< Previous Section Home Next Section >


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. $y = f(2.99) + 0.5dx, 0 ≤ dx ≤ 0.03$

  4. $dy = 0.5dx$

  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 $2.99 ≤ x ≤ 3.02$.