Difference between revisions of "009A Sample Final 1, Problem 7"
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− | |First, we find the slope of the tangent line at the point <math style="vertical-align: - | + | |First, we find the slope of the tangent line at the point  <math style="vertical-align: -5px">(3,3).</math> |
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− | |We plug | + | |We plug <math style="vertical-align: -5px">(3,3)</math>  into the formula for  <math style="vertical-align: -12px">\frac{dy}{dx}</math>  we found in part '''(a)'''. |
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|So, we get | |So, we get | ||
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− | ::<math>m=\frac{3(3)^2-6(3)}{6(3)-3(3)^2}=\frac{9}{-9}=-1.</math> | + | ::<math>m\,=\,\frac{3(3)^2-6(3)}{6(3)-3(3)^2}\,=\,\frac{9}{-9}\,=\,-1.</math> |
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!Step 2: | !Step 2: | ||
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− | |Now, we have the slope of the tangent line at <math style="vertical-align: - | + | |Now, we have the slope of the tangent line at <math style="vertical-align: -5px">(3,3)</math>  and a point. |
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|Thus, we can write the equation of the line. | |Thus, we can write the equation of the line. | ||
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− | |So, the equation of the tangent line at <math style="vertical-align: - | + | |So, the equation of the tangent line at <math style="vertical-align: -5px">(3,3)</math>  is |
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− | ::<math>y=-1(x-3)+3.</math> | + | ::<math>y\,=\,-1(x-3)+3.</math> |
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== 4 == | == 4 == | ||
{| class="mw-collapsible mw-collapsed" style = "text-align:left;" | {| class="mw-collapsible mw-collapsed" style = "text-align:left;" |
Revision as of 13:31, 4 March 2016
A curve is defined implicitly by the equation
- 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=6xy.}
a) Using implicit differentiation, compute 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 \frac{dy}{dx}} .
b) Find an equation of the tangent line to the curve 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=6xy} 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 (3,3)} .
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Foundations: |
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1. What is the result of implicit differentiation of 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?} |
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2. What two pieces of information do you need to write the equation of a line? |
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3. What is the slope of the tangent line of a curve? |
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Solution:
2
(a)
Step 1: |
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Using implicit differentiation on the equation 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=6xy,} we get |
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Step 2: |
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Now, we move all the 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 \frac{dy}{dx}} terms to one side of the equation. |
So, we have |
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We solve to 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 \frac{dy}{dx}=\frac{3x^2-6y}{6x-3y^2}.} |
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(b)
Step 1: |
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First, we find 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 (3,3).} |
We plug 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 (3,3)} into the formula 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 \frac{dy}{dx}} we found in part (a). |
So, we get |
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Step 2: |
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Now, we have the slope of the tangent line at 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 (3,3)} and a point. |
Thus, we can write the equation of the line. |
So, the equation of the tangent line at 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 (3,3)} is |
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4
Final Answer: |
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(a) 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 \frac{dy}{dx}=\frac{3x^2-6y}{6x-3y^2}} |
(b) 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=-1(x-3)+3} |