Demokurs: Introduction to Engineering Mathematics: Power Functions, Polynomials and Power Series

1. Functions

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.

Zeroes

Calculate the zeroes of the function $$f$$ by the substitution $$f (x) = 0$$ . For polynomials with degree $$p>2$$ polynomial long division is sometimes necessary to reduce the degree.

Answer. We have $\begin{aligned}f(x)=0 \Leftrightarrow \frac{1}{4}x^4 - 2x^2 - \frac{9}{4} &=0 \quad |\cdot 4 \\ \Leftrightarrow x^4 - 8x^2 - 9 &=0\end{aligned}$ This biquartic equation is transformed into a quadratic equation by the substitution $$x^2 =: u$$ which yields the solutions $$u_1 = 9$$ and $$u_2 = -1$$ . The reverse substitution $x = \sqrt{u}$ gives the pair of solutions $$x_1 = -3$$ and $$x_2 = 3$$ . It also gives the zeroes $$N_1 = (-3,0)$$ und $$N_2 = (3,0)$$ .

Extrema (Points with horizontal tangents) and monotonicity

Determine the extrema of the function by setting $$f'(x)=0$$ .

Answer. We set the first derivation of $$f$$ to zero: $\begin{aligned}f'(x)=0 \Leftrightarrow x^3-4x &=0 \\ \Leftrightarrow x(x^2-4) & =0\end{aligned}$ This equation has the solutions $$x_1=0, x_2=-2, x_3=2$$ .

Substituting these values into the second derivative $$f''(x)=3x^2-4$$ gives a local maximum $$x_1=0$$ (since $$f''(0)=-4 < 0$$ ) and local minima at $$x_2=-2$$ (since $$f''(-2) = 8 > 0$$ ) and $$x_3=2$$ (since $$f''(2) = 8 > 0$$ or because of the symmetry).

Substituting into the functional equation $$f(x)$$ gives the corresponding y-coordinate of the extrema: $$MAX=(0,-2.25)$$ , $$MIN_1=(-2,-6.25)$$ , $$MIN_2=(2,-6.25)$$ . From the behaviour tending towards $\pm \infty$ and the position and the type of extrema we can deduce that $$f$$ is strictly monotonous decreasing in the intervals $(-\infty;-2)$ and $$(0;2)$$ and strictly monotonous increasing in the intervals $$(-2;0)$$ and $(2;\infty)$ .

Points of inflexion and curvature behaviour

Determine the points of inflexion of the function by setting $$f''(x)=0$$ .

Answer. We set the second derivation of $$f$$ to zero: $f''(x)=0 \Leftrightarrow 3x^2-4=0$ This equation has the solutions $x_1=\sqrt{\frac{4}{3}}, x_2=-\sqrt{\frac{4}{3}}$ .
For both values the third derivative is $f'''(x)=6x\neq 0$ . Substitution into the functional equation $$f(x)$$ gives us the corresponding y-coordinates of the point of inflexion: $IFP_1=\big(-\sqrt{\frac{4}{3}},-\frac{161}{36}\big)$ , $IFP_2=\big(\sqrt{\frac{4}{3}},-\frac{161}{36}\big)$ .
From the behaviour tending to $\pm \infty$ , the monotonic nature and from the position of the point of inflexion we can deduce, that $$f$$ is concave up in the intervals $\big(-\infty;-\sqrt{\frac{4}{3}}\big)$ and $\big(\sqrt{\frac{4}{3}};\infty\big)$ and concave down in the interval $\big(-\sqrt{\frac{4}{3}};\sqrt{\frac{4}{3}}\big)$ .

Table of values and a graph

Now plot on the coordinate system the roots, asymptotes, extrema and inflexion points already determined. To plot the graph more exactly, a table of values should be constructed, in which the values of the function are calculated for certain values of x. Now plot these values on the coordinate system and sketch the graph taking the nature of the monotonicity and curvature into account.

We compute significant additional points to plot on the graph (here we can also use the symmetry of the graph), e.g. $\begin{aligned}f(1) &= f(-1) = \textstyle{\frac{1}{4}\cdot1^4-2\cdot1^2-\frac{9}{4}} = -4; \\ f(3.25) &= f(-3.25) = \textstyle{\frac{1}{4}\cdot (3.25)^4 - 2\cdot(3.25)^2 - \frac{9}{4} \approx 4.52};\end{aligned}$

Finally we plot the graph:

When only even exponents appear in a polynomial, that polynomial is axially symmetric about the y-axis. Polynomials with odd exponents only, are point-symmetric about the origin.

Polynomials play an important role in analysis, as they are a (relatively) easy class of functions to represent. In particular they are easy to differentiate and integrate. Additionally they can be used to approximate more complicated functions.

Optional reading: Power series

This time we consider infinite sums of power functions instead of finite sums, and obtain power series.

Definition 7: Power series

Let $$(a_n)_n$$ be a sequence of real numbers and $x,a\in\mathbb{R}$ . Then the series $\sum_{n=0}^\infty a_n(x-a)^n$ is called a power series with coefficients $$a_n$$ and centre a.

Polynomials are therefore particular examples of power series, where an infinite number of coefficients equal 0.

Power series represent powerful tools in analysis. Below we give two examples of power series. These are covered extensively in analysis lectures at the beginning of the university course.

Example 2: Geometric series

The geometric series $f(x) = \sum_{n=0}^\infty x^n$ converges for all $x\in (-1;1)$ . We can write this particular function $$f$$ as: $f(x) = \frac{1}{1-x}.$

Exercise.

Show this for specific values of x, by adding up the sum of the first 2, 3, 4, 5,... terms of the series.

Example 3: Exponential series

The function $$f$$ where $f(x) = \sum_{n=0}^\infty \frac{1}{n!}x^n$ is well-defined over the whole of $\mathbb{R}$ is called exponential series. We learn more about this particular function in the next chapter...

Exercise.

For specific values of x add up the sum of the first 2, 3, 4, 5,... terms of the series.

New terms:

Power function
Polynomials
Power series

Short revision exercises

Exercise 1.

Sketch the following power functions: $$x^3$$ , $x^{\frac{1}{3}}$ , $x^{-3}$ , $x^{-\frac{1}{3}}$

Exercise 2.

Specify the type and position of the extrema, inflection points and the roots for the polynomial function $f(x)=-\frac{1}{4}x^4+\frac{3}{2}x^2-2x-2.$

1. Functions

1.2. Power Functions, Polynomials and Power Series

Power functions

Definition 1: Power function with positive integer exponent

Exercise.

Definition 2: Power function with integer exponents

Parabolas of order $\(n\)$ - the building blocks of polynomials

Definition 3: Power functions with fractions in the exponents

Exercise.

Example 1: Root function

Definition 4: Power functions with negative fractions in the exponent

Exercise.

Definition 5: Power functions with rational number exponents

Exercise.

Power functions with rational number exponents

Polynomials

Definition 6: Polynomial of degree $\(n\)$

Optional reading: Description of curve sketching by an example

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

Symmetries

Zeroes

Extrema (Points with horizontal tangents) and monotonicity

Points of inflexion and curvature behaviour

Table of values and a graph

Optional reading: Power series

Definition 7: Power series

Example 2: Geometric series

Exercise.

Example 3: Exponential series

Exercise.

New terms:

Short revision exercises

Exercise 1.

Exercise 2.

1. Functions

1.2. Power Functions, Polynomials and Power Series

Power functions

Definition 1: Power function with positive integer exponent

Exercise.

Definition 2: Power function with integer exponents

Parabolas of order \(n\) - the building blocks of polynomials

Definition 3: Power functions with fractions in the exponents

Exercise.

Example 1: Root function

Definition 4: Power functions with negative fractions in the exponent

Exercise.

Definition 5: Power functions with rational number exponents

Exercise.

Power functions with rational number exponents

Polynomials

Definition 6: Polynomial of degree \(n\)

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

Symmetries

Zeroes

Extrema (Points with horizontal tangents) and monotonicity

Points of inflexion and curvature behaviour

Table of values and a graph

Definition 7: Power series

Example 2: Geometric series

Exercise.

Example 3: Exponential series

Exercise.

New terms:

Short revision exercises

Exercise 1.

Exercise 2.

Parabolas of order $\(n\)$ - the building blocks of polynomials

Definition 6: Polynomial of degree $\(n\)$