Difference between revisions of "009A Sample Final 1, Problem 8"

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!Foundations:    
 
!Foundations:    
 
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|What is the differential <math style="vertical-align: -4px">dy</math> of <math style="vertical-align: -4px">y=x^2</math> at <math style="vertical-align: -1px">x=1?</math>
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|What is the differential <math style="vertical-align: -4px">dy</math>&nbsp; of <math style="vertical-align: -4px">y=x^2</math> at <math style="vertical-align: -1px">x=1?</math>
 
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::Since &thinsp;<math style="vertical-align: -4px">x=1,</math>&thinsp; the differential is &thinsp;<math style="vertical-align: -4px">dy=2xdx=2dx.</math>
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&nbsp; &nbsp; &nbsp; &nbsp; Since &nbsp;<math style="vertical-align: -4px">x=1,</math>&nbsp; the differential is &nbsp;<math style="vertical-align: -4px">dy=2xdx=2dx.</math>
 
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!Step 1: &nbsp;  
 
!Step 1: &nbsp;  
 
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|-
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|First, we find the differential <math style="vertical-align: -4px">dy.</math>
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|First, we find the differential &nbsp; <math style="vertical-align: -4px">dy.</math>
 
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|-
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|Since <math style="vertical-align: -5px">y=x^3,</math>&thinsp; we have
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|Since <math style="vertical-align: -5px">y=x^3,</math>&nbsp; we have
 
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|-
 
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::<math>dy\,=\,3x^2\,dx.</math>
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&nbsp; &nbsp; &nbsp; &nbsp;<math>dy\,=\,3x^2\,dx.</math>
 
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!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
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|-
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|Now, we plug <math style="vertical-align: 0px">x=2</math>&thinsp; into the differential from Step 1.
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|Now, we plug <math style="vertical-align: 0px">x=2</math>&nbsp; into the differential from Step 1.
 
|-
 
|-
 
|So, we get  
 
|So, we get  
 
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|-
 
|
 
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::<math>dy\,=\,3(2)^2\,dx\,=\,12\,dx.</math>
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&nbsp; &nbsp; &nbsp; &nbsp;<math>dy\,=\,3(2)^2\,dx\,=\,12\,dx.</math>
 
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!Step 1: &nbsp;  
 
!Step 1: &nbsp;  
 
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|-
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|First, we find <math style="vertical-align: 0px">dx</math>. We have &thinsp;<math style="vertical-align: -1px">dx=1.9-2=-0.1.</math>
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|First, we find <math style="vertical-align: 0px">dx.</math>&nbsp;  We have &nbsp;<math style="vertical-align: -1px">dx=1.9-2=-0.1.</math>
 
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|-
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|Then, we plug this into the differential from part '''(a)'''.
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|Then, we plug this into the differential from part (a).
 
|-
 
|-
 
|So, we have  
 
|So, we have  
 
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|-
 
|
 
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::<math>dy\,=\,12(-0.1)\,=\,-1.2.</math>
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&nbsp; &nbsp; &nbsp; &nbsp;<math>dy\,=\,12(-0.1)\,=\,-1.2.</math>
 
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!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
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|-
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|Now, we add the value for <math style="vertical-align: -4px">dy</math> to &thinsp;<math style="vertical-align: 0px">2^3</math>&thinsp; to get an
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|Now, we add the value for <math style="vertical-align: -4px">dy</math>&nbsp; to &nbsp;<math style="vertical-align: 0px">2^3</math>&nbsp; to get an
 
|-
 
|-
 
|approximate value of <math style="vertical-align: -1px">1.9^3.</math>
 
|approximate value of <math style="vertical-align: -1px">1.9^3.</math>
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|-
 
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::<math>1.9^3\,\approx \, 2^3+-1.2\,=\,6.8.</math>
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&nbsp; &nbsp; &nbsp; &nbsp;<math>1.9^3\,\approx \, 2^3+-1.2\,=\,6.8.</math>
 
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!Final Answer: &nbsp;  
 
!Final Answer: &nbsp;  
 
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|'''(a)''' &thinsp;<math style="vertical-align: -5px">dy=12\,dx</math>
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|&nbsp; &nbsp; '''(a)''' &nbsp; &nbsp; <math style="vertical-align: -5px">dy=12\,dx</math>
 
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|'''(b)''' &thinsp;<math style="vertical-align: -1px">6.8</math>   
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|&nbsp; &nbsp; '''(b)''' &nbsp; &nbsp; <math style="vertical-align: -1px">6.8</math>   
 
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|}
 
[[009A_Sample_Final_1|'''<u>Return to Sample Exam</u>''']]
 
[[009A_Sample_Final_1|'''<u>Return to Sample Exam</u>''']]

Revision as of 16:59, 25 February 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 y=x^3.}

(a) Find the differential 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 dy} 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 y=x^3} at 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} .

(b) Use differentials to find an approximate value 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 1.9^3} .

Foundations:  
What is the differential 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 dy}   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 y=x^2} at 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=1?}

        Since  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=1,}   the differential 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 dy=2xdx=2dx.}


Solution:

(a)

Step 1:  
First, we find the differential   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 dy.}
Since 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=x^3,}   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 dy\,=\,3x^2\,dx.}

Step 2:  
Now, we plug 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}   into the differential from Step 1.
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 dy\,=\,3(2)^2\,dx\,=\,12\,dx.}

(b)

Step 1:  
First, we 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 dx.}   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 dx=1.9-2=-0.1.}
Then, we plug this into the differential from part (a).
So, 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 dy\,=\,12(-0.1)\,=\,-1.2.}

Step 2:  
Now, we add the value 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 dy}   to  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 2^3}   to get an
approximate value 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 1.9^3.}
Hence, 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 1.9^3\,\approx \, 2^3+-1.2\,=\,6.8.}


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 dy=12\,dx}
    (b)     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 6.8}

Return to Sample Exam

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