Chapter 3: Applications of Differentiation
Section 3.7: What Derivatives Reveal about Graphs
Example 3.7.3
Graph fx=x6−10x5−15x4+140x3+160x2−528x−800 for x∈−4,11; then use the tools of the calculus to analyze the features of this graph.
Although f is a polynomial, it presents two distinct problems. First, it is of degree six, so neither f=0 nor f′=0 will have exact solutions. Moreover, even though f″=0 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, f is very large in the specified domain, so in any reasonably sized graph the relevant features where f 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.
Solution
Initialize
Tools≻Load Package: Student Calculus 1
Loading Student:-Calculus1
Control-drag fx=…
Context Panel: Assign Function
fx=x6−10x5−15x4+140x3+160x2−528 x−800→assign as functionf
Curve Analysis Tutor
Figure 3.7.12, an image of the tutor, illustrates the features of the graph of fx=x6−10x5−15x4+140x3+160x2−528 x−800 that can be determined from f itself, and from the derivatives f′ and f″.
Where f is increasing or decreasing, its graph is drawn in red or black, respectively,
Intervals where the graph of f is concave up or down are shaded in gray or yellow, respectively.
Relative extrema and inflection points are shown in green.
Selecting one of the eight radio-buttons and clicking the "Calculate" button yields the information listed in Table 3.7.4.
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.
The x-intercepts are marked with circles; the inflection points, with crosses; and the extreme points with diamonds. These distinctions are not visible in the tutor.
Figure 3.7.3(a) Curve Analysis tutor applied to fx
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)]);
Figure 3.7.3(b) Graph via the FunctionChart command
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 0.541 × 104 = 5410.
The local maxima occur at:
[-4., .541e4]
[3., -80.]
[11., .141e6]
The local minima occur at:
[.866, -.106e4]
[8.47, -.526e5]
The function is increasing on the intervals:
[.866, 3.]
[8.47, 11.]
The function is decreasing on the intervals:
[-4., -2.]
[-2., .866]
[3., 8.47]
The function is concave up on the intervals:
[-.365, 2.11]
[6.92, 11.]
The function is concave down on the intervals:
[-2., -.365]
[2.11, 6.92]
The points of inflection occur at:
[-2., -76.]
[-.365, -593.]
[2.11, -510]
[6.92, -.338e5]
The zeros occur at x=:
-2.59
10.0
Table 3.7.3(a) Data generated by the Curve Analysis tutor for fx=x6−10x5−15x4+140x3+160x2−528 x−800,x∈−4,11
Figures 3.7.3(c) and 3.7.3(d), by limiting the domains to −4,4, and 4,11, respectively, show greater detail in the regions where f is first small, then large.
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)]);
Figure 3.7.3(c) Graph of f on −4,4
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)]);
Figure 3.7.3(d) Graph of f on 4,11
Obtain the Critical Numbers
Obtain the critical numbers ck,k=1,…,5, by solving the equation f′x=0
Write f′x=0 and press the Enter key.
Context Panel: Solve≻Numerically Solve
Context Panel: Conversions≻To List
Context Panel: Assign Name≻c
f′x=0
6x5−50x4−60x3+420x2+320x−528=0
→solve
−2.,−2.,0.8660819163,3.,8.467251417
→to list
→assign to a name
c
Second-Derivative Test
Apply the Second-Derivative test
Since f″c1=0, the point c1,fc1 is a candidate for an inflection, a conclusion that is consistent with Figure 3.7.3(c).
f″c1 = 0.
fc1 = −80.
Since f″c3 is positive, the point c3,fc3 is a relative minimum, a conclusion that is consistent with Figure 3.7.3(c).
f″c3 = 799.4413525
fc3 = −1059.215765
Since f″c4 is negative, the point c4,fc4 is a relative maximum, a conclusion that is consistent with Figure 3.7.3(c).
f″c4 = −1750.
fc4 = −80.
Since f″c5 is positive, the point c5,fc5 is a relative minimum, a conclusion that is consistent with Figure 3.7.3(d).
f″c5 = 27319.07716
fc5 = −52622.04625
Candidates for Inflection
Obtain candidates for inflection points by solving the equation f″x=0
Write f″x=0 and press the Enter key.
Context Panel: Solve≻Obtain Solutions for≻x
Context Panel: Assign Name≻p
f″x=0
30x4−200x3−180x2+840x+320=0
−2.,−0.3646357398,2.114685393,6.916617013
p
Zeros of the Function
Find the x-intercepts by solving the equation fx=0
Write fx=0 and press the Enter key.
fx=0
x6−10x5−15x4+140x3+160x2−528x−800=0
−2.588203081,10.00094434
Conclusions
Figures 3.7.3(e) and 3.7.3(f), graphs of f′ and f″, respectively, are useful for clarifying intervals of increase/decrease, and concavity, and for determining if the candidates for inflection are indeed inflection points.
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");
Figure 3.7.3(e) First derivative of f
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");
Figure 3.7.3(f) Second derivative of f
Wherever the red curve in Figure 3.7.3(e) is below the x-axis, the function f is decreasing; above the x-axis, increasing. Wherever the green curve in Figure 3.7.3(f) is below the x-axis, the function f is concave downward; above the x-axis, concave upward.
p1,fp1 = −2.,−80. = is an inflection point, the concavity of f changing across x=−2, as verified by the results in Table 3.7.3(a) and the graph in Figure 3.7.3(c).
The endpoint −4,f−4 = −4,5408 is a relative maximum for the restricted domain x∈−4,11. The point c4,fc4 = 3.,−80. is also a relative maximum, corroborated by Figure 3.7.3(c). The endpoint 11,f11 = 11,140528 is a relative maximum, and the absolute maximum, as corroborated by Figure 3.7.3(b).
The point c3,fc3 = 0.8660819163,−1059.215765 is a relative minimum. The point c5,fc5 = 8.467251417,−52622.04625 is also a relative minimum. From Figure 3.7.3(b), it is the absolute minimum.
From Table 3.7.3(a) and Figures 3.7.3(c-e), the function increases on the intervals c3,3 = 0.8660819163,3 and c5,11 = 8.467251417,11.
From Table 3.7.3(a) and Figures 3.7.3(c-e), the function decreases on the intervals −4,−2, −2,c3 = −2,0.8660819163, and 3,c5 = 3,8.467251417.
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 −4,−2, p2,p3 = −0.3646357398,2.114685393, and p4,11 = 6.916617013,11; it is concave downward on the intervals −2,p2 = −2,−0.3646357398, and p3,p4 = 2.114685393,6.916617013 .
From Table 3.7.3(a) and Figures 3.7.3(c-d), and 3.7.3(f), the points p1,fp1 = −2.,−80., p2,fp2 = −0.3646357398,−593.1846506, p3,fp3 = 2.114685393,−510.550592, and p4,fp4 = 6.916617013,−33611.27165 are all inflection points.
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 fx is unbounded as x→±∞.
Some Useful Commands
Applicable Commands
CriticalPointsfx,x=−4..11,numeric = −2.,0.8660819163,3.,8.467251417
ExtremePointsfx,x=−4..11,numeric = −4.,0.8660819163,3.,8.467251417,11.
InflectionPointsfx,x=−4..11,numeric = −2.,−0.3646357398,2.114685393,6.916617013
Rootsfx,x=−4..11,numeric = −2.588203081,10.00094434
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