007A Sample Final 2, Problem 4 Detailed Solution

Use implicit differentiation to find an equation of the tangent line to the curve at the given point.

$3x^{2}+xy+y^{2}=5$  at the point  $(1,-2)$

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Background Information:
The equation of the tangent line to  $f(x)$  at the point  $(a,b)$  is
$y=m(x-a)+b$  where  $m=f'(a).$

Solution:

Step 1:
We use implicit differentiation to find the derivative of the given curve.
Using the product and chain rule, we get
$6x+xy'+y+2yy'=0.$
We rearrange the terms and solve for  $y'.$
Therefore,
$xy'+2yy'=-6x-y$
and
$y'={\frac {-6x-y}{x+2y}}.$

Step 2:
Therefore, the slope of the tangent line 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,-2)}   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 \begin{array}{rcl} \displaystyle{m} & = & \displaystyle{\frac{-6(1)-(-2)}{1-4}}\\ &&\\ & = & \displaystyle{\frac{4}{3}.} \end{array}}
Hence, the equation of the tangent line to the curve 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,-2)}   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=\frac{4}{3}(x-1)-2.}

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=\frac{4}{3}(x-1)-2.}

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