Difference between revisions of "009A Sample Final 2, Problem 4"
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(Created page with "<span class="exam">Use implicit differentiation to find an equation of the tangent line to the curve at the given point. ::<span class="exam"><math style="vertical-align: -4p...") |
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|Using the product and chain rule, we get | |Using the product and chain rule, we get | ||
|- | |- | ||
| − | | <math>6x+xy'+y+2yy'= | + | | <math>6x+xy'+y+2yy'=0.</math> |
|- | |- | ||
|We rearrange the terms and solve for <math style="vertical-align: -5px">y'.</math> | |We rearrange the terms and solve for <math style="vertical-align: -5px">y'.</math> | ||
| Line 27: | Line 27: | ||
|Therefore, | |Therefore, | ||
|- | |- | ||
| − | | <math>xy'+2yy'= | + | | <math>xy'+2yy'=-6x-y</math> |
|- | |- | ||
|and | |and | ||
|- | |- | ||
| − | | <math>y'=\frac{ | + | | <math>y'=\frac{-6x-y}{x+2y}.</math> |
|} | |} | ||
| Line 40: | Line 40: | ||
|- | |- | ||
| <math>\begin{array}{rcl} | | <math>\begin{array}{rcl} | ||
| − | \displaystyle{m} & = & \displaystyle{\frac{ | + | \displaystyle{m} & = & \displaystyle{\frac{-6(1)-(-2)}{1-4}}\\ |
&&\\ | &&\\ | ||
| − | & = & \displaystyle{ | + | & = & \displaystyle{\frac{4}{3}.} |
\end{array}</math> | \end{array}</math> | ||
|- | |- | ||
|Hence, the equation of the tangent line to the curve at the point <math style="vertical-align: -5px">(1,-2)</math> is | |Hence, the equation of the tangent line to the curve at the point <math style="vertical-align: -5px">(1,-2)</math> is | ||
|- | |- | ||
| − | | <math>y= | + | | <math>y=\frac{4}{3}(x-1)-2.</math> |
|- | |- | ||
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!Final Answer: | !Final Answer: | ||
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| − | | <math>y= | + | | <math>y=\frac{4}{3}(x-1)-2.</math> |
|} | |} | ||
[[009A_Sample_Final_2|'''<u>Return to Sample Exam</u>''']] | [[009A_Sample_Final_2|'''<u>Return to Sample Exam</u>''']] | ||
Latest revision as of 17:37, 14 September 2017
Use implicit differentiation to find an equation of the tangent line to the curve at the given 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 3x^2+xy+y^2=5} 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)}
| 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 6x+xy'+y+2yy'=0.} |
| 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 xy'+2yy'=-6x-y} |
| 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{-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.} |