Difference between revisions of "009A Sample Final A, Problem 4"
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|which has as a derivative | |which has as a derivative | ||
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| − | | <math>2\cdot y+2x\cdot y' = 2y +2x\cdot y'.</math> | + | | <math>2\cdot y+2x\cdot y' = 2y +2x\cdot y'.</math><br> |
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| <math>-3x^{2}-2y-2x\cdot y'+3y^{2}\cdot y'=0.</math> | | <math>-3x^{2}-2y-2x\cdot y'+3y^{2}\cdot y'=0.</math> | ||
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| − | |From here, I would immediately plug in (1,1) to find | + | |From here, I would immediately plug in <math style="vertical-align: -22%">(1,1)</math> to find <math style="vertical-align: -22%">y'</math>: |
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| − | | <math style="vertical-align: -20%">-3-2-2y'+3y'=0</math>, or <math style="vertical-align: -20%">y' = 5.</math> | + | | <math style="vertical-align: -20%">-3-2-2y'+3y'=0</math>, or <math style="vertical-align: -20%">y' = 5.</math><br> |
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|or in slope-intercept form | |or in slope-intercept form | ||
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| − | | <math>y=5x-4.</math> | + | | <math>y=5x-4.</math><br> |
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[[009A_Sample_Final_A|'''<u>Return to Sample Exam</u>''']] | [[009A_Sample_Final_A|'''<u>Return to Sample Exam</u>''']] | ||
Revision as of 12:49, 27 March 2015
4. Find an equation for the tangent
line to the function at the point .
| Foundations: |
|---|
| Since only two variables are present, we are going to differentiate everything with respect 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 x} in order to find an expression for 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 = y' = dy/dx} . Then we can use the point-slope equation form 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 \left(x_1,y_1\right) = (1,1)} to find the equation of the tangent line. |
| Note that implicit differentiation will require the product rule and the chain rule. In particular, differentiating 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 2xy} can be treated as |
| 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 (2x)\cdot (y),} |
| which has as a derivative |
| 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\cdot y+2x\cdot y' = 2y +2x\cdot y'.}
|
| Finding the slope: |
|---|
| We use implicit differentiation on our original equation 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 -3x^{2}-2y-2x\cdot y'+3y^{2}\cdot y'=0.} |
| From here, I would immediately plug in 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)} 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 y'} : |
| 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 -3-2-2y'+3y'=0}
, 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' = 5.}
|
| Writing the Equation of the Tangent Line: |
|---|
| Now, we simply plug our 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 = y = 1} 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 m = 5} into the point-slope form to find 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 (1,1)} 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-1=5(x-1),} |
| or in slope-intercept form |
| 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=5x-4.}
|