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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|-
 
−
|
 
 
|-
 
|-
−
|
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|Using the Chain Rule, we have
 
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|-
 
|
 
|
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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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|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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|'''(a)'''
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|&nbsp; &nbsp;'''(a)'''&nbsp; &nbsp;&nbsp;<math>(0,0),(2,4),(4,0).</math>
 
|-
 
|-
 
|'''(b)'''
 
|'''(b)'''
 
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|}
 
[[009A_Sample_Final_3|'''<u>Return to Sample Exam</u>''']]
 
[[009A_Sample_Final_3|'''<u>Return to Sample Exam</u>''']]

Revision as of 12:12, 7 March 2017

Let

(a) Find all critical points of    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  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)=0}   and solve 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.}

        Also, we include the values 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 x}   where  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)}   is undefined.

2. To find the absolute maximum and 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 f(x)}   on an 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 [a,b],}

        we need to compare 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 y}   values of our critical points with  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(a)}   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 f(b).}


Solution:

(a)

Step 1:  
To find the critical points, first we need to find  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).}
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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