Difference between revisions of "009A Sample Final 1, Problem 10"

Latest revision as of 16:17, 2 December 2017

If a resistor of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} ohms is connected across a battery of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E} volts with internal resistance Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} ohms, then the power (in watts) in the external resistor is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P=\frac{E^2R}{(R+r)^2}.}

If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} are fixed but Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} varies, what is the maximum value of the power?

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@@ Line 1: / Line 1: @@
-<span class="exam">Consider the following continuous function:
+<span class="exam">If a resistor of &nbsp;<math style="vertical-align: 0px">R</math>&nbsp; ohms is connected across a battery of &nbsp;<math style="vertical-align: 0px">E</math>&nbsp; volts with internal resistance &nbsp;<math style="vertical-align: 0px">r</math>&nbsp; ohms, then the power (in watts) in the external resistor is
-::<math>f(x)=x^{1/3}(x-8)</math>
-<span class="exam">defined on the closed, bounded interval &nbsp;<math style="vertical-align: -5px">[-8,8]</math>.
+::<math>P=\frac{E^2R}{(R+r)^2}.</math>
-<span class="exam">(a) Find all the critical points for &nbsp;<math style="vertical-align: -5px">f(x)</math>.
+<span class="exam">If &nbsp;<math style="vertical-align: 0px">E</math>&nbsp; and &nbsp;<math style="vertical-align: 0px">r</math>&nbsp; are fixed but &nbsp;<math style="vertical-align: 0px">R</math>&nbsp; varies, what is the maximum value of the power?
-<span class="exam">(b) Determine the absolute maximum and absolute minimum values for &nbsp;<math style="vertical-align: -5px">f(x)</math>&nbsp; on the interval &nbsp;<math style="vertical-align: -5px">[-8,8]</math>.
+<hr>
+[[009A Sample Final 1, Problem 10 Solution|'''<u>Solution</u>''']]
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-!Foundations: &nbsp;
-|-
-|'''1.''' To find the critical points for &nbsp;<math style="vertical-align: -5px">f(x),</math>&nbsp; we set &nbsp;<math style="vertical-align: -5px">f'(x)=0</math>&nbsp; and solve for &nbsp;<math style="vertical-align: -1px">x.</math>
-|-
-|
-&nbsp; &nbsp; &nbsp; &nbsp; Also, we include the values of &nbsp;<math style="vertical-align: -1px">x</math>&nbsp; where &nbsp;<math style="vertical-align: -5px">f'(x)</math>&nbsp; is undefined.
-|-
-|'''2.''' To find the absolute maximum and minimum of &nbsp;<math style="vertical-align: -5px">f(x)</math>&nbsp; on an interval &nbsp;<math>[a,b],</math>
-|-
-|
-&nbsp; &nbsp; &nbsp; &nbsp; we need to compare the &nbsp;<math style="vertical-align: -5px">y</math>&nbsp; values of our critical points with &nbsp;<math style="vertical-align: -5px">f(a)</math>&nbsp; and &nbsp;<math style="vertical-align: -5px">f(b).</math>
-|}
+[[009A Sample Final 1, Problem 10 Detailed Solution|'''<u>Detailed Solution</u>''']]
-'''Solution:'''
-'''(a)'''
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-!Step 1: &nbsp;
-|-
-|To find the critical points, first we need to find &nbsp;<math style="vertical-align: -5px">f'(x).</math>
-|-
-|Using the Product Rule, we have
-|-
-|
-&nbsp; &nbsp; &nbsp; &nbsp;<math>\begin{array}{rcl}
-\displaystyle{f'(x)} & = & \displaystyle{\frac{1}{3}x^{-\frac{2}{3}}(x-8)+x^{\frac{1}{3}}}\\
-&&\\
-& = & \displaystyle{\frac{x-8}{3x^{\frac{2}{3}}}+x^{\frac{1}{3}}.}\\
-\end{array}</math>
-|}
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-!Step 2: &nbsp;
-|-
-|Notice &nbsp;<math style="vertical-align: -5px">f'(x)</math>&nbsp; is undefined when &nbsp;<math style="vertical-align: -1px">x=0.</math>
-|-
-|Now, we need to set &nbsp;<math style="vertical-align: -5px">f'(x)=0.</math>
-|-
-|So, we get
-|-
-|
-&nbsp; &nbsp; &nbsp; &nbsp;<math>-x^{\frac{1}{3}}\,=\,\frac{x-8}{3x^{\frac{2}{3}}}.</math>
-|-
-|We cross multiply to get &nbsp;<math style="vertical-align: 1px">-3x=x-8.</math>
-|-
-|Solving, we get &nbsp;<math style="vertical-align: -1px">x=2.</math>
-|-
-|Thus, the critical points for &nbsp;<math style="vertical-align: -5px">f(x)</math>&nbsp; are &nbsp;<math style="vertical-align: -5px">(0,0)</math>&nbsp; and &nbsp;<math style="vertical-align: -4px">(2,2^{\frac{1}{3}}(-6)).</math>
-|}
-'''(b)'''
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-!Step 1: &nbsp;
-|-
-|We need to compare the values of &nbsp;<math style="vertical-align: -5px">f(x)</math>&nbsp; at the critical points and at the endpoints of the interval.
-|-
-|Using the equation given, we have &nbsp;<math style="vertical-align: -5px">f(-8)=32</math>&nbsp; and &nbsp;<math style="vertical-align: -5px">f(8)=0.</math>
-|}
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-!Step 2: &nbsp;
-|-
-|Comparing the values in Step 1 with the critical points in (a), the absolute maximum value for &nbsp;<math style="vertical-align: -5px">f(x)</math>&nbsp; is &nbsp;<math style="vertical-align: -1px">32</math>
-|-
-|and the absolute minimum value for &nbsp;<math style="vertical-align: -5px">f(x)</math>&nbsp; is &nbsp;<math style="vertical-align: -5px">2^{\frac{1}{3}}(-6).</math>
-|}
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-!Final Answer: &nbsp;
-|-
-|&nbsp; &nbsp; '''(a)'''&nbsp; &nbsp; <math style="vertical-align: -4px">(0,0)</math>&nbsp; and &nbsp;<math style="vertical-align: -4px">(2,2^{\frac{1}{3}}(-6))</math>
-|-
-|&nbsp; &nbsp; '''(b)'''&nbsp; &nbsp; The absolute maximum value for &nbsp;<math style="vertical-align: -5px">f(x)</math>&nbsp; is &nbsp;<math style="vertical-align: -1px">32</math>&nbsp; and the absolute minimum value for &nbsp;<math style="vertical-align: -5px">f(x)</math>&nbsp; is &nbsp;<math style="vertical-align: -5px">2^{\frac{1}{3}}(-6).</math>
-|}
 [[009A_Sample_Final_1|'''<u>Return to Sample Exam</u>''']]

Difference between revisions of "009A Sample Final 1, Problem 10"

Latest revision as of 16:17, 2 December 2017

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