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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

Values of

Figure 4.2.1 shows the changes in d

Figure 4.2.2 shows the same changes in d

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 $2.99 ≤ x ≤ 3.02$.