Page:Calculus Made Easy.pdf/103

If a curve has the form of Fig. 18, the value of ${\dfrac {dy}{dx}}$ will be negative in the upper part, and positive in the lower part; while at the nose of the curve where it becomes actually perpendicular, the value of ${\dfrac {dy}{dx}}$ will be infinitely great.

Fig. 18.

Now that we understand that ${\dfrac {dy}{dx}}$ measures the steepness of a curve at any point, let us turn to some of the equations which we have already learned how to differentiate.

(1) As the simplest case take this:

$y=x+b$.

It is plotted out in Fig. 19, using equal scales for $x$ and $y$. If we put $x=0$, then the corresponding ordinate will be $y=b$; that is to say, the “curve” crosses the $y$-axis at the height $b$. From here it