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.



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