Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/31

of every number between $\scriptstyle {a}$ and $\scriptstyle {b}$ in the order of their magnitudes. This may be illustrated geometrically as follows:

The origin being at $\scriptstyle {O}$, lay off on the straight line the points $\scriptstyle {A}$ and $\scriptstyle {B}$ corresponding to the numbers $\scriptstyle {a}$ and $\scriptstyle {b}$. Also let the point $\scriptstyle {P}$ correspond to a particular value of the variable $\scriptstyle {x}$. Evidently the interval $\scriptstyle {[a,b]}$ is represented by the segment $\scriptstyle {AB}$. Now as $\scriptstyle {x}$ varies continuously from $\scriptstyle {a}$ to $\scriptstyle {b}$ inclusive, i.e. through the interval $\scriptstyle {[a,b]}$, the point $\scriptstyle {P}$ generates the segment $\scriptstyle {AB}$.

9. Functions. When two variables are so related that the value of the first variable depends on the value of the second variable, then the first variable is said to be a function of the second variable.

Nearly all scientific problems deal with quantities and relations of this sort, and in the experiences of everyday life we are continually meeting conditions illustrating the dependence of one quantity on another. For instance, the weight a man is able to lift depends on his strength, other things being equal. Similarly, the distance a boy can run may be considered as depending on the time. Or, we may say that the area of a square is a function of the length of a side, and the volume of a sphere is a function of its diameter.

10. Independent and dependent variables. The second variable, to which values may be assigned at pleasure within limits depending on the particular problem, is called the independent variable, or argument; and the first variable, whose value is determined as soon as the value of the independent variable is fixed, is called the dependent variable, or function.

Frequently, when we are considering two related variables, it is in our power to fix upon whichever we please as the independent variable; but having once made the choice, no change of independent variable is allowed without certain precautions and transformations.

One quantity (the dependent variable) may be a function of two or more other quantities (the independent variables, or arguments). For example, the cost of cloth is a function of both the quality and quantity; the area of a triangle is a function of the base and altitude; the volume of a rectangular parallelepiped is a function of its three dimensions.