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Differentials as Linear Functions

Two quantities that vary at 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 vary 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 variations in $dx$ and $dy$ as $dx$ varies from 0 to 0.03. However, those variations are hardly perceptible at the scale in which they are presented. If you look closely, you can barely see $dx$ and $dy$ varying.

Figure 4.2.2 shows the same variations in $dx$ and $dy$ as in Figure 4.2.1, but zoomed in. This closeup shows that there is quite a bit going on as $dx$ varies from 0 to 0.03. Figure 4.2.2 suggests that $dy$ varies linearly with $dx$.

We can make the following statements about Figure 4.2.2.

- As $dx$ varies from 0 to 0.03, $dy$ varies from 0 to 0.5∙0.03,
- As
*x*varies from 2.99 to 3.02,*y*varies from 3 to 3.015, - $y = f(2.99)+ 0.5dx, 0 ≤ dx ≤ 0.03$
- $dy = 0.5dx$
- The function
*f*varies at an essentially constant rate of 0.5 with respect to variations in*x*over the interval $2.99\lt x \lt 3.02$.

**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* varies 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 varying 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* varies at a constant rate of 0.5 with respect to variations in *x* over the interval $2.99 ≤ x ≤ 3.02$.

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