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Activity4.4Higher Order Derivatives

Seeing as the first derivative of \(f\) is a function in its own right, \(\fd{f}\) must have its own first derivative. The first derivative of \(\fd{f}\) is the second derivative of \(f\) and is symbolized as \(\sd{f}\) (\(f\) double-prime). Likewise, \(\td{f}\) (\(f\) triple-prime) is the first derivative of \(\sd{f}\), the second derivative of \(\fd{f}\), and the third derivative of \(f\).

All of the graphical relationships you've established between \(f\) and \(\fd{f}\) work their way down the derivative chain; this is illustrated in Table 4.4.1.

When \(\fd{f}\) is … \(f\) is …
When \(\sd{f}\) is … \(\fd{f}\) is …
When \(\td{f}\) is … \(\sd{f}\) is …
Positive Increasing
Negative Decreasing
Constantly Zero Constant
Increasing Concave Up
Decreasing Concave Down
Constant Linear
Table4.4.1Relationships between \(f\), \(\fd{f}\), \(\sd{f}\) and \(\td{f}\)

Subsection4.4.1Exercises

1

Extrapolating from Table 4.4.1, what must be true about \(f\) over intervals where \(\sd{f}\) is, respectively: positive, negative, or constantly zero?

2

A function, \(g\), and its first three derivatives are shown in Figures 4.4.24.4.5, although not in that order. Determine which curve is which function (\(g,\fd{g},\sd{g},\td{g}\)).

Figure4.4.2
Figure4.4.3
Figure4.4.4
Figure4.4.5

Three containers are shown in Figures 4.4.64.4.8. Each of the following questions are in reference to these containers.

Figure4.4.6
Figure4.4.7
Figure4.4.8
3

Suppose that water is being poured into each of the containers at a constant rate. Let \(h_a\), \(h_b\), and \(h_c\) be the heights (measured in cm) of the liquid in containers 4.4.64.4.8 respectively, \(t\) seconds after the water began to fill the containers. What would you expect the sign to be on the second derivative functions \(\sd{h_a}\), \(\sd{h_b}\), and \(\sd{h_c}\) while the containers are being filled? (Hint: Think about the shape of the curves \(y=\fe{h_a}{t}\), \(y=\fe{h_b}{t}\), and \(y=\fe{h_c}{t}\).)

4

Suppose that water is being drained from each of the containers at a constant rate. Let \(h_a\), \(h_b\), and \(h_c\) be the heights (measured in cm) of the liquid remaining in the containers \(t\) seconds after the water began to drain. What would you expect the sign to be on the second derivative functions \(\sd{h_a}\), \(\sd{h_b}\), and \(\sd{h_c}\) while the containers are being drained?

5

During the recession of 2008–2009, the total number of employed Americans decreased every month. One month a talking head on the television made the observation that “at least the second derivative was positive this month.” Why was it a good thing that the second derivative was positive?

6

During the early 1980s the problem was inflation. Every month the average price for a gallon of milk was higher than the month before. Was it a good thing when the second derivative of this function was positive? Explain.