Page:Calculus Made Easy.pdf/106

where ${\displaystyle x=0}$, the curve (Fig.22) has no steepness–that is, it is level. On the left of the origin, where ${\displaystyle x}$ has negative values, ${\displaystyle {\dfrac {dy}{dx}}}$ will also have negative values, or will descend from left to right, as in the Figure.

Let us illustrate this by working out a particular instance. Taking the equation

and differentiating it, we get

Now assign a few successive values, say from ${\displaystyle 0}$ to ${\displaystyle 5}$, to ${\displaystyle x}$; and calculate the corresponding values of ${\displaystyle y}$ by the first equation; and of ${\displaystyle {\dfrac {dy}{dx}}}$ from the second equation. Tabulating results, we have:

${\displaystyle x}$	${\displaystyle 0}$	${\displaystyle 1}$	${\displaystyle 2}$	${\displaystyle 3}$	${\displaystyle 4}$	${\displaystyle 5}$
${\displaystyle y}$	${\displaystyle 3}$	${\displaystyle 3{\tfrac {1}{4}}}$	${\displaystyle 4}$	${\displaystyle 5{\tfrac {1}{4}}}$	${\displaystyle 7}$	${\displaystyle 9{\tfrac {1}{4}}}$
${\displaystyle {\tfrac {dy}{dx}}}$	${\displaystyle 0}$	${\displaystyle {\tfrac {1}{2}}}$	${\displaystyle 1}$	${\displaystyle 1{\tfrac {1}{2}}}$	${\displaystyle 2}$	${\displaystyle 2{\tfrac {1}{2}}}$

Then plot them out in two curves, in Figs. 23 and 24 in Fig. 23 plotting the values of ${\displaystyle y}$ against those of ${\displaystyle x}$ and Fig. 24 those of ${\displaystyle {\dfrac {dy}{dx}}}$ against those of ${\displaystyle x}$. For