Page:Calculus Made Easy.pdf/205

Slopes of Curves, and the Curves themselves.

Let us make a little preliminary enquiry about the slopes of curves. For we have seen that differentiating a curve means finding an expression for its slope (or for its slopes at different points). Can we perform the reverse process of reconstructing the whole curve if the slope (or slopes) are prescribed for us?

Go back to case (2) on p. 84. Here we have the simplest of curves, a sloping line with the equation

$y=ax+b$.

Fig. 47.

We know that here $b$ represents the initial height of $y$ when $x=0$, and that $a$, which is the same as ${\dfrac {dy}{dx}}$, is the “slope” of the line. The line has a constant slope. All along it the elementary triangles have the same proportion between height and base. Suppose we were to take the $dx$’s, and $dy$’s of finite