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# Section 4.2Differentials 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 f varies at a constant rate of 0.5 with respect to variations 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 variations in x as x varies from 2.99 to 3.02. Therefore, with $y = f(x)$, $dy$ varies 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 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 4.2.1. dy varies 0.5 times as much as dx varies, but you can barely see any variations.

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 $dx$.

Figure 4.2.2. dy (the variation in y) is 0.5 times as large as dx (the variation 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 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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