BASIC MATHEMATICS FOR NATURAL SCIENCES UNDERGRADUATE STUDENT TEXTBOOK

After completing this section, the student should be able to:

For example, \(2^{5} = 2 \times 2 \times 2 \times 2 \times 2 = \text{32}\text{.}\)

Based of the definition of \(x^{n}\text{,}\) \(n\) must be a natural number. It does not make sense for \(n\) to be negative or zero. However, we can extend the definition of exponents to include 0 and negative exponents.

As a result of the above definition, we have \(\frac{1}{x^{- n}} = x^{n}\text{.}\) We have the following rules of exponents for integer exponents:

Next we extend the definition of exponents even further to include rational number exponents. To do this, we assume that we want the rules for integer exponents also to apply to rational exponents and then use the rules to show us to define a rational exponent. For example, how do we define \(a^{\frac{1}{2}}\text{?}\) Consider \(9^{\frac{1}{2}}\text{.}\)

If we apply rule 2 and square \(9^{\frac{1}{2}}\text{,}\) we get \(\left( 9^{\frac{1}{2}} \right)^{2} = 9^{\frac{1}{2}} = 9\text{.}\) Thus, \(9^{\frac{1}{2}}\) is a number that, when squared, yields 9. There are two possible answers: 3 and – 3, since squaring either number will yield 9. To avoid ambiguity, we define \(a^{\frac{1}{2}}\)(called the principal square root of \(a\)) as the non-negative quantity that, when squared, yield \(a\text{.}\) Thus, \(9^{\frac{1}{2}} = 3\text{.}\)

We will arrive at the definition of \(a^{\frac{1}{3}}\) in the same way as we did for \(a^{\frac{1}{2}}\text{.}\) For example, if we cube \(8^{\frac{1}{3}}\text{,}\) we get \(\left( 8^{\frac{1}{3}} \right)^{3} = 8^{\frac{3}{3}} = 8\text{.}\) Thus, \(8^{\frac{1}{3}}\) is the number that, when cubed, yields 8. Since \(2^{3} = 8\) we have \(\text{.}\) Similarly, \(\left( - \text{27} \right)^{\frac{1}{3}} = - 3\text{.}\) Thus, we define \(a^{\frac{1}{3}}\)(called the cube root of \(a\)) as the quantity that, when cubed yields \(a\text{.}\)

We call \(a^{\frac{1}{n}}\) the principal \(n^{\text{th}}\) root of \(a\text{.}\) Hence, \(a^{\frac{1}{n}}\) is the real number (nonnegative when \(n\) is even) that, when raised to the \(n^{\text{th}}\) power, yields \(a\text{.}\) Therefore,

Thus far, we have defined \(a^{\frac{1}{n}}\text{,}\) where \(n\) is a natural number. With the help of the second rule for exponent, we can define the expression \(a^{\frac{m}{n}}\text{,}\) where \(m\) and \(n\) are natural numbers and \(\frac{m}{n}\) is reduced to lowest terms.

We can also define negative rational exponents:

Radical notation is an alternative way of writing an expression with rational exponents. We define for real number \(a\text{,}\) the \(n^{\text{th}}\) root of \(a\) as follows:

The number \(\sqrt[n]{a}\) is also called the principal \(n^{\text{th}}\) root of \(a\text{.}\) If the \(n^{\text{th}}\) root of \(a\) exists, we have:

For example, \(3\sqrt{5^3}=5 \, \text{and}\, 4\sqrt{(-3)^4}=3\text{.}\)

In the previous sections we examined functions of the form \(f(x) = x^{n}\text{,}\) where \(n\) is a constant. How is this function different from \(f(x) = n^{x}\text{.}\)

In the previous subsection we noted that the exponential function \(f(x) = b^{x}\) (where \(b > 0\) and \(b \neq 1\)) is one to one. Thus, the exponential function has an inverse function. What is the inverse of \(f(x) = b^{x}\text{?}\)

To find the inverse of \(f(x) = b^{x}\text{,}\) let’s review the process for finding an inverse function by comparing the process for the polynomial function \(y = x^{3}\) and the exponential function \(y = 3^{x}\text{.}\) Keep in mind that \(x\) is our independent variable and \(y\) is the dependent variable and so whenever possible we want a function solved explicitly for \(y\text{.}\)

There is no algebraic procedure we can use to solve \(x = 3^{y}\) for \(y\text{.}\) By introducing radical notations we could express the inverse of \(y = x^{3}\) explicitly in the form \(y = \sqrt[3]{x}\text{.}\) In words, \(y^{3} = x\) and \(y = \sqrt[3]{x}\) both mean exactly the same thing: \(y\) is the number whose cube is \(x\text{.}\) Similarly, if we want to express \(x = 3^{y}\) explicitly as a function of \(x\text{,}\) we need to invent a special notation for this. The key idea is to take the equation \(x = 3^{y}\) and express it verbally.

We introduce the following notation, which expresses this idea in a much more compact form.

We read \(y = \log_{b} x\) as “\(y\) equals the logarithm of \(x\) to the base \(b\)”.

\(\textbf{REMEMBER:}\) \(y = \log_{b} x\) is an alternative way of writing \(x = b^{y}\)

When an expression is written in the form \(x=b^{y}\text{,}\) it is said to be in exponential form. When an expression is written in the form \(y=\log_{b} x\text{,}\) it is said to be in logarithmic form. The table below illustrates the equivalence of the exponential and logarithmic forms.

As was pointed out at the beginning of this subsection, logarithm notation was invented to express the inverse of the exponential function. Thus, \(\log_{b} x\) is a function of \(x\text{.}\) We usually write \(f ( x ) = \log_{b} x\) rather than writing \(f ( x ) = \log_{b} ( x )\) and use parenthesis only when needed to clarify the input to the log function. For example,

If \(f ( x ) = \log_{5} ( 4 - x )\text{,}\) then \(f ( -1) = \log_{5} (4 - ( -1)) = \log_{5} 5 = 1\text{,}\) whereas if \(f ( x ) = 4 - \log_{5} x\text{,}\) then \(f ( -1) = 4 - \log_{5} ( -1)\text{,}\) which is undefined.

Acknowledging that the logarithmic and exponential functions are inverses, we can derive a great deal of information about the logarithmic function and its graph from the exponential function and its graph.

Taking note of the features of the two graphs we have the following important informations about the graph of the log function:

The following table contains the basic properties of logarithm:

The following table contains the basic properties of logarithm:

The logarithmic function was introduced without stressing the particular base chosen. However, there are two bases of special importance in science and mathematics, namely, \(b = 10\) and \(b = e\text{.}\)

The inverse of the natural exponential function is called the natural logarithmic function and has its own special notation.

For the functions we have encountered so far, namely polynomial, rational and exponential functions, as the independent variable goes to infinity the graph of each of these three functions either goes to infinity(very quickly) for exponential functions or approaches a finite horizontal asymptote. None of these functions can model the regular periodic patterns that play an important role in the social, biological, and physical sciences: business cycles, agricultural seasons, heart rhythms, and hormone level fluctuations, and tides and planetary motions. The basic functions for studying regular periodic behaviour are the trigonometric functions. The domain of the trigonometric functions is more naturally the set of all geometric angles.

A periodic function keeps repeating the same set of \(y\) - values over and over again. The graph of a periodic function shows the same basic segment of its graph being repeated. In the case of sine and cosine functions, the period is \(2\pi\text{.}\) The period of the tangent function is \(\pi\text{.}\)

Thus, the amplitude of the basic sine and cosine function is \(1\text{.}\) The portion of the graph of a sine or cosine function over one period is called a complete cycle of the graph. In other words, the minimal portion of a sine or cosine graph that keeps repeating itself is called a complete cycle of the graph.

The frequency of the basic sine curve \(y = \sin x\) and the basic cosine curve \(y = \cos x\) is \(1\text{,}\) because each graph makes \(1\) complete cycle in the interval \([0, 2\pi]\text{.}\)

If a sine function has period of \(\frac{\pi}{2}\) (see the figure below), then the number of complete cycles its graph will make in an interval of length \(2\pi\) is \(\frac{2 \pi}{\frac{\pi}{2}}=4\text{.}\)

Thus if a sine function has a period of \(\frac{\pi}{2}\text{,}\) its frequency is \(4\) and its graph will make \(4\) complete cycles in an interval of length \(2\pi\text{.}\)

For convenience we summarize our discussion on the domains of the trigonometric functions in the table.

In the course of our discussion of the trigonometric functions, we have discussed two types of trigonometric relationships: the reciprocal and quotient relationships. These relationships are examples of trigonometric identities. In the table below we list identities that are satisfied by the trigonometric functions.

The hyperbolic functions are certain combinations of exponential functions, that occur in various applications, with properties similar to those of the trigonometric functions. Among many other applications they are used to describe the formation of satellite rings around the planets, to describe the shape of a rope hanging from two points, and have application in relativity theory. The two basic hyperbolic functions are the hyperbolic sine and hyperbolic cosine functions. They are defined as follows:

Since \(\cosh x\) is an even function, its graph is symmetric about the y-axis. Its \(y-\) intercept is \((0,1)\text{,}\) because \(\cosh(0)=1\) . As \(x\) tends to infinity, \(\cosh x =\frac{e^x}{2}+\frac{e^{-x}}{2}\) tends to infinity because \(\frac{e^x}{2}\) goes to infinity and \(\frac{e^{-x}}{2}\) approaches to 0. When \(x\) is a large negative number \(\cosh x\) acts like \(\frac{e^{-x}}{2}\text{,}\) because \(\frac{e^x}{2}\) gets close to \(0\text{.}\) Thus the graph of \(y = \cosh x\) looks like:

This graph can also be obtained by geometrically adding the two curves \(y = e^x\) and \(y =e^{-x}\text{,}\) and taking half of each resulting \(y\)-value. Observe that range of \(\cosh x\) is \([1, \infty)\text{.}\)

Since \(\sinh\,x\) is an odd function, its graph is symmetric about the origin. The graph passes through the origin because \(\sinh(0) = 0\text{.}\) As \(x\) gets large, \(\sinh \,x\) acts like \(\frac{e^x}{2}\) and when \(x\) is a large negative number, \(\sinh \,x\) acts like \(-\frac{e^{-x}}{2}\text{.}\) Thus, the graph of \(y = \sinh \,x\) looks like:

The remaining four hyperbolic functions are defined in terms of cosh x and sinh x by analogy with trigonometry.

\(\tanh\,x=\frac{\sinh\,x}{\cosh\,x}=\frac{e^x-e^{-x}}{e^x+e^{-x}}\) (The domain of \(\tanh x\) is \(\Re\)).

\(\coth\,x=\frac{\cosh\,x}{\sinh\,x}=\frac{e^x+e^{-x}}{e^x-e^{-x}}\) (The domain of \(\coth x\) is \(\Re\setminus\{0\}\)).

\(\text{sech}\,x=\frac{1}{\cosh\,x}=\frac{2}{e^x+e^{-x}}\) (The domain of \(\text{sech}\,x\) is \(\Re\)).

\(\text{csch}\,x=\frac{\sinh\,x}{\cosh\,x}=\frac{e^x-e^{-x}}{e^x+e^{-x}}\) (The domain of \(\text{csch}\,x\) is \(\Re\setminus\{0\}\)).

You may sketch the graphs of these four hyperbolic functions (see exercise \(19\)).

The trigonometric terminology and notation for the hyperbolic functions stem from the fact that they satisfy a list of identities that much resemble the familiar trigonometric identities, apart from an occasional difference of sign.

The trigonometric functions are sometimes called circular functions because the point (\(\cos \theta\,,\sin\,\theta\)) lies on the circle \(x^2+y^2=1\) for all \(\theta\) . Similarly, identity (1) tells us that the point (\(\cosh \theta\,,\sinh\,\theta\)) lies on the hyperbola \(x^2-y^2=1\text{,}\) and this is the reason for the name hyperbolic functions.