Chapter 3: Applications of Differentiation
Section 3.7: What Derivatives Reveal about Graphs
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Example 3.7.1
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For the function , use first principles to obtain the data in Table 3.7.1.
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Solution
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Define the Function
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Define the function
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Control-drag (or type)
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Context Panel: Assign Function
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Graph the Function and its Derivatives
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Graph
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Figure 3.7.1(a) Graph of
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Figure 3.7.1(b) Graph of
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Figure 3.7.1(c) Graph of
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Obtain the Critical Numbers
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Obtain the critical numbers and by solving the equation
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Tools≻Load Package: Student Calculus 1
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Loading Student:-Calculus1
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Write the equation ; press the Enter key.
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Context Panel:
Student Calculus1≻Solve≻Find Roots
Complete the Roots dialog as per Figure 3.7.1(d)
Press OK.
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Context Panel: Assign to a Name≻
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Figure 3.7.1(d) Roots dialog
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The two solutions of in the interval are in the list whose name is .
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The individual critical numbers can now be referenced by typing and .
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Second-Derivative Test
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Apply the Second-Derivative test
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Since is negative, the point is a relative maximum, a conclusion that is consistent with Figure 3.7.1(a).
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=
=
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Since is positive, the point is a relative minimum, a conclusion that is consistent with Figure 3.7.1(a).
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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:
Student Calculus1≻Solve≻Find Roots
See Figure 3.7.1(d); bound roots in .
Press OK.
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Context Panel: Assign to a Name≻
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The individual candidates can now be referenced by typing and .
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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:
Student Calculus1≻Solve≻Find Roots
See Figure 3.7.1(d); bound roots in .
Uncheck "calculate numerically".
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Conclusions
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The point = is a relative maximum. The point
= is also a relative maximum. From Figure 3.7.1(a), it is also the absolute maximum
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The point is a relative minimum. The point = is also a relative minimum. From Figure 3.7.1(a), this point is also the absolute minimum.
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From Figures 3.7.1(a) and 3.7.1(b), the function increases on the intervals = and = . The function decreases on the interval = .
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From Figures 3.7.1(a) and 3.7.1(c), the function is concave upward on the interval = , and concave downward on the intervals = , and = .
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From Figures 3.7.1(a) and 3.7.1(c), the points = , and = are inflection points.
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Some Useful Commands
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Tools≻Load Package: Student Calculus 1
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Loading Student:-Calculus1
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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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=
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The numeric option was necessary in the first three commands because of the complexity of the equations being solved. For functions whose "special points" can be given analytically, the option can be omitted. For the given function, the zeros can be determined analytically, as shown by the second call to the Roots command.
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