Practical Mathematical Examples

Section 2.1 Practical Mathematical Examples

We start out with a few problems as examples that arise in mathematics where numerical analysis plays an important role.

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Example 2.1.1.

Using only the four basic operations \(+,-,\cdot, /\) on a calculator, find \(\sqrt{65}\) to within 0.001.

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Also, answer the following questions:

Write down the steps (the algorithm) that you used.
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How do you know your answer is correct to 3 decimal places?
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Can you use this technique to find \(\sqrt{a}\) to \(d\) decimal places for any positive number \(a\) and positive integer \(d\text{?}\) Explain.
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The following is a classic example of an example where numerical analysis helps another mathematical field, specifically Statistics. Typically users in an introductory class are given tables of distributions or use software to calculate probabilities. Where do those tables come from? How does the software perform the calculations? The following specifically asks some questions related to the normal distribution.

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Example 2.1.2.

If a random variable, \(X\) is normally distributed with mean \(\mu\) and standard deviation \(\sigma\text{,}\) the probability density function can be written as

\begin{equation*} f(x;\mu,\sigma) = \frac{1}{\sigma \sqrt{2\pi}}e^{-(x-\mu)^2/\sigma^2} \end{equation*}

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The probability that \(X\) is less than \(a\) is given by

\begin{align} P(X \lt a) \amp = \int_{-\infty}^{a} f(x;\mu,\sigma) \, dx \notag\\ \amp = \int_{-\infty}^{a} \frac{1}{\sigma \sqrt{2\pi}}e^{-(x-\mu)^2/\sigma^2}\, dx \tag{2.1.1} \end{align}

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Here’s a few standard questions:

If we are given \(\mu\text{,}\) \(\sigma\) and \(a\text{,}\) how do we calculate \(P(X<a)\text{?}\)
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For a given \(\mu\) and \(\sigma\) and \(k\) for \(0<k<1\text{,}\) how do we solve \(P(X<a)=k\text{?}\)
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The following shows another common example in which data is given and information needs to be extracted from it.

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Example 2.1.3.

The following table lists the price of hard drive space from 1980 to 2015.

This was originally found at http://www.jcmit.com/diskprice.htm, however, it is no longer available there.

The price is given in USD per GB.

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Table 2.1.4. Historical Price Data

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Year	Price	Year	Price	Year	Price	Year	Price
’80	6,328,125	’89	124,877	’98	1,129	’07	45.29
’81	4,374,218	’90	95,947	’99	847.93	’08	21.45
’82	1,933,593	’91	43,457	’00	823.82	’09	11.20
’83	1,933,593	’92	31,738	’01	297.54	’10	19.07
’84	1,299,479	’93	29,296	’02	202.52	’11	8.34
’85	703,125	’94	33,203	’03	112.53	’12	4.76
’86	292,480	’95	30,151	’04	171.66	’13	7.15
’87	172,037	’96	10,986	’05	113.48	’14	8.34
’88	178,001	’97	4,028	’06	80.09	’15	4.17

Find a function that gives a prediction of the price per GB for any year after 1980.

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Subsection 2.1.1 What is Numerical Analysis?

In general, this course and this subject in many ways is about solving problems. Often in math classes, only reasonable problems are given in classes as examples, homework and exam problems and by reasonable, one means a problem that can be solved in a few minutes to maybe an hour using relatively standard techniques. In the real world, however, the problems aren’t always so nice, and in short this course will teach you how to attack these harder problems.

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The problems in Example 2.1.1 and Example 2.1.3 either arise on occasion or were difficult problems to solve historically. In more detail:

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As in Example 2.1.1, finding the square root of any number seems trivial today with calculators and computers, but before 1950 it was difficult. In general this falls into rootfinding which is the subject of Chapter 3. Extensions of rootfinding can be found in optimization problems which are sometimes full courses in and of themselves.
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Example 2.1.2 falls into two different categories: probability and statistics as well as numerical integration. This problem uses the definite integral of a function that doesn’t have an antiderivative (as a elementary function). We will examine this in Chapter 5.
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Example 2.1.3 falls into the realm of either regression or interpolation. Often a set of data is given and you want to find a function that either passes through all of the data (interpolation) or is closest to some known function (regression). Interpolation will be presented in Chapter 4 and approximation theory like regression will be studied in Chapter 6.
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