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

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!Foundations:    
 
!Foundations:    
 
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|Review tangent lines of polar curves
 
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!Step 1:    
 
!Step 1:    
 
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|First, we need to find the slope of the tangent line. Since <math>\frac{dy}{dt}=-4\sin t</math> and <math>\frac{dx}{dt}=3\cos t</math>,
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|First, we need to find the slope of the tangent line.  
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|Since <math>\frac{dy}{dt}=-4\sin t</math> and <math>\frac{dx}{dt}=3\cos t</math>,
 
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|we have <math>\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{-4\sin t}{3\cos t}</math>.
 
|we have <math>\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{-4\sin t}{3\cos t}</math>.
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|If we plug in <math>t_0=\frac{\pi}{4}</math> into the equations for <math>x(t)</math> and <math>y(t)</math>, we get
 
|If we plug in <math>t_0=\frac{\pi}{4}</math> into the equations for <math>x(t)</math> and <math>y(t)</math>, we get
 
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|<math>x\bigg(\frac{\pi}{4}\bigg)=3\sin\bigg(\frac{\pi}{4}\bigg)=\frac{3\sqrt{2}}{2} and
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|<math>x\bigg(\frac{\pi}{4}\bigg)=3\sin\bigg(\frac{\pi}{4}\bigg)=\frac{3\sqrt{2}}{2}</math> and
 
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|<math>y\bigg(\frac{\pi}{4}\bigg)=4\cos\bigg(\frac{\pi}{4}\bigg)=2\sqrt{2}</math>.
 
|<math>y\bigg(\frac{\pi}{4}\bigg)=4\cos\bigg(\frac{\pi}{4}\bigg)=2\sqrt{2}</math>.

Revision as of 12:21, 10 February 2016

A curve is given in polar parametrically 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 y(t)=4\cos t}
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 0\leq t \leq 2\pi}

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:  
Review tangent lines of polar curves

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 (a) 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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