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Activity4.2Graphical Derivatives

Much information about a function's first derivative can be gleaned simply by looking at a graph of the function. In fact, a person with good visual skills can “see” the graph of the derivative while looking at the graph of the function. This activity focuses on helping you develop that skill.

Subsection4.2.1Exercises

A parabolic function is shown in Figure 4.2.1. The next several questions are in reference to that function.

\(x\) \(y\)
\(-5\) \(6\)
\(-3\)
\(-1\) \(2\)
\(1\)
\(3\) \(-2\)
\(5\) \(-4\)
\(7\)
Figure4.2.1\(y=\fe{g}{x}\)
Figure4.2.2\(y=\fe{g}{x}\)
Table4.2.3\(y=\fe{\fd{g}}{x}\)
1

Several values of \(\fd{g}\) are given in Table 4.2.3. For each given value of \(x\) draw onto Figure 4.2.1 a nice long line segment at the corresponding point on \(g\) whose slope is equal to the value of \(\fd{g}\). If we think of these line segments as actual lines, what do we call the lines?

2

What is the value of \(\fe{\fd{g}}{1}\)? How do you know? Enter that value into Table 4.2.3.

3

The function \(g\) is symmetric across the line \(x=1\); that is, if we move equal distance to the left and right from this line the corresponding \(y\)-coordinates on \(g\) are always equal. Notice that the slopes of the tangent lines are “equal but opposite” at points that are equally removed from the axis of symmetry; this is reflected in the values of \(\fe{\fd{g}}{-1}\) and \(\fe{\fd{g}}{3}\). Use the idea of “equal but opposite slope equidistant from the axis of symmetry” to complete Table 4.2.3.

4

Plot the points from Table 4.2.3 onto Figure 4.2.2 and connect the dots. Determine the formula for the resultant linear function.

5

The formula for \(\fe{g}{x}\) is \(-0.5x^2+x+5.5\). Use Definition 3.3.1 to determine the formula for \(\fe{\fd{g}}{x}\).

6

The line you drew onto Figure 4.2.2 is not a tangent line to \(g\). Just what exactly is this line?

A function \(f\) is shown in Figure 4.2.4 and the corresponding first derivative function \(\fd{f}\) is shown in Figure 4.2.5. Answer each of the following questions referencing these two functions.

Figure4.2.4\(y=\fe{f}{x}\)
Figure4.2.5\(y=\fe{\fd{f}}{x}\)
7

Draw the tangent line to \(f\) at the three points indicated in Figure 4.2.4 after first using the graph of \(\fd{f}\) to determine the exact slope of the respective tangent lines.

8

Write a sentence relating the slope of the tangent line to \(f\) with the corresponding \(y\)-coordinate on \(\fd{f}\).

9

Copy each of the following sentences onto your paper and supply the words or phrases that correctly complete each sentence.

  • Over the interval where \(\fd{f}\) is negative, \(f\) is …

  • Over the interval where \(\fd{f}\) is positive, \(f\) is …

  • Over the interval where \(\fd{f}\) is increasing, \(f\) is …

  • Over the interval where \(\fd{f}\) is decreasing, \(f\) is …

10

In each of Figures 4.2.6 and 4.2.7 a function (the solid curve) is given; both of these functions are symmetric about the \(y\)-axis. The first derivative of each function (the dash-dotted curves) have been drawn over the interval \(\ointerval{0}{7}\). Use the given portion of the first derivative together with the symmetry of the function to help you draw each first derivative over the interval \(\ointerval{-7}{0}\). Please note that a function that is symmetric across a vertical line has a first derivative that is symmetric about a point. Similarly, a function that is symmetric about a point has a first derivative that is symmetric across a vertical line.

Figure4.2.6\(y=\fe{f}{x}\) and part of the curve \(y=\fe{\fd{f}}{x}\)
Figure4.2.7\(y=\fe{f}{x}\) and part of the curve \(y=\fe{\fd{f}}{x}\)

A graph of the function \(f\) given by \(\fe{f}{x}=\frac{1}{x}\) is shown in Figure 4.2.8.

Figure4.2.8\(y=\fe{f}{x}=\frac{1}{x}\)
Figure4.2.9\(y=\fe{\fd{f}}{x}\)
11

Except at \(0\), there is something that is always true about the value of \(\fd{f}\); what is the common trait?

14

Use the formula for \(\fe{\fd{f}}{x}\) to determine the horizontal and vertical asymptotes to the graph of \(y=\fe{\fd{f}}{x}\).

15

Keeping it simple, draw onto Figure 4.2.9 a curve with the asymptotes found in Exercise 4.2.1.14 and the property determined in Exercise 4.2.1.11. Does the curve you drew have the properties you would expect to see in the first derivative of \(f\)? For example, \(f\) is concave down over \(\ointerval{-\infty}{0}\), and concave up over \(\ointerval{0}{\infty}\); what are the corresponding differences in the behavior of \(\fd{f}\) over those two intervals?

16

A graph of the function \(g\) is shown in Figure 4.2.10. The absolute minimum value ever obtained by \(\fd{g}\) is \(-3\). With that in mind, draw \(\fd{g}\) onto Figure 4.2.11. Make sure that you draw and label any and all necessary asymptotes. Please note that \(g\) is symmetric about the point \(\point{1}{2}\).

Figure4.2.10\(y=\fe{g}{x}\)
Figure4.2.11\(y=\fe{\fd{g}}{x}\)

Answer the following questions in reference to the function \(w\), shown in Figure 4.2.12.

Figure4.2.12\(y=\fe{w}{x}\)
17

An inflection point is a point where the function is continuous and the concavity of the function changes.

The inflection points on \(w\) occur at \(2\), \(3.25\), and \(6\). With that in mind, over each interval stated in Table 4.2.13 exactly two of the properties from its caption apply to \(\fd{w}\). Complete Table 4.2.13 with the appropriate pairs of properties.

Interval Properties of \(\fd{w}\)
\(\ointerval{-\infty}{2}\) \(\phantom{\text{positive/negative, increasing/decreasing}}\)
\(\ointerval{2}{3.25}\)
\(\ointerval{3.25}{4}\)
\(\ointerval{4}{6}\)
\(\ointerval{6}{7}\)
\(\ointerval{7}{\infty}\)
Table4.2.13Properties of \(\fd{w}\) (positive, negative, increasing, or decreasing)
18

In Table 4.2.14, three possible values are given for \(\fd{w}\) at several values of \(x\). In each case, one of the values is correct. Use tangent lines to \(w\) to determine each of the correct values.

\(x\) Proposed values
\(0\) \(\sfrac{2}{3}\) or \(\sfrac{8}{3}\) or \(\sfrac{28}{3}\)
\(1\) \(\sfrac{1}{2}\) or \(\sfrac{3}{2}\) or \(\sfrac{5}{2}\)
\(3\) \(\sfrac{1}{3}\) or \(1\) or \(3\)
\(5\) \(-\sfrac{1}{2}\) or \(-1\) or \(-\sfrac{3}{2}\)
\(6\) \(-\sfrac{4}{3}\) or \(-\sfrac{8}{3}\) or \(-4\)
\(8\) \(1\) or \(6\) or \(12\)
Table4.2.14Choose the correct values for \(\fd{w}\)
Figure4.2.15\(y=\fe{\fd{w}}{x}\)
19

The value of \(\fd{w}\) is the same at \(2\), \(4\), and \(7\). What is this common value?