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. |
|- | |- | ||
| − | | | + | |Using the product and chain rule, we get |
|- | |- | ||
| − | | | + | | <math>3x^2+3y^2y'=2y+2xy'.</math> |
|- | |- | ||
| − | | | + | |We rearrange the terms and solve for <math style="vertical-align: -5px">y'.</math> |
| + | |- | ||
| + | |Therefore, | ||
| + | |- | ||
| + | | <math>3x^2-2y=2xy'-3y^2y'</math> | ||
| + | |- | ||
| + | |and | ||
| + | |- | ||
| + | | <math>y'=\frac{3x^2-2y}{2x-3y^2}.</math> | ||
|} | |} | ||
Revision as of 11:57, 6 March 2017
Calculate the equation of the tangent line to the curve defined by 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^3+y^3=2xy} 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: |
|---|