Difference between revisions of "009A Sample Final 3, Problem 10"

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<span class="exam">Let <math>y=\tan(x).</math>
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<span class="exam">Let &nbsp;<math style="vertical-align: -5px">y=\tan(x).</math>
  
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<span class="exam">(a) Find the differential <math>dy</math> of <math>y=\tan (x)</math> at <math>x=\frac{\pi}{4}.</math>  
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<span class="exam">(a) Find the differential &nbsp;<math style="vertical-align: -4px">dy</math>&nbsp; of &nbsp;<math style="vertical-align: -5px">y=\tan (x)</math>&nbsp; at &nbsp;<math style="vertical-align: -15px">x=\frac{\pi}{4}.</math>  
  
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<span class="exam">(b) Use differentials to find an approximate value for <math>\tan(0.885).</math> Hint: <math>\frac{\pi}{4}\approx 0.785.</math>
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<span class="exam">(b) Use differentials to find an approximate value for &nbsp;<math style="vertical-align: -5px">\tan(0.885).</math>&nbsp; Hint: &nbsp;<math style="vertical-align: -15px">\frac{\pi}{4}\approx 0.785.</math>
  
 
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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!Step 1: &nbsp;  
 
!Step 1: &nbsp;  
 
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|First, we find &nbsp;<math style="vertical-align: -1px">dx.</math>&nbsp; We have
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!Step 2: &nbsp;
 
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'''(c)'''
 
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!Step 1: &nbsp;  
 
 
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|&nbsp; &nbsp; &nbsp; &nbsp;<math style="vertical-align: -1px">dx=0.885-\frac{\pi}{4}\approx 0.885-0.785=0.1.</math>
 
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|Then, we plug this into the differential from part (a).
 
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|So, we have
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&nbsp; &nbsp; &nbsp; &nbsp;<math>dy\,=\,2(0.1)\,=\,0.2.</math>
 
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!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
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|Now, we add the value for &nbsp;<math style="vertical-align: -4px">dy</math>&nbsp; to &nbsp;<math style="vertical-align: -16px">\tan\bigg(\frac{\pi}{4}\bigg)</math>&nbsp; to get an
 
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|approximate value of &nbsp;<math style="vertical-align: -5px">\tan(0.885).</math>
 
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|Hence, we have
 
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&nbsp; &nbsp; &nbsp; &nbsp;<math>\tan(0.885)\,\approx \, \tan\bigg(\frac{\pi}{4}\bigg)+0.2\,=\,1.2.</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; <math style="vertical-align: -5px">dy=2\,dx</math>
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|'''(b)'''
 
 
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|'''(c)'''
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|&nbsp; &nbsp; '''(b)''' &nbsp; &nbsp; <math style="vertical-align: -1px">1.2</math> 
 
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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 08:39, 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 y=\tan(x).}

(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=\tan (x)}   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=\frac{\pi}{4}.}

(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 \tan(0.885).}   Hint:  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 \frac{\pi}{4}\approx 0.785.}

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=\tan x,}   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\,=\,\sec^2 x\,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=\frac{\pi}{4}}   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\,=\,\bigg(\sec\bigg(\frac{\pi}{4}\bigg)\bigg)^2\,dx\,=\,2\,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=0.885-\frac{\pi}{4}\approx 0.885-0.785=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\,=\,2(0.1)\,=\,0.2.}

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 \tan\bigg(\frac{\pi}{4}\bigg)}   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 \tan(0.885).}
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 \tan(0.885)\,\approx \, \tan\bigg(\frac{\pi}{4}\bigg)+0.2\,=\,1.2.}


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 dy=2\,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 1.2}

Return to Sample Exam

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