Difference between revisions of "022 Exam 1 Sample A, Problem 7"

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(Created page with "<span class="exam">Find the slope of the tangent line to the graph of <math style="vertical-align: -14%">f(x)=x^{3}-3x^{2}-5x+7</math> at the point <math style="vertical-align...")
 
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! Foundations: &nbsp;  
 
! Foundations: &nbsp;  
 
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|Recall that for a given value, <math style="vertical-align: -16%">f'(x)</math> is precisely the point of the tangent line through the point <math style="vertical-align: -16%">\left(x,f(x)\right)</math>. Once we have the slope, we can then use the point-slope form for a line:
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|Recall that for a given value, <math style="vertical-align: -18%">f'(x)</math> is precisely the point of the tangent line through the point <math style="vertical-align: -16%">\left(x,f(x)\right)</math>. Once we have the slope, we can then use the point-slope form for a line:
 
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!Write the Equation of the Line: &nbsp;  
 
!Write the Equation of the Line: &nbsp;  
 
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|Using the point-slope form listed in foundations, along with the point <math style="vertical-align: -20%">(3,-8)</math> and the slope <math style="vertical-align: 0%">m=4</math>, we find  
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|Using the point-slope form listed in foundations, along with the point <math style="vertical-align: -20%">(3,-8)</math> and the slope <math style="vertical-align: -3%">m=4</math>, we find  
 
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Latest revision as of 19:26, 13 April 2015

Find the slope of the tangent line to the graph 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)=x^{3}-3x^{2}-5x+7} at the point 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,-8)} .

Foundations:  
Recall that for a given value, 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 precisely the point of the tangent line through the point 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 \left(x,f(x)\right)} . Once we have the slope, we can then use the point-slope form for a line:
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-y_0 = m(x-x_0),}
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 m} is the known slope 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 \left(x_0,y_0\right)} is a point on the line.

 Solution:

Finding the slope:  
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 f'(x)\,\,=\,\,3x^2-6x-5,}
so the tangent line through 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,-8)} has slope
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 m\,\,=\,\,f'(3)\,\,=\,\,3(3)^2-6(3)-5\,\,=\,\,4.}
Write the Equation of the Line:  
Using the point-slope form listed in foundations, along with the point 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,-8)} and the slope 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 m=4} , 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 y-(-8)\,\,=\,\,4(x-3),}
or
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\,\,=\,\,4x-20.}
Final Answer:  
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\,\,=\,\,4x-20.}

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