ELEMENTARY FUNCTIONS

CHAPTER I

FUNCTIONS, EQUATIONS, AND GRAPHS

1. Comparison of the Reasoning in Natural Science and in Mathematics. In science, the method of procedure in determining the law of phenomena is as follows:

1. Observations are recorded, compared, and classified.
2. An induction is made and the generalization resulting
is stated as a hypothesis.

3. Deduction from the hypothesis leads to a conclusion. 4. The conclusions are tested and verified by experiment. Mathematical reasoning is not inductive, and hence it is not of the nature of stages 1 and 2 of the scientific method. But when a science has advanced to the point where the data are expressed in terms of magnitude, the generalization can be expressed mathematically in simple and compact form, and the deductive process of the scientific method can be carried out by the direct and powerful methods of mathematics.

The reasoning in mathematics is purely deductive in character, and the conclusion reached contains no more than the hypothesis from which it was derived. If the conclusion were more general than the hypothesis, then it would be certain that the deductive reasoning was not performed correctly. While a correctly deduced conclusion states nothing which was not included in the hypothesis, it is in a form which can be more easily comprehended and more readily used. The process of going from hypothesis to conclusion may be likened to the unwrapping of a compact bundle; there is no more pertaining to the bundle at the end of the operation

than there was at the beginning, but the contents are disclosed to the mind, and can be examined. What is implicit in a hypothesis becomes explicit in the conclusion.

In natural science, if the conclusion when tested does not check with experience, the observations are reëxamined, increased in number, and in some cases after a long interval of time and much work by many men, a new hypothesis is stated, and the rest of the scientific process repeated. In mathematics, the question of the truth or usefulness of the hypothesis may not be raised, but the accuracy of the deduction is an everpresent concern, and should be constantly tested by checks.

The verification in the scientific method involves reference to the field of observation and is not deductive in character, and hence not mathematical.

2. Example of the Utility of Mathematics in Science. A classical example of the scientific method and the part which mathematics plays in natural science is furnished by the steps leading to the discovery of the planet Neptune.

Observations on the motions of the planets of the solar system were recorded in great number by the astronomer Tycho Brahe (1546–1601).

His assistant, Johannes Kepler (1571-1630), generalized from the observations and stated the hypotheses known as Kepler's laws.

Sir Isaac Newton (1642-1727), by means of mathematics, condensed Kepler's three laws into one, the law of gravitation.

Up to 1846 Uranus was the outermost planet of the solar system then known. The irregularities in its orbit led astronomers to suspect that there was another planet outside Uranus which caused these disturbances.

LeVerrier and Adams, independently, using Newton's law and the facts of the disturbances, deduced mathematically the position that an outer planet must occupy to produce these perturbations. Adams spent months in carrying through the intricate calculations necessary and in checking the accuracy of his deductions, which he did by solving the problem a number of times in different ways.

The last stage, the verification, was performed by Galle, who turned his telescope in the direction indicated by LeVerrier, and on September 23, 1846, discovered Neptune. Here was a wonderful verification of Newton's law and a monument to the power and value of mathematics in making deductions. 3. Forms in which Data are Recorded. The observations of the scientist are usually recorded in the form of a table of two related sets of magnitudes. The biologist compares the width of a leaf with the number of specimens examined having that width; the physicist, the expansion of a substance with the temperature; the chemist, the amount of substance in solution with the time.

The following tables illustrate the manner of recording the data.

The data are sometimes recorded by an instrument. The temperature and barometer records of the Weather Bureau are so recorded. A pen is connected with the thermometer and rests against a sheet of paper fastened to a revolving cylinder. The cylinder revolves steadily and the pen rising and falling with the temperature traces a curve on the paper, from which the temperature at any time may be determined.

The variation in the tide at an important port is similarly recorded on a chart fastened to a revolving cylinder by a pen attached to a float which rises and falls with the tide (Figure 6).

The data may be presented most conveniently in the form of a generalization, expressed either in words or in mathematical symbols. For example, a comparison of the pairs of values in the second table above, neglecting the first two pairs of values, leads to the following generalization, expressed in words:

For a temperature of 400°, or greater, an increase of 200° in the temperature causes an expansion of approximately 0.1 of a millimeter in the length of the rod.

The generalization may also be expressed in symbols. If every increase of 200° causes an expansion of 0.1 of a millimeter,

then an increase of 1° would cause an expansion of 0.1/200, or 0.0005; hence for an increase of temperature of t degrees the expansion e would be

e = 0.0005t.

This equation expresses the generalization in more compact form than the sentence above. In this illustration we have considered only a part of the table, a part for which the generalization is very simple.

The determination of the mathematical expression of the generalization from a table of values will be one of the objects of this course. The generalization from a mathematical point of view is considered in the following section.

4. Variable. Function. The following table gives the lengths of an iron bar suspended from one end when carrying different loads.

Any one of the numbers in each of these sets of numbers is conveniently represented by a single symbol, as x, y, t, etc. Thus x may be taken to represent one of the numbers 0, 500, 1000, 1500, 2000 and any other number that might be included in the half of the table giving loads, while y may represent the corresponding number giving lengths in the second part of the table.

The symbols x and y are called variables, in accordance with the

DEFINITION. A variable is a symbol for any one of a set of numbers.

In the experiment giving rise to the above table, the load was changed arbitrarily by the experimenter and the length of the bar for the chosen load measured. In consequence of this order of measurement the variable representing the load is called the independent variable and the one representing the length of the bar the dependent variable.

It is customary to denote the independent variable by x and the dependent variable by y.

If the experiment were repeated, under the same conditions, it would be found that for a specified load the length of the bar would be the same, that is, there is a law connecting the load and the length of the bar. This relation between the variables is expressed by saying that the length, y, is a function of the load, x, in accordance with the

DEFINITION. A function is a variable so related to another variable (called the independent variable) that for every admissible value of the independent variable, one or more values of the function are determined. The function is also called the dependent variable.

The idea of a function arises wherever there is a relation between magnitudes which are changing, and it underlies all magnitude relations which mankind has discovered.

EXAMPLE 1. The algebraic expression 2x + 3 is a variable whose value is determined whenever a definite value is assigned to x. If x be given the value 1, then 2x + 3 has the value of 5, and if x has the value 2, 2x + 3 has the value 7. Hence 2x + 3 is a function of x.