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| | <span class="exam">(b) Find the equation of the tangent line to the curve at the origin. | | <span class="exam">(b) Find the equation of the tangent line to the curve at the origin. |
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Foundations:
| + | [[009C Sample Final 3, Problem 10 Solution|'''<u>Solution</u>''']] |
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| − | |'''1.''' What two pieces of information do you need to write the equation of a line?
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| − | You need the slope of the line and a point on the line.
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| − | |'''2.''' What is the slope of the tangent line of a parametric curve? | |
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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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| − | '''Solution:''' | + | [[009C Sample Final 3, Problem 10 Detailed Solution|'''<u>Detailed Solution</u>''']] |
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| − | !(a)
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| − | |Insert graph
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| − | '''(b)'''
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Step 1:
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| − | |First, we need to find the slope of the tangent line.
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| − | |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>
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Step 2:
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| − | |Now, the origin corresponds to <math>x=0</math> and <math>y=0.</math>
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| − | |This gives us two equations. When we solve for <math>t,</math> we get <math>t=0.</math>
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| − | |Plugging in <math>t=0</math> into
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| − | | <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>\frac{dy}{dx}</math> is undefined at <math>t=0.</math>
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| − | |So, there is no tangent line at the origin.
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Final Answer:
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| − | | '''(a)''' See above
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| − | | '''(b)''' There is no tangent line at the origin.
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| − | |}
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| | [[009C_Sample_Final_3|'''<u>Return to Sample Exam</u>''']] | | [[009C_Sample_Final_3|'''<u>Return to Sample Exam</u>''']] |
