On Relative Growings
Excerpts
On Relative Growings
Those which we regard as of fixed value, and call *constants*, we generally denote algebraically by letters from the beginning of the alphabet, such as $a$, $b$, or $c$; while those which we consider as capable of growing, or (as mathematicians say) of “varying,” we denote by letters from the end of the alphabet, such as $x$, $y$, $z$, $u$, $v$, $w$, or sometimes $t$.
On Relative Growings
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?
On Relative Growings
It is a solemn scientific name for this very simple thing. But we are not going to be frightened by solemn names, when the things themselves are so easy.
On Relative Growings
You have now to learn to go hunting in a new way; the fox being now neither $x$ nor $y$. Instead of this you have to hunt for this curious cub called $\dfrac{dy}{dx}$.
On Relative Growings
We classify all quantities into two classes: *constants* and *variables*. Those which we regard as of fixed value, and call *constants*, we generally denote algebraically by letters from the beginning of the alphabet, such as $a$, $b$, or $c$; while those which we consider as capable of growing, or (as mathematicians say) of “varying,” we denote by letters from the end of the alphabet, such as $x$, $y$, $z$, $u$, $v$, $w$, or sometimes $t$.
On Relative Growings
So we see that making $dx$ an increase of $1$ inch has resulted in making $dy$ a decrease of $0.11$ inch.
On Relative Growings
For example $x^2 + 3 = 2y - 7$ is an implicit function in $x$ and $y$; it may be written $y = \dfrac{x^2 + 10}{2}$ (explicit function of $x$) or $x = \sqrt{2y - 10}$ (explicit function of $y$).
On Relative Growings
Now right through the differential calculus we are hunting, hunting, hunting for a curious thing, a mere ratio, namely, the proportion which $dy$ bears to $dx$ when both of them are indefinitely small.
On Relative Growings
If, while $x$ is, as before, the distance of the foot of the ladder from the wall, $y$ is, instead of the height reached, the horizontal length of the wall, or the number of bricks in it, or the number of years since it was built, any change in $x$ would naturally cause no change whatever in $y$; in this case $\dfrac{dy}{dx}$ has no meaning whatever, and it is not possible to find an expression for it.
On Relative Growings
It will never do to fall into the schoolboy error of thinking that $dx$ means $d$ times $x$, for $d$ is not a factor---it means “an element of” or “a bit of” whatever follows.
Equations
On Relative Growings
\frac{dy}{dx} = \frac{1}{1.73}For a right triangle whose other side slopes at a fixed 30°, the ratio of a small increase in height to the matching small increase in base is 1/1.73.
On Relative Growings
\frac{dy}{dx} = - \frac{0.11}{1}In the ladder example, moving the foot of the ladder 1 inch further from the wall lowers the top by 0.11 inch, so dy/dx is -0.11 per inch.
On Relative Growings
x^2 + y^2 = l^2The foot distance x, the height y and the fixed ladder length l of a ladder against a wall satisfy the Pythagorean relation.
On Relative Growings
y &= x \tan 30°The height of the triangle written explicitly as a function of its base, for a 30° angle.
On Relative Growings
y &= \sqrt{ l^2 - x^2}The height of the ladder on the wall written explicitly as a function of the foot distance from the wall.
On Relative Growings
x^2 + 3 = 2y - 7An example of an implicit function relating x and y, which can be rewritten either way as an explicit function.
On Relative Growings
y = \dfrac{x^2 + 10}{2}The example implicit relation x^2 + 3 = 2y - 7 written as y explicitly in terms of x.
On Relative Growings
u = x^2 \sin \thetaAn example function in which u depends on x and theta, so x and theta are the independent variables and u the dependent variable.
Problems
No exercises in this chapter.