Demokurs: Introduction to Engineering Mathematics

Website:	WueCampus
Kurs:	vhb - Demokurs Lehramt
Buch:	Demokurs: Introduction to Engineering Mathematics

Gedruckt von:	Gast
Datum:	Donnerstag, 21. November 2024, 23:50

Beschreibung

Inhaltsverzeichnis

1. Functions
2. Outlook: Content of the Course

1. Functions

Funktionen

IEM: Functions

The chapter Functions includes:

Linear and Quadratic Functions
Power Functions, Polynomials and Power Series
The Exponential Function and Growth
Trigonometric Functions
Multivariable Functions
Curves and Maps
Exercises

This chapter revises some of the functions learnt about in school, namely linear and quadratic functions, power functions, the exponential function and trigonometric functions. Their basic characteristics are highlighted but most importantly they are illustrated by means of interactive diagrams. A foretaste of analysis is then given by examining both multivariable functions and a number of mapping-equations of specific curves.

1.1. Linear and Quadratic Functions

This chapter contains the definition of the term function and examines linear and quadratic functions.

What is a function?

Example 1.

The following relationship exists between the speed of sound in metres per second, $$c$$ , and the air temperature in degrees Celsius, $\theta$ : $c=331,5\cdot\sqrt{1+\frac{\theta}{273}} \qquad \text{for } \theta>-273$ $c=c(\theta)$ is a function of $\theta$ .

Definition 1: Function

For arbitrary sets $$A$$ and $$B$$ , a function (or map) $f: A\rightarrow B$ is a rule which assigns a unique value $y=f(x) \in B$ to $x \in A$ . Here $$A$$ is called domain and $$B$$ the codomain. The graph of $$f$$ is the set $G_f = \{(x,y) \in A\times B : y=f(x)\}.$

Explanation: This means that a function is a unique assignment rule.

nonunique_assignment

This example, assigns to each natural number $$x$$ its square $$y=f(x)=x^2$$ :

Zuordnung

Usually, the functions that are relevant within this course are represented (plotted) as a graph in a coordinate system. A graph is – as defined above – a set of points, which results in the common curve depiction. Those points have the shape $$(x|f(x))$$ .

Linear functions

The functional equation of a linear function $f:x\in\mathbb{R}\mapsto f(x)\in\mathbb{R}$ can be written as: $f(x) = m\cdot x + t, \qquad m,t\in\mathbb{R}$

Here, $$m$$ is called the slope or gradient and $$t$$ the $$y$$ -intercept of the function (more precisely: of the graph of the function).

For a positive slope/ gradient $$m>0$$ the line increases (rises), for a negative slope/ gradient $$m<0$$ the line decreases (falls). The larger the absolute value of $$m$$ , the steeper the line rises or falls: The larger a positive gradient is, the steeper the line increases; the smaller a negative gradient is, the steeper the line decreases.

$$m=0$$ results in the special case of a line which is parallel to the $$x$$ -axis, that is, horizontal: $\[f(x)=t\]$

One should note that lines parallel to the $$y$$ -axis cannot be depicted by linear functions, because the slope would have to be infinitely large. In that case infinitely many values would have to be assigned to one x-value (which would contradict the definition of a function).

Such linear functions can be represented with the following equation (which is not a function!):

x=a, a\in\mathbb{R}

As $$t$$ increases or decreases the value of the function independently from $$x$$ (this means $$t$$ does not change for different $$x$$ values/ is the same for all $$x$$ ), we can move the whole graph along the $$y$$ -axis. For positive $$t$$ the graph moves up and for negative $$t$$ it moves down. If $$t=0$$ , then the line passes through the origin; in this case we have $$f(0)=m∙0+t=t=0$$ .

Interactivity. Explore the influence of the parameters $$m$$ and $$t$$ on the linear function.

This notation means that the function rule $$f$$ assigns to each value $$x$$ (the „pre-image“) a function value $$f(x)$$ (the „image“). Both $$x$$ and $$f(x)$$ lie in $\mathbb{R}$ in this case.

The absolute value $$|a|$$ of a number $$a$$ is its value without the sign: $\[|3|=3\]$ $\[|-3|=3\]$

Acccordingly, one can define the absolute value function: $f(x)=|x|=\begin{cases} x, \text{for } x\geq 0\\ -x, \text{for } x < 0\end{cases}$

Calculating the gradient and the $$y$$ -intercept

The gradient of a linear function is calculated using the so-called 'gradient triangle' as follows: $m=\frac{\Delta y}{\Delta x} = \frac{\Delta f(x)}{\Delta x} = \frac{f(x_2)-f(x_1)}{x_2-x_1}$

Here, the vertical distance between two points on the graph is divided by the horizontal distance of those two points (to be more precise, the values of the distances). So, the gradient depicts how the vertical value changes when the horizontal value changes.

As an example, road gradients are measured in a similar way. Here, the difference in altitude covered by the car is measured while it moves 100m in the vertical direction. (So, the actual distance is the hypotenuse of the resulting right-angle triangle) With this in mind - $$x$$ as the difference in altitude per 100m - the gradient is expressed as a percentage.

The $$y$$ -intercept $$t$$ equals the value the function has for $$x=0$$ , which is calculated by evaluating the function for $$0$$ : $\[t = f(0)\]$

Plotting the graph of the function

To plot the graph of the function you should memorize the following steps:

Starting at the origin move along the $$y$$ -axis until you reach $$t$$ . We call this point $$P$$ .
Starting at $$P$$ move up, vertically, the value of the numerator of $$m$$ and move right, horizontally, the value of the denominator of m. We call this point $$Q$$ .
The graph is a straight line through $$P$$ and $$Q$$ .

\(m\)

has already been calculated,

\(m\)

can be used as the numerator, and

\(1\)

as the denominator, since

m=\frac{m}{1}

Calculating the angle $\alpha$ between the straight line and the $$x$$ -axis

With the definition for the trigonometric function of the tangent in right-angled triangles ((the length of) the opposite side divided by (the length of) the adjacent side) it is true that: $tan(\alpha ) = \frac{\Delta f(x)}{\Delta x} = m$

according to the definition of the gradient m and therefore: $\alpha = arctan(m)$

Zeroes

A linear function has exactly one zero ( $$f(x) = 0$$ ), that means there is only one intersection with the $$x$$ -axis. The exception being the case where $$m=0$$ . Here $$f$$ is a constant function and thus the graph is a line parallel to the $$x$$ -axis: $$f(x) = t$$ . Such a function does not have a zero for $t\neq 0$ . For $$t=0$$ it holds: $$f(x)=0$$ . In this case the function has infinitely many zeros.

To calculate the zero $$x_0$$ set $f(x_0)=m\cdot x_0+t = 0$ and solve the equation for $$x_0$$ : $x_0 = -\frac{t}{m}$

Relationships between graphs of linear functions

Two lines $$G_f$$ and $$G_g$$ are parallel if, and only, if their gradients $$m_1$$ and $$m_2$$ are equal: $m_1 = m_2 \Leftrightarrow G_f \parallel G_g$

So, parallel lines $f_1, f_2, f_3,\ldots$ only differ through their $$y$$ -intercepts $t_1, t_2, t_3,\ldots$ This means the functions emanate from each other through translations (parallel shifts) in the vertical direction (upwards and downwards).

Two lines $$G_f$$ and $$G_g$$ are perpendicular, if and only if, the product of their gradients $$m_1$$ and $$m_2$$ equals -1: $m_1\cdot m_2 = -1 \Leftrightarrow G_f \perp G_g$

Exercise.

Justify this relationship between $$m_1$$ and $$m_2$$ .

Quadratic functions

The general form of the quadratic function $f:x\mapsto f(x)$ is: $f(x) = a\cdot x^2 + b\cdot x + c, \qquad a \neq 0,b,c\in\mathbb{R}$

Graphs of quadratic functions are parabolas.

The coefficient of $$x^2$$ i.e. a, determines the form of the parabola. If $$a$$ is a positive number, the parabola opens upwards; if negative, it opens downwards. The bigger the value of $$|a|$$ is chosen, the graph appears more closed. Conversely the shape of the parabola becomes less closed for a smaller value of $$|a|$$ .

A change in the parameter $$c$$ shifts the parabola in the $$y$$ -direction, because $$c$$ is independent from $$x$$ . As a result changes in $$x$$ have no influence on $$c$$ . If $$c$$ is increased by one, the graph is displaced upwards by one. On the other hand if $$c$$ is decreased by one, the graph is displaced downwards by one.

$$c$$ is therefore basically the $$y$$ -intercept which we encountered by looking at straight lines.

$$|a|$$ denotes the absolute value of $$a$$ , that is the value of $$a$$ "without the sign".

Identifying vertices

The vertex of the parabola is either the absolute minimum point (if $$a$$ is a positive number) or the absolute maximum point (if $$a$$ is a negative number). If the function is converted to the vertex form below by completing the square and using the binomial formula, the coordinates of the vertex can be read off directly: $f(x) = a\cdot (x-x_s)^2 + y_s.$

With this form of the equation the graph of $$f$$ can be seen as a variation of the (standardized) parabola with the form $$g(x)=x^2$$ . $$f$$ generates from this by shifting it to the right by $$x_S$$ , up by $$y_S$$ and by dilation/compression by the factor $$a$$ .

From the general form of the quadratic equation (see above), we get the coordinates of the vertex as: $x_s = -\frac{b}{2a}$ $y_s = \frac{4ac-b^2}{4a}$

The vertex then has the coordinates $$(x_s, y_s)$$ . The graph is axially symmetric to the line parallel to the $$y$$ -axis and passing through $$x_s$$ .

Exercise.

Show that the general form $f(x)=ax^2+bx+c,\quad a, b, c\in\mathbb{R}, a\neq 0$ can be converted into the vertex form $f(x) = a\cdot (x-x_s)^2 + y_s.$

A turning point is labeled as "absolute" (or global) when it is the largest/smallest value in the whole domain. In contrast a "local" Maximum/Minimum is only a maximum/minimum in a particular area.

The coefficient $$a$$ does not change after this transformation!

Optional reading: Finding the vertex of a parabola by completing the square

Finding the vertex of a parabola by completing the square may be the current method for calculating the extreme values on the lower secondary-school level. It is required, that the given problem can be represented by a quadratic function where the vertex is the wanted extreme value.

Every quadratic function with the form

$\[y=p(x)=ax^2+bx+c\]$

belongs to a parabola (graph) with its corresponding vertex which is the extreme value of the graph. For $$a>0$$ it is a minimum and for $$a<0$$ a maximum.

In order to find these, one can use the method of completing the square, with which the equation above can be changed by using the binomial formulas, in order to see the $$x$$ -value and $$y$$ -value of the vertex directly.

$y=a(x^2+\frac{b}{a}\cdot x + (\frac{b}{2a})^2 - (\frac{b}{2a})^2) + c$

$y=a(x+\frac{b}{2a})^2 - \frac{ab^2}{4a^2} +c$

$y=a(x+\frac{b}{2a})^2 +c - \frac{b^2}{4a}$

The resulting function is a dilated/compressed standard parabola moved in the horizontal of vertical direction of the form $y=\alpha (x- \beta )^2+ \gamma$ . The vertex can be located at the point $(\beta|\gamma)$ .

So, the function $$p$$ attains its maximum, and accordingly its minimum, for $x=-\frac{b}{2a}$ and $y=c-\frac{b^2}{4a}$ .

Roots

A quadratic function has zero, one or two roots, depending on the position of the vertex $$S$$ and the sign of $$a$$ .

To calculate the roots we set $$f(x) = 0$$ and solve the equation using the quadratic formula: $x_{1/2} = \frac{-b\pm\sqrt{b^2-4ac}}{2a}$

New terms:

Function
Linear function
Quadratic function.

Short revision exercises

Exercise 1.

Calculate the roots of the function f and plot its graph for $$f(x) = -3x + 6$$

Exercise 2.

Let $$P = (4, 3)$$ and $$Q = (0, -2)$$ be two points. Construct the equation of the straight line g passing through $$P$$ and $$Q$$ and calculate the angle at which g intersects with the x-axis.

Exercise 3.

Determine the vertex $$S$$ of the parabola given by $$f(x) = -5x^2 + 30x + 47$$ by completing the square.

Exercise 4.

Calculate the roots of the parabola given by $f(x) = -\frac{1}{2}x^2+\frac{1}{2}x+3.$

1.2. Power Functions, Polynomials and Power Series

In this chapter we revise the characteristics of power functions and polynomials. Then, in an optional section, we introduce another analysis concept -- the power series.

Power functions

Definition 1: Power function with positive integer exponent

A power function with a positive integer exponent is a function of the form: $f : x\mapsto a\cdot x^n \qquad a\in\mathbb{R},\ n\in\mathbb{N},\ D=\mathbb{R}$

Setting $$n=0$$ provides a constant function $$f(x)=a.$$

Note: We have already met two specific examples of power functions in the last chapter: For $$n=1$$ we have the pure linear function $$f$$ where $$f(x) = ax$$ and for $$n=2$$ we have the pure quadratic function $$f$$ where $$f(x) = ax^2$$ . Additionally, for $$n=0$$ we get the special case of a constant function $$f(x) = a∙x^0 = a∙1 = a$$ .

Exercise.

Vary the parameter $$a$$ on the adjacent example for $f(x) = a\cdot x^3$ . Comment on how the graph changes.

Definition 2: Power function with integer exponents

A power function with an integer exponent is a function of the form: $f : x\mapsto a\cdot x^z \qquad a\in\mathbb{R},\ z\in\mathbb{Z},\ D=\mathbb{R}\backslash\{0\}$

They can be separated into two groups, which have opposite characteristics:

$f(x) = a\cdot x^n$ (positive integer exponent, $$n>0$$ ): These functions have one root at $$x=0$$ and grow without bounds for large $$|x|$$ (tend towards plus or minus infinity). They are continuous on the whole of $\mathbb{R}$ . Their graphs always have one branch and are called parabolas of order $$n$$ .
$f(x) = a\cdot x^{-n}$ (negative integer exponent, $$-n<0$$ ): These functions have a pole at $$x=0$$ and approach 0 for large $$|x|$$ . There are continuous on $\mathbb{R}\backslash\{0\}$ . We call their graphs hyperbole of order $$n$$ . Because of the pole, they have two branches.

The graphs of power functions with odd exponents are point-symmetric about the origin, while those of power functions with even exponents are axially-symmetric about the $$y$$ -axis.

$\[f(-x) = (-x)^2 = x^2 = f(x)\]$

$\[f(-x) = (-x)^3 = -(x^3) = -f(x)\]$

Parabolas of order $$n$$ - the building blocks of polynomials

For growing $$x$$ the absolute value of $$x^n$$ increases continuously and diverges towards $\infty$ .

For growing $$x$$ the denominator (actually its absolute value) increases and the fraction converges towards $$0$$ .

Definition 3: Power functions with fractions in the exponents

A power function with a fraction in the exponent is a function of the form: $f : x\mapsto a\cdot x^{\frac{1}{n}} \qquad a\in\mathbb{R},\ n\in\mathbb{N},\ D=\mathbb{R}^+_0$

Exercise.

On the right are the graphs of the function $$f$$ where $f(x)=x^{\frac{1}{n}}$ for $n \in \{2,3,4,5,6\}$ . Describe these graphs, paying particular attention to the domain, monotonicity, and the largest and smallest values of the function. Are there points common to all graphs of these functions? Justify this!

Example 1: Root function

A particular example of a power function is the function $f : \mathbb{R}^+_0\rightarrow\mathbb{R}^+_0, x\mapsto\sqrt{x}=x^{\frac{1}{2}}$ which takes the square root of every positive number. Analogously the $$k$$ -th roots $\sqrt[k]{x}$ are also power functions thanks to the identity $\sqrt[k]{x}=x^{\frac{1}{k}}.$ The $$k$$ -th root function $$f$$ where $f(x)=x^{\frac{1}{k}}$ is the inverse of the power function $$g$$ where $$g(x) = x^k$$ . Also $$g$$ is the inverse of $$f$$ .

Definition 4: Power functions with negative fractions in the exponent

A power function with a negative fraction in the exponent is a function of the form: $f : x\mapsto a\cdot x^{-\frac{1}{n}} \qquad a\in\mathbb{R},\ n\in\mathbb{N},\ D=\mathbb{R}^+$

Exercise.

Compare the graphs of the functions with negative fractions in the exponents (blue) with the graphs of the functions that were introduced earlier with positive fractions in the exponents (red dotted). Pay particular attention to the domain, symmetry, monotonicity and the largest and smallest values of the function. Are there points which all graphs have in common? Justify this.

Definition 5: Power functions with rational number exponents

A power function with a rational number exponent is a function of the form: $f : x\mapsto a\cdot x^{\frac{p}{q}} \qquad a\in\mathbb{R},\ p\in\mathbb{Z},\ q\in\mathbb{N},\ D=\mathbb{R}^+$

Exercise.

Compare the graphs of power function with rational exponents (blue) with those we met in the above sections (red and purple dotted); by using the scroll bar you can alter the exponents. Describe the similarities and differences of the graphs, paying particular attention to the domain and symmetry.

Power functions with rational number exponents

The representation of functions of the form $f(x) = (\sqrt[k]{x})^p$ as $f(x) = x^{\frac{p}{k}}$ plays a role when differentiating functions as the common rules for derivatives are transferred to root-functions.

Various power functions.

Polynomials

Polynomials are constructed by adding up power functions $a \cdot x^n$ where $$n$$ is a natural number. They are defined and continuous over the whole of $\mathbb{R}$ .

Definition 6: Polynomial of degree $$n$$

For $a_0, a_1,\ldots, a_n \in\mathbb{R},$ $p : \mathbb{R}\rightarrow\mathbb{R}, x\mapsto\sum^n_{k=0}a_kx^k = a_0+a_1^x+a_2x^2+\ldots+a_nx^n$ is called a polynomial of degree $$n$$ , when $a_n\neq 0.$

So, per definition constant, linear, quadratic and power functions are polynomials.

The behaviour of the graphs of polynomials with degree $$>2$$ is not as easy to determine as for graphs in special cases like constant, linear and quadratic functions. To determine the characteristics of these polynomials and their graphs, we use the method of curve sketching taught in school (see below).

Various polynomial functions.

Optional reading: Description of curve sketching by an example

In this section we explain the general procedure and then give the example of a function $$f$$ where $f(x) = \frac{1}{4}x^4-2x^2-\frac{9}{4}.$

Determine the domain, i.e., all numbers which can be inputted into the function without leading to a forbidden operation (e.g., division by 0, square root of a negative number,...). Also check that all discontinuities are removable. In this case you can perform many calculations with the continuous continuation, which is conform to the original function at all points, except the discontinuities.

Answer. The function $$f(x)$$ is a polynomial. All $x\in\mathbb{R}$ can be inputted. The domain $$D$$ is therefore $\mathbb{R}$ .

Behaviour on the boundaries of the domain, determination of the asymptotes

Explore the behaviour of the function at the boundaries of the domain by generating limits. If the domain is not bounded, explore the behaviour of the function tending to infinity.

Answer. For $x \to -\infty$ is $\lim_{x\rightarrow -\infty} f(x) = \lim_{x\rightarrow -\infty} \frac{1}{4}x^4-2x^2-\frac{9}{4} = \infty$ and for $x \to \infty$ we have $\lim_{x\rightarrow \infty} f(x) = \lim_{x\rightarrow \infty} \frac{1}{4}x^4-2x^2-\frac{9}{4} = \infty$

Symmetries

Ascertain whether the function is axially-symmetric about the y-axis or point-symmetric about the origin.

Answer. The function $$f$$ above has only even exponents; it is therefore axially-symmetric about the y-axis.