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| | ::<span class="exam"><math style="vertical-align: -4px">3x^2+xy+y^2=5</math> at the point <math style="vertical-align: -5px">(1,-2)</math> | | ::<span class="exam"><math style="vertical-align: -4px">3x^2+xy+y^2=5</math> at the point <math style="vertical-align: -5px">(1,-2)</math> |
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
| + | <hr> |
| − | !Foundations:
| + | [[009A Sample Final 2, Problem 4 Solution|'''<u>Solution</u>''']] |
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| − | |The equation of the tangent line to <math style="vertical-align: -5px">f(x)</math> at the point <math style="vertical-align: -5px">(a,b)</math> is
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| − | |- | |
| − | | <math style="vertical-align: -5px">y=m(x-a)+b</math> where <math style="vertical-align: -5px">m=f'(a).</math>
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| − | |}
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| − | '''Solution:''' | + | [[009A Sample Final 2, Problem 4 Detailed Solution|'''<u>Detailed Solution</u>''']] |
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !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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| − | | <math>6x+xy'+y+2yy'=5.</math>
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| − | |We rearrange the terms and solve for <math style="vertical-align: -5px">y'.</math>
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| − | |Therefore,
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| − | | <math>xy'+2yy'=5-6x-y</math>
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| − | |and
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| − | | <math>y'=\frac{5-6x-y}{x+2y}.</math>
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| − | |}
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Step 2:
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| − | |Therefore, the slope of the tangent line at the point <math style="vertical-align: -5px">(1,-2)</math> is
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| − | | <math>\begin{array}{rcl}
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| − | \displaystyle{m} & = & \displaystyle{\frac{5-6(1)-(-2)}{1-4}}\\
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| − | &&\\
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| − | & = & \displaystyle{\frac{1}{-3}.}
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| − | \end{array}</math>
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| − | |Hence, the equation of the tangent line to the curve at the point <math style="vertical-align: -5px">(1,-2)</math> is
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| − | |-
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| − | | <math>f(x)=-\frac{1}{3}(x-1)-2.</math>
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| − | |}
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Final Answer:
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| − | | <math>f(x)=-\frac{1}{3}(x-1)-2.</math>
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| − | |}
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| | [[009A_Sample_Final_2|'''<u>Return to Sample Exam</u>''']] | | [[009A_Sample_Final_2|'''<u>Return to Sample Exam</u>''']] |
