Two-Dimensional Coordinate System

10. Basic Concepts of Descriptive Vector Geometry

11. Language of Descriptive Statistics

Chapter 6 Elementary Functions

6.1.3 Functions in Mathematics and Applications

Example 6.1.12

Exercise 6.1.13