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:

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