Difference between revisions of "009C Sample Final 1, Problem 10"

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::The slope is <math style="vertical-align: -21px">m=\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}</math>.
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::The slope is <math style="vertical-align: -21px">m=\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}.</math>
 
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|First, we need to find the slope of the tangent line.  
 
|First, we need to find the slope of the tangent line.  
 
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|Since <math style="vertical-align: -14px">\frac{dy}{dt}=-4\sin t</math> and <math style="vertical-align: -14px">\frac{dx}{dt}=3\cos t</math>, we have
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|Since <math style="vertical-align: -14px">\frac{dy}{dt}=-4\sin t</math> and <math style="vertical-align: -14px">\frac{dx}{dt}=3\cos t,</math> we have
 
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::<math>\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{-4\sin t}{3\cos t}</math>.
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::<math>\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{-4\sin t}{3\cos t}.</math>
 
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|So, at <math>t_0=\frac{\pi}{4}</math>, the slope of the tangent line is  
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|So, at <math>t_0=\frac{\pi}{4},</math> the slope of the tangent line is  
 
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::<math>m=\frac{-4\sin\bigg(\frac{\pi}{4}\bigg)}{3\cos\bigg(\frac{\pi}{4}\bigg)}=\frac{-4}{3}</math>.
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::<math>m=\frac{-4\sin\bigg(\frac{\pi}{4}\bigg)}{3\cos\bigg(\frac{\pi}{4}\bigg)}=\frac{-4}{3}.</math>
 
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|Since we have the slope of the tangent line, we just need a find a point on the line in order to write the equation.
 
|Since we have the slope of the tangent line, we just need a find a point on the line in order to write the equation.
 
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|If we plug in <math>t_0=\frac{\pi}{4}</math> into the equations for <math style="vertical-align: -5px">x(t)</math> and <math style="vertical-align: -5px">y(t)</math>, we get
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|If we plug in <math>t_0=\frac{\pi}{4}</math> into the equations for <math style="vertical-align: -5px">x(t)</math> and <math style="vertical-align: -5px">y(t),</math> we get
 
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::<math>y\bigg(\frac{\pi}{4}\bigg)=4\cos\bigg(\frac{\pi}{4}\bigg)=2\sqrt{2}</math>.
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::<math>y\bigg(\frac{\pi}{4}\bigg)=4\cos\bigg(\frac{\pi}{4}\bigg)=2\sqrt{2}.</math>
 
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|Thus, the point <math>\bigg(\frac{3\sqrt{2}}{2},2\sqrt{2}\bigg)</math> is on the tangent line.
 
|Thus, the point <math>\bigg(\frac{3\sqrt{2}}{2},2\sqrt{2}\bigg)</math> is on the tangent line.
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::<math>y=\frac{-4}{3}\bigg(x-\frac{3\sqrt{2}}{2}\bigg)+2\sqrt{2}</math>.
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::<math>y=\frac{-4}{3}\bigg(x-\frac{3\sqrt{2}}{2}\bigg)+2\sqrt{2}.</math>
 
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Revision as of 11:46, 29 February 2016

A curve is given in polar parametrically by

a) Sketch the curve.

b) Compute 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 t_0=\frac{\pi}{4}} .

Foundations:  
1. What two pieces of information do you need to write the equation of a line?
You need the slope of the line and a point on the line.
2. What is the slope of the tangent line of a parametric curve?
The slope 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 m=\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}.}

Solution:

(a)

Step 1:  
Insert sketch of curve

(b)

Step 1:  
First, we need to find the slope of the tangent line.
Since 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}{dt}=-4\sin t} 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 \frac{dx}{dt}=3\cos t,} we have
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{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{-4\sin t}{3\cos t}.}
So, 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 t_0=\frac{\pi}{4},} the slope of the tangent line 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 m=\frac{-4\sin\bigg(\frac{\pi}{4}\bigg)}{3\cos\bigg(\frac{\pi}{4}\bigg)}=\frac{-4}{3}.}
Step 2:  
Since we have the slope of the tangent line, we just need a find a point on the line in order to write the equation.
If we plug in 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 t_0=\frac{\pi}{4}} into the equations 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 x(t)} 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(t),} 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 x\bigg(\frac{\pi}{4}\bigg)=3\sin\bigg(\frac{\pi}{4}\bigg)=\frac{3\sqrt{2}}{2}} 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\bigg(\frac{\pi}{4}\bigg)=4\cos\bigg(\frac{\pi}{4}\bigg)=2\sqrt{2}.}
Thus, 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 \bigg(\frac{3\sqrt{2}}{2},2\sqrt{2}\bigg)} is on the tangent line.
Step 3:  
Using the point found in Step 2, 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 t_0=\frac{\pi}{4}} 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}\bigg(x-\frac{3\sqrt{2}}{2}\bigg)+2\sqrt{2}.}
Final Answer:  
(a) See Step 1 above for the graph.
(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=\frac{-4}{3}\bigg(x-\frac{3\sqrt{2}}{2}\bigg)+2\sqrt{2}}

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