Chapter 3: Applications of Differentiation
Section 3.7: What Derivatives Reveal about Graphs
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Example 3.7.6
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Determine how the value of affects the graph of the rational function .
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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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Context Panel: Assign Function
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Preliminary Analysis
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Figure 3.7.6(a) provides a graph of in which the value of the parameter is controlled by a slider. As the parameter is varied by the slider, the bounds on the vertical axis are allowed to change as needed so that the resulting graph is completely representative. The graphs are drawn with the RationalFunctionPlot command from the Student Precalculus package; this command displays the horizontal asymptote with a dotted green line and any vertical asymptotes with dotted black lines.
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Figure 3.7.6(a) Slider-controlled graph of
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Table 3.7.6(a) lists several observations that can be extracted from Figure 3.7.6(a), observations that can be verified by the appropriate analytic calculations of the calculus.
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The graph of appears to have as a horizontal asymptote for all values of .
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For values of less than some threshold value near , the graph of appears to have two vertical asymptotes, and a single relative maximum that is negative.
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For values of greater than this threshold, the graph of appears to have no vertical asymptotes, and a single relative maximum that is positive, and is also an absolute maximum.
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At the threshold value of , it appears that the two vertical asymptotes in the graph of merge into a single vertical asymptote, and there are no extrema.
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Table 3.7.6(a) Observations extracted from Figure 3.7.6(a)
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The first observation in Table 3.7.6(a) is verified by limit calculations in Table 3.7.6(b). Each limit is independent of the value of , so the limits hold for all real .
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Table 3.7.6(b) Existence of the horizontal asymptote
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To continue the analysis, complete the square in in the denominator of . This is done interactively in Table 3.7.6(c).
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Context Panel: Evaluate and Display Inline
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Context Panel: Complete Square≻
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Table 3.7.6(c) Completing the square in
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From the result in Table 3.7.6(c), deduce that for
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, the denominator of is positive for all real ; there are no vertical asymptotes.
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, the denominator vanishes for ; each of these locates a vertical asymptote.
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, the function reduces to ; there is a single vertical asymptote at .
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Representative Graphs
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Figure 3.7.6(b) provides representative graphs of , and . The code that activates the radio buttons at the right of the figure is initiated by pressing the button marked "Figure 3.7.6(b)". Selecting one of the three regions for the parameter changes the figure to show graphs relevant to that region.
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Representative graphs of , , and
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Table 3.7.6(d) lists conclusions that can be drawn from the graphs in Figure 3.7.6(b).
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There are two vertical asymptotes; equations are and , with .
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For , is increasing and concave upward because and are positive.
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For , is decreasing but concave upward because is negative and is positive.
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For , increases to a local maximum (at ), then decreases;
(The first derivative goes from positive to negative.)
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For , is concave downward because is strictly negative.
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On any finite domain that included the two vertical asymptotes, the endpoints would be local minima.
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The single vertical asymptote has the equation .
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For , is increasing and concave upward because both and are positive.
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For , is decreasing and concave upward because is negative while is positive.
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On any finite domain that included the vertical asymptote, the endpoints would be local minima.
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For all , there is a local maximum at , and two inflection points symmetrically placed, one on either side of .
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For , is increasing because is positive. However, changes concavity from upward to downward because changes from positive to negative.
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For , is decreasing because is negative. However, changes concavity from downward to upward because changes from negative to positive.
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On any finite domain that included , the endpoints would be local minima.
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Table 3.7.6(d) Conclusions drawn from Figure 3.7.6(b)
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Analytic Computations
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Obtain critical numbers by solving
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Write
Context Panel: Evaluate and Display Inline
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Context Panel: Complete Square≻
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Context Panel: Solve≻Obtain Solutions for≻
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Obtain when
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Expression palette: Evaluation template
Evaluate for
Context Panel: Evaluate and Display Inline
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For , but at , the first derivative is not defined.
Obtain candidates for inflection by solving
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Write ; press the Enter key.
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Context Panel: Simplify≻Simplify
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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 to a Name≻
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Evaluate at each candidate for inflection
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The candidates for inflection, namely , are real only if . Figure 3.7.6(b) then shows each candidate for inflection is actually an inflection point when .
At the critical number ,
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so that for , the point
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Curve Analysis Tutor
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Figures 3.7.6(c-e) are representative graphs generated by the FunctionChart (a.k.a. FunctionPlot) command.
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f1:=10/(x^2+6*x+5):
Student:-SetColors(red,black,green,gray,yellow):
Student:-Calculus1:-FunctionChart(f1,x=-9..5,concavity=[filled(gray,yellow)],caption="");
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Figure 3.7.6(c)
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f2:=10/(x^2+6*x+9):
Student:-SetColors(red,black,green,gray,yellow):
Student:-Calculus1:-FunctionChart(f2,x=-9..4,concavity=[filled(gray,yellow)],caption="",view=[DEFAULT,0..10]);
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Figure 3.7.6(d)
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f3:=10/(x^2+6*x+12):
Student:-SetColors(red,black,green,gray,yellow):
Student:-Calculus1:-FunctionChart(f3,x=-10..5,concavity=[filled(gray,yellow)],caption="");
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Figure 3.7.6(e)
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The graph in Figure 3.7.6(f) is generated by the FunctionChart command applied to , with the value of controlled by a slider.
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From Figures 3.7.6(c-e) and the Curve Analysis tutor, the data in Table 3.7.6(e) can be constructed, provided , and are calculated analytically as the locations of the vertical asymptotes, and inflection points, respectively. Table 3.7.6(e) assumes that the domain of is as large as possible. Note that has no zeros or local minima.
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To launch the Curve Analysis tutor for :
Enter a value of : ;
then press:
tutor.
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Figure 3.7.6(f) Slider-controlled graph of
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Increasing
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Decreasing
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Concave Up
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Concave Down
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Inflections
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Maxima
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Table 3.7.16 Analytic data inferred from Figures 3.7.6(c-e) and the Curve Analysis tutor
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