Difference between revisions of "009A Sample Final 3, Problem 9"

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!Step 1:    
 
!Step 1:    
 
|-
 
|-
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|
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|To find the critical points, first we need to find &nbsp;<math style="vertical-align: -5px">g'(x).</math>
−
|-
 
−
|
 
 
|-
 
|-
−
|
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|Using the Chain Rule, we have
 
|-
 
|-
 
|
 
|
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&nbsp; &nbsp; &nbsp; &nbsp;<math>\begin{array}{rcl}
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\displaystyle{g'(x)} & = & \displaystyle{\frac{2}{3}(2x^2-8x)^{-\frac{1}{3}}(2x^2-8x)'}\\
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&&\\
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& = & \displaystyle{\frac{2}{3}(2x^2-8x)^{-\frac{1}{3}}(4x-8)}\\
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&&\\
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& = & \displaystyle{\frac{8x-16}{3\sqrt[3]{2x^2-8x}}.}
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\end{array}</math>
 
|}
 
|}
  
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Step 2: &nbsp;
 
!Step 2: &nbsp;
 +
|-
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|First, we note that &nbsp;<math style="vertical-align: -5px">g'(x)</math>&nbsp; is undefined when
 +
|-
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|&nbsp; &nbsp; &nbsp; &nbsp;<math>3\sqrt[3]{2x^2-8x}=0.</math>
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|-
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|Solving for &nbsp;<math style="vertical-align: -4px">x,</math>&nbsp; we get
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|-
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|&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
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\displaystyle{0} & = & \displaystyle{2x^2-8x}\\
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&&\\
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& = & \displaystyle{x(2x-8).}
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\end{array}</math>
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|-
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|Therefore, &nbsp;<math style="vertical-align: -5px">g'(x)</math>&nbsp; is undefined when &nbsp;<math style="vertical-align: -4px">x=0,4.</math>&nbsp;
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|-
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|Now, we need to set &nbsp;<math style="vertical-align: -5px">g'(x)=0.</math>
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|-
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|So, we get
 
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|-
 
|
 
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&nbsp; &nbsp; &nbsp; &nbsp;<math>8x-16=0.</math>
 
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|-
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|Solving, we get &nbsp;<math style="vertical-align: 0px">x=2.</math>
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|-
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|Thus, the critical points for &nbsp;<math style="vertical-align: -5px">f(x)</math>&nbsp; are &nbsp;<math style="vertical-align: -5px">(0,0),(2,4),(4,0).</math>
 
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!Final Answer: &nbsp;  
 
!Final Answer: &nbsp;  
 
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|-
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|'''(a)'''
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|&nbsp; &nbsp;'''(a)'''&nbsp; &nbsp;&nbsp;<math>(0,0),(2,4),(4,0).</math>
 
|-
 
|-
 
|'''(b)'''
 
|'''(b)'''
 
|}
 
|}
 
[[009A_Sample_Final_3|'''<u>Return to Sample Exam</u>''']]
 
[[009A_Sample_Final_3|'''<u>Return to Sample Exam</u>''']]

Revision as of 11:12, 7 March 2017

Let

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 g(x)=(2x^2-8x)^{\frac{2}{3}}}

(a) Find all critical points 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 g}   over the  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 x} -interval  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 [0,8].}

(b) Find absolute maximum and absolute minimum 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 g}   over  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 [0,8].}

Foundations:  
1. To find the critical points for  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 f(x),}   we set    and solve for  

        Also, we include the values of    where    is undefined.

2. To find the absolute maximum and minimum of    on an interval  

        we need to compare the    values of our critical points with    and  


Solution:

(a)

Step 1:  
To find the critical points, first we need to find  
Using the Chain Rule, we have

       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 \begin{array}{rcl} \displaystyle{g'(x)} & = & \displaystyle{\frac{2}{3}(2x^2-8x)^{-\frac{1}{3}}(2x^2-8x)'}\\ &&\\ & = & \displaystyle{\frac{2}{3}(2x^2-8x)^{-\frac{1}{3}}(4x-8)}\\ &&\\ & = & \displaystyle{\frac{8x-16}{3\sqrt[3]{2x^2-8x}}.} \end{array}}

Step 2:  
First, we note that  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 g'(x)}   is undefined when
       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 3\sqrt[3]{2x^2-8x}=0.}
Solving for  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 x,}   we get
        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 \begin{array}{rcl} \displaystyle{0} & = & \displaystyle{2x^2-8x}\\ &&\\ & = & \displaystyle{x(2x-8).} \end{array}}
Therefore,  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 g'(x)}   is undefined when  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 x=0,4.}  
Now, we need to set  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 g'(x)=0.}
So, we get

       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 8x-16=0.}

Solving, we get  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 x=2.}
Thus, the critical points for  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 f(x)}   are  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 (0,0),(2,4),(4,0).}

(b)

Step 1:  
Step 2:  


Final Answer:  
   (a)    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 (0,0),(2,4),(4,0).}
(b)

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