Difference between revisions of "009A Sample Final 3, Problem 5"

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!Step 1:    
 
!Step 1:    
 
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|We use implicit differentiation to find the derivative of the given curve.
 
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|Using the product and chain rule, we get
 
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|&nbsp; &nbsp; &nbsp; &nbsp; <math>3x^2+3y^2y'=2y+2xy'.</math>
 
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|We rearrange the terms and solve for &nbsp;<math style="vertical-align: -5px">y'.</math>
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|Therefore,
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|&nbsp; &nbsp; &nbsp; &nbsp; <math>3x^2-2y=2xy'-3y^2y'</math>
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|and
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|&nbsp; &nbsp; &nbsp; &nbsp; <math>y'=\frac{3x^2-2y}{2x-3y^2}.</math>
 
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Revision as of 10:57, 6 March 2017

Calculate the equation of the tangent line to the curve defined by    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 (1,1).}

Foundations:  
The equation of the tangent line 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 f(x)}   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 (a,b)}   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 y=m(x-a)+b}   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=f'(a).}


Solution:

Step 1:  
We use implicit differentiation to find the derivative of the given curve.
Using the product and chain rule, 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 3x^2+3y^2y'=2y+2xy'.}
We rearrange the terms 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 y'.}
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 3x^2-2y=2xy'-3y^2y'}
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 y'=\frac{3x^2-2y}{2x-3y^2}.}
Step 2:  


Final Answer:  

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