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2017-03-19T19:11:35Z

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A curve is described parametrically by
::<math>x=t^2</math>
::<math>y=t^3-t</math>

(a) Sketch the curve for  <math style="vertical-align: -2px">-2\le t \le 2.</math>

(b) Find the equation of the tangent line to the curve at the origin.

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!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  <math style="vertical-align: -21px">m=\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}.</math>
|}

'''Solution:'''

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!(a)  
|-
| 
|-
|[[File:9CSF3_10_GP.png|center|277px]]
|}

'''(b)'''

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 1:  
|-
|First, we need to find the slope of the tangent line.
|-
|Since   <math style="vertical-align: -14px">\frac{dy}{dt}=3t^2-1</math>   and   <math style="vertical-align: -14px">\frac{dx}{dt}=2t,</math>  we have
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|
       <math>\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{3t^2-1}{2t}.</math>
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 2:  
|-
|Now, the origin corresponds to  <math style="vertical-align: 0px">x=0</math>  and  <math style="vertical-align: -4px">y=0.</math>
|-
|This gives us two equations. When we solve for  <math style="vertical-align: -4px">t,</math>  we get  <math style="vertical-align: -1px">t=0.</math>
|-
|Plugging in  <math style="vertical-align: -1px">t=0</math>  into
|-
|       <math>\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{3t^2-1}{2t},</math>
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|we see that  <math style="vertical-align: -14px">\frac{dy}{dx}</math>  is undefined at  <math style="vertical-align: -1px">t=0.</math>
|-
|So, there is no tangent line at the origin.
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Final Answer:  
|-
|    '''(a)'''    See above
|-
|    '''(b)'''    There is no tangent line at the origin.
|}
[[009C_Sample_Final_3|'''Return to Sample Exam''']]

009C Sample Final 3, Problem 10 - Revision history

MathAdmin: Created page with "A curve is described parametrically by ::x=t^2 ::y=t^3-t (a) Sketch the..."

MathAdmin: Created page with "A curve is described parametrically by :: $x=t^2$ :: $y=t^3-t$ (a) Sketch the..."