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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 $y = f(x)$, dy changes at a constant rate of 0.5 with respect to dx when x = 2.99 and dx 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 dx varying from 0 to 0.03.

Figure 4.2.1 shows the changes in dx and dy as dx 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 dx and dy 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 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.
 

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.
  1. As dx varies from 0 to 0.03, dy varies from 0 to 0.5∙0.03,

  2. As x varies from 2.99 to 3.02, y varies from 3 to 3.015,

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

  4. $\mathrm{d}y = 0.5\mathrm{d}x$

  5. 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 $2.99 ≤ x ≤ 3.02$.