Page:Calculus Made Easy.pdf/207

 all alike, clearly $$\dfrac{dy}{dx} = a$$, if we reckon $$y$$ as the total of all the $$dy$$’s, and $$x$$ as the total of all the $$dx$$’s. But whereabouts are we to put this sloping line? Are we to start at the origin $$O$$, or higher up? As the only information we have is as to the slope, we are without any instructions as to the particular height above $$O$$; in fact the initial height is undetermined. The slope will be the same, whatever the initial height. Let us therefore make a shot at what may be wanted, and start the sloping line at a height $$C$$ above $$O$$. That is, we have the equation

It becomes evident now that in this case the added constant means the particular value that $$y$$ has when $$x=0$$.

Now let us take a harder case, that of a line, the slope of which is not constant, but turns up more and more. Let us assume that the upward slope gets greater and greater in proportion as $$x$$ grows. In symbols this is:

Or, to give a concrete case, take $$a = \tfrac{1}{5}$$, so that

Then we had best begin by calculating a few of the values of the slope at different values of $$x$$, and also draw little diagrams of them.