Page:Calculus Made Easy.pdf/32

 positive increment, the increment which results to $$y$$ is negative.

Yes, but how much? Suppose the ladder was so long that when the bottom end $$A$$ was $$19$$ inches from the wall the top end $$B$$ reached just $$15$$ feet from the ground. Now, if you were to pull the bottom end out $$1$$ inch more, how much would the top end come down? Put it all into inches: $$x=19$$ inches, $$y=180$$ inches. Now the increment of $$x$$ which we call $$dx$$, is $$1$$ inch: or $$x+dx=20$$ inches.

How much will $$y$$ be diminished? The new height will be $$y-dy$$. If we work out the height by Euclid I. 47, then we shall be able to find how much $$dy$$ will be. The length of the ladder is $\sqrt{(180)^2+(19)^2}=181$ inches. Clearly then, the new height, which is $$y-dy$$, will be such that

Now $$y$$ is $$180$$, so that $$dy$$ is $$180-179.89=0.11$$ inch.

So we see that making $$dx$$ an increase of $$1$$ inch has resulted in making $$dy$$ a decrease of $$0.11$$ inch.

And the ratio of $$dy$$ to $$dx$$ may be stated thus:

It is also easy to see that (except in one particular position) $$dy$$ will be of a different size from $$dx$$.

Now right through the differential calculus we are hunting, hunting, hunting for a curious thing,