Page:Calculus Made Easy.pdf/106

where $x=0$, the curve (Fig.22) has no steepness–that is, it is level. On the left of the origin, where $x$ has negative values, ${\dfrac {dy}{dx}}$ 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

$y={\tfrac {1}{4}}x^{2}+3$,

and differentiating it, we get

${\dfrac {dy}{dx}}={\tfrac {1}{2}}x$.

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

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

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