Difference between revisions of "009A Sample Final 3, Problem 5"
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!Step 2: | !Step 2: | ||
|- | |- | ||
| − | | | + | |Therefore, the slope of the tangent line at the point <math style="vertical-align: -5px">(1,1)</math> is |
|- | |- | ||
| − | | | + | | <math>\begin{array}{rcl} |
| + | \displaystyle{m} & = & \displaystyle{\frac{3(1)^2-2(1)}{2(1)-3(1)^2}}\\ | ||
| + | &&\\ | ||
| + | & = & \displaystyle{\frac{3-2}{2-3}}\\ | ||
| + | &&\\ | ||
| + | & = & \displaystyle{-1.} | ||
| + | \end{array}</math> | ||
| + | |- | ||
| + | |Hence, the equation of the tangent line to the curve at the point <math style="vertical-align: -5px">(1,1)</math> is | ||
| + | |- | ||
| + | | <math>f(x)=-1(x-1)+1.</math> | ||
|} | |} | ||
| Line 44: | Line 54: | ||
!Final Answer: | !Final Answer: | ||
|- | |- | ||
| − | | | + | | <math>f(x)=-1(x-1)+1</math> |
|} | |} | ||
[[009A_Sample_Final_3|'''<u>Return to Sample Exam</u>''']] | [[009A_Sample_Final_3|'''<u>Return to Sample Exam</u>''']] | ||
Revision as of 11:01, 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: |
|---|
| 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 (1,1)} 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 \begin{array}{rcl} \displaystyle{m} & = & \displaystyle{\frac{3(1)^2-2(1)}{2(1)-3(1)^2}}\\ &&\\ & = & \displaystyle{\frac{3-2}{2-3}}\\ &&\\ & = & \displaystyle{-1.} \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 (1,1)} 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 f(x)=-1(x-1)+1.} |
| 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 f(x)=-1(x-1)+1} |