# Section
4.2 Differentials as Linear Functions

Two quantities that are related by a constant rate with respect to each
other are said to be related linearly. One reason for saying that they are
related linearly is that the graph of one as a function of the other, in
Cartesian coordinates, is a line. By definition, differentials change at a
constant rate with respect to each other. Therefore, differentials are
related linearly. A graph in Cartesian coordinates of either one in relation
to the other will be a line.

A function *f* changes at a constant rate of 0.5 with respect to
changes in *x* over the interval $2.99 ≤ x ≤ 3.02$, and
$f(2.99) = 3$. We will examine the behavior of $y = f(x)$ over this
interval.

Values of *f*(*x*) vary at a constant rate of 0.5 with respect to
changes in *x* as *x* changes from 2.99 to 3.02. Therefore, with
$y = f(x)$, d*y* changes at a constant rate of 0.5 with respect to d*x*
when *x* = 2.99 and d*x* varies from 0 to 0.03. That
is, instead of thinking of *x* varying from 2.99 to 3.02, we can think
of *x* being fixed at 2.99 and d*x* varying from 0 to 0.03.

Figure 4.2.1 shows the changes in d*x* and d*y* as d*x*
varies from 0 to 0.03. However, those changes are hardly perceptible at the
scale in which they are presented. If you look closely, you can barely see d*x*
and d*y* changing.

*Figure 4.2.1. dy changes 0.5 times as
much as dx changes, but you can barely see any changes.*

Figure 4.2.2 shows the same changes in d*x* and d*y* as in Figure
4.2.1, but zoomed in. This closeup shows that there is quite a bit going on
as d*x* varies from 0 to 0.03. Figure 4.2.2 suggests that d*y*
varies linearly with d*x*.

*Figure 4.2.2. dy (the change in y) is
0.5 times as large as dx (the change in x). As dx varies from 0 to 0.03,
dy varies from 0 to 0.5∙0.03*

We can make the following statements about Figure 4.2.2.

- As d
*x* varies from 0 to 0.03, d*y* varies from 0 to
0.5∙0.03,

- As
*x* varies from 2.99 to 3.02, *y* varies from 3 to
3.015,

- $y = f(2.99) + 0.5\mathrm{d}x, 0 ≤ \mathrm{d}x ≤ 0.03$

- $\mathrm{d}y = 0.5\mathrm{d}x$

- The function
*f* changes at a constant rate of 0.5 with respect
to changes in * x* over the interval
$2.99 ≤ x ≤ 3.02$.

*Reflection 4.2.1.* While all of statements a-e are true, they are not
equivalent—they do not have exactly the same meanings. Organize these
statements so that statements in a group have the same meanings? Are the
meanings in different groups close to being the same?

Reflection 4.2.2. We were told that *f* changes at a constant
rate with respect to *x* over the interval $2.99 \le x \le 3.02$, but
we were not told anything about values of *f * elsewhere. Sketch a
possible graph for *f* that shows it changing at a non-constant rate
everywhere other than $2.99 ≤ x ≤ 3.02$. The only constraints
on your graph are that $f(2.99) = 3$ and that *f* changes at
a constant rate of 0.5 with respect to changes in *x* over the
interval $2.99 ≤ x ≤ 3.02$.