Difference between revisions of "009C Sample Final 3, Problem 10"

Latest revision as of 17:57, 2 December 2017

A curve is described parametrically by

x=t^{2}

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^3-t}

(a) Sketch the curve 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 -2\le t \le 2.}

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

Solution

Detailed Solution

Return to Sample Exam

@@ Line 7: / Line 7: @@
 <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: &nbsp;
+[[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?
-|-
-|
-&nbsp; &nbsp; &nbsp; &nbsp;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?
-|-
-|
-&nbsp; &nbsp; &nbsp; &nbsp;The slope is &nbsp;<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) &nbsp;
-|-
-|Insert graph
-|-
-|
-|-
-|
-|}
-'''(b)'''
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-!Step 1: &nbsp;
-|-
-|First, we need to find the slope of the tangent line.
-|-
-|Since &nbsp; <math style="vertical-align: -14px">\frac{dy}{dt}=3t^2-1</math> &nbsp; and &nbsp; <math style="vertical-align: -14px">\frac{dx}{dt}=2t,</math>&nbsp; we have
-|-
-|
-&nbsp; &nbsp; &nbsp; &nbsp;<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: &nbsp;
-|-
-|Now, the origin corresponds to <math>x=0</math> and <math>y=0.</math>
-|-
-|This gives us two equations. When we solve for <math>t,</math> we get <math>t=0.</math>
-|-
-|Plugging in <math>t=0</math> into
-|-
-|&nbsp; &nbsp; &nbsp; &nbsp;<math>\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{3t^2-1}{2t},</math>
-|-
-|we see that <math>\frac{dy}{dx}</math> is undefined at <math>t=0.</math>
-|-
-|So, there is no tangent line at the origin.
-|}
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-!Final Answer: &nbsp;
-|-
-|&nbsp; &nbsp; '''(a)'''&nbsp; &nbsp; See above
-|-
-|&nbsp; &nbsp; '''(b)'''&nbsp; &nbsp;  There is no tangent line at the origin.
-|}
 [[009C_Sample_Final_3|'''<u>Return to Sample Exam</u>''']]

Difference between revisions of "009C Sample Final 3, Problem 10"

Latest revision as of 17:57, 2 December 2017

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