|
|
| Line 7: |
Line 7: |
| | <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. |
| | | | |
| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
| + | <hr> |
| − | !Foundations:
| + | [[009C Sample Final 3, Problem 10 Solution|'''<u>Solution</u>''']] |
| − | |-
| |
| − | |'''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:''' | + | [[009C Sample Final 3, Problem 10 Detailed Solution|'''<u>Detailed Solution</u>''']] |
| | | | |
| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
| |
| − | !(a)
| |
| − | |-
| |
| − | |Insert graph
| |
| − | |-
| |
| − | |
| |
| − | |-
| |
| − | |
| |
| − | |}
| |
| | | | |
| − | '''(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
| |
| − | |-
| |
| − | |
| |
| − | <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>
| |
| − | |-
| |
| − | |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|'''<u>Return to Sample Exam</u>''']] | | [[009C_Sample_Final_3|'''<u>Return to Sample Exam</u>''']] |
