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Chapter 3: Applications of Differentiation
Section 3.7: What Derivatives Reveal about Graphs
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Example 3.7.3
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Graph for ; then use the tools of the calculus to analyze the features of this graph.
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Although is a polynomial, it presents two distinct problems. First, it is of degree six, so neither nor will have exact solutions. Moreover, even though has exact solutions, they would most likely be so cumbersome as to be useless. Hence, the analysis of the graph has to be based on numeric calculations. Second, is very large in the specified domain, so in any reasonably sized graph the relevant features where is not large will be "swamped" by the region where this magnitude is large. Hence, the domain must be divided accordingly when analyzing the features of the required graph.
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Solution
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Initialize
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Tools≻Load Package:
Student Calculus 1
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Loading Student:-Calculus1
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Control-drag
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Context Panel: Assign Function
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Curve Analysis Tutor
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Figure 3.7.12, an image of the
tutor, illustrates the features of the graph of that can be determined from itself, and from the derivatives and .
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Where is increasing or decreasing, its graph is drawn in red or black, respectively,
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Intervals where the graph of is concave up or down are shaded in gray or yellow, respectively.
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Relative extrema and inflection points are shown in green.
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Selecting one of the eight radio-buttons and clicking the "Calculate" button yields the information listed in Table 3.7.4.
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Figure 3.7.3(b) uses the FunctionChart (a.k.a. FunctionPlot) command to draw the graph contained in Figure 3.7.3(a). The command provides slightly more control over the features of the graph. The symbols for the seven green points can be made larger, and arrows are used to indicate concavity.
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The -intercepts are marked with circles; the inflection points, with crosses; and the extreme points with diamonds. These distinctions are not visible in the tutor.
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Figure 3.7.3(a) Curve Analysis tutor applied to
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Student:-SetColors(red,black,green,gray,yellow):
Student:-Calculus1:-FunctionChart(x^6-10*x^5-15*x^4+140*x^3+160*x^2-528*x-800,x=-4..11,pointoptions=[symbolsize=20],caption=[],concavity=[filled(gray,yellow)]);
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Figure 3.7.3(b) Graph via the FunctionChart command
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The graph in Figure 3.7.3(b) provides a bit more insight than the graph in the Curve Analysis tutor (Figure 3.7.3(a)). However, the Curve Analysis tutor does provide useful calculations. Table 3.7.3(a) displays the information that would be provided by the "Calculate" button in the tutor. Note that a number such as .541e4 represents × = 5410.
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The local maxima occur at:
[-4., .541e4]
[3., -80.]
[11., .141e6]
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The local minima occur at:
[.866, -.106e4]
[8.47, -.526e5]
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The function is increasing on the intervals:
[.866, 3.]
[8.47, 11.]
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The function is decreasing on the intervals:
[-4., -2.]
[-2., .866]
[3., 8.47]
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The function is concave up on the intervals:
[-4., -2.]
[-.365, 2.11]
[6.92, 11.]
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The function is concave down on the intervals:
[-2., -.365]
[2.11, 6.92]
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The points of inflection occur at:
[-2., -76.]
[-.365, -593.]
[2.11, -510]
[6.92, -.338e5]
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The zeros occur at :
-2.59
10.0
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Table 3.7.3(a) Data generated by the Curve Analysis tutor for
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Figures 3.7.3(c) and 3.7.3(d), by limiting the domains to , and , respectively, show greater detail in the regions where is first small, then large.
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Student:-SetColors(red,black,green,gray,yellow):
Student:-Calculus1:-FunctionChart(x^6-10*x^5-15*x^4+140*x^3+160*x^2-528*x-800,x=-4..4,pointoptions=[symbolsize=20],caption=[],concavity=[filled(gray,yellow)]);
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Figure 3.7.3(c) Graph of on
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Student:-SetColors(red,black,green,gray,yellow):
Student:-Calculus1:-FunctionChart(x^6-10*x^5-15*x^4+140*x^3+160*x^2-528*x-800,x=4..11,pointoptions=[symbolsize=20],caption=[],concavity=[filled(gray,yellow)]);
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Figure 3.7.3(d) Graph of on
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Obtain the Critical Numbers
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Obtain the critical numbers , by solving the equation
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Write and press the Enter key.
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Context Panel: Solve≻Numerically Solve
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Context Panel: Conversions≻To List
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Context Panel: Assign Name≻c
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Second-Derivative Test
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Apply the Second-Derivative test
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Since , the point is a candidate for an inflection, a conclusion that is consistent with Figure 3.7.3(c).
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=
=
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Since is positive, the point is a relative minimum, a conclusion that is consistent with Figure 3.7.3(c).
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=
=
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Since is negative, the point is a relative maximum, a conclusion that is consistent with Figure 3.7.3(c).
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=
=
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Since is positive, the point is a relative minimum, a conclusion that is consistent with Figure 3.7.3(d).
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=
=
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Candidates for Inflection
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Obtain candidates for inflection points by solving the equation
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Write and press the Enter key.
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Context Panel: Solve≻Obtain Solutions for≻
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Context Panel: Conversions≻To List
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Context Panel: Assign Name≻
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Zeros of the Function
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Find the -intercepts by solving the equation
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Write and press the Enter key.
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Context Panel: Solve≻Numerically Solve
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Conclusions
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Figures 3.7.3(e) and 3.7.3(f), graphs of and , respectively, are useful for clarifying intervals of increase/decrease, and concavity, and for determining if the candidates for inflection are indeed inflection points.
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F:=x^6-10*x^5-15*x^4+140*x^3+160*x^2-528*x-800:
plot(diff(F,x),x=-4..9,color=red,title="First Derivative");
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Figure 3.7.3(e) First derivative of
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F:=x^6-10*x^5-15*x^4+140*x^3+160*x^2-528*x-800:
plot(diff(F,x,x),x=-4..8,color=green,title="Second Derivative");
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Figure 3.7.3(f) Second derivative of
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Wherever the red curve in Figure 3.7.3(e) is below the -axis, the function is decreasing; above the -axis, increasing. Wherever the green curve in Figure 3.7.3(f) is below the -axis, the function is concave downward; above the -axis, concave upward.
= = is an inflection point, the concavity of changing across , as verified by the results in Table 3.7.3(a) and the graph in Figure 3.7.3(c).
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The endpoint
= is a relative maximum for the restricted domain . The point = is also a relative maximum, corroborated by Figure 3.7.3(c). The endpoint = is a relative maximum, and the absolute maximum, as corroborated by Figure 3.7.3(b).
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The point = is a relative minimum. The point = is also a relative minimum. From Figure 3.7.3(b), it is the absolute minimum.
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From Table 3.7.3(a) and Figures 3.7.3(c-e), the function increases on the intervals = and = .
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From Table 3.7.3(a) and Figures 3.7.3(c-e), the function decreases on the intervals , = , and = .
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From Table 3.7.3(a) and Figures 3.7.3(c-d), and 3.7.3(f), the function is concave upward on the intervals , = , and = ; it is concave downward on the intervals = , and =
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From Table 3.7.3(a) and Figures 3.7.3(c-d), and 3.7.3(f), the points = , = , = , and = are all inflection points.
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The endpoints of a finite domain for the function have to be considered when searching for extrema. If the domain is unrestricted, that is, if it is the full set of real numbers for which the rule of the function is defined, then for this function, there would not be a global maximum or minimum because is unbounded as .
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Some Useful Commands
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Applicable Commands
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=
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=
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=
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=
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