Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/50

26 Assuming $$\scriptstyle{x=10}$$ for the initial value of $$\scriptstyle{x}$$ fixes $$\scriptstyle{y=100}$$ as the initial value of $$\scriptstyle{y}$$.

It may happen that as $$\scriptstyle{x}$$ increases, $$\scriptstyle{y}$$ decreases, or the reverse; in either case $$\scriptstyle{\Delta x}$$ and $$\scriptstyle{\Delta y}$$ will have opposite signs.

It is also clear (as illustrated in the above example) that if $$\scriptstyle{y=f(x)}$$ is a continuous function and $$\scriptstyle{\Delta x}$$ is decreasing in numerical value, then $$\scriptstyle{\Delta y}$$ also decreases in numerical value.

Consider the function Rh

Assuming a fixed initial value for $$\scriptstyle{x}$$, let $$\scriptstyle{x}$$ take on an increment $$\scriptstyle{\Delta x}$$. Then $$\scriptstyle{y}$$ will take on a corresponding increment $$\scriptstyle{\Delta y}$$, and we have we get the increment $$\scriptstyle{\Delta y}$$ in terms of $$\scriptstyle{x}$$ and $$\scriptstyle{\Delta x}$$.

To find the ratio of the increments, divide (B) by $$\scriptstyle{\Delta x}$$, giving

If the initial value of $$\scriptstyle{x}$$ is $$\scriptstyle{4}$$, it is evident that

Let us carefully note the behavior of the ratio of the increments of $$\scriptstyle{x}$$ and $$\scriptstyle{y}$$ as the increment of $$\scriptstyle{x}$$ diminishes.