009C Sample Final 3, Problem 10 - Revision history

Kayla Murray at 01:57, 3 December 2017

2017-12-03T01:57:34Z

2017-03-05T21:50:29Z

2017-03-05T21:46:17Z

2017-03-05T01:24:40Z

2017-03-05T01:01:46Z

2017-02-18T19:07:59Z

@@ 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>''']]
-'''Solution:'''
+[[009C Sample Final 3, Problem 10 Detailed Solution|'''<u>Detailed Solution</u>''']]
 [[009C_Sample_Final_3|'''<u>Return to Sample Exam</u>''']]

@@ Line 50: / Line 50: @@
 !Step 2: &nbsp;
 |-
-|Now, the origin corresponds to <math>x=0</math> and <math>y=0.</math>
+|Now, the origin corresponds to &nbsp;<math style="vertical-align: 0px">x=0</math>&nbsp; and &nbsp;<math style="vertical-align: -4px">y=0.</math>
 |-
-|This gives us two equations. When we solve for <math>t,</math> we get <math>t=0.</math>
+|This gives us two equations. When we solve for &nbsp;<math style="vertical-align: -4px">t,</math>&nbsp; we get &nbsp;<math style="vertical-align: -1px">t=0.</math>
 |-
-|Plugging in <math>t=0</math> into
+|Plugging in &nbsp;<math style="vertical-align: -1px">t=0</math>&nbsp; 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>
+|we see that &nbsp;<math style="vertical-align: -14px">\frac{dy}{dx}</math>&nbsp; is undefined at &nbsp;<math style="vertical-align: -1px">t=0.</math>
 |-
 |So, there is no tangent line at the origin.

@@ Line 3: / Line 3: @@
 ::<span class="exam"><math>y=t^3-t</math>
-<span class="exam">(a) Sketch the curve for &nbsp;<math>-2\le t \le 2.</math>
+<span class="exam">(a) Sketch the curve for &nbsp;<math style="vertical-align: -2px">-2\le t \le 2.</math>
 <span class="exam">(b) Find the equation of the tangent line to the curve at the origin.

@@ Line 1: / Line 1: @@
 <span class="exam">A curve is described parametrically by
-::::<span class="exam"><math>x=t^2</math>
+::<span class="exam"><math>x=t^2</math>
-::::<span class="exam"><math>y=t^3-t</math>
+::<span class="exam"><math>y=t^3-t</math>
-::<span class="exam">a) Sketch the curve for <math>-2\le t \le 2.</math>
+<span class="exam">(a) Sketch the curve for &nbsp;<math>-2\le t \le 2.</math>
-::<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;"
@@ Line 20: / Line 20: @@
+|
 |}
 '''Solution:'''
@@ Line 62: / Line 63: @@
+|
 |}
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"

@@ Line 1: / Line 1: @@
-<span class="exam">Compute
+<span class="exam">A curve is described parametrically by
-::<span class="exam">a) <math style="vertical-align: -12px">\lim_{n\rightarrow \infty} \frac{3-2n^2}{5n^2+n+1}</math>
+::<span class="exam">a) Sketch the curve for <math>-2\le t \le 2.</math>
-::<span class="exam">b) <math style="vertical-align: -12px">\lim_{n\rightarrow \infty} \frac{\ln n}{\ln 3n}</math>
+::<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;"

@@ Line 10: / Line 10: @@
 !Foundations: &nbsp;
 |-
+|'''1.''' What two pieces of information do you need to write the equation of a line?
 |-
+|
 |-
+|'''2.''' What is the slope of the tangent line of a parametric curve?
 |-
+|
 |}
 '''Solution:'''
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-!Step 1: &nbsp;
+!(a) &nbsp;
 |-
+|Insert graph
 |-
+|
@@ Line 49: / Line 39: @@
 !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
 |-
+|
 |}
@@ Line 59: / Line 50: @@
 !Step 2: &nbsp;
 |-
+|Now, the origin corresponds to <math>x=0</math> and <math>y=0.</math>
 |-
+|So, there is no tangent line at the origin.
 |}
@@ Line 68: / Line 67: @@
 !Final Answer: &nbsp;
 |-
-|&nbsp;&nbsp; '''(a)'''
+|&nbsp; &nbsp; '''(a)'''&nbsp; &nbsp; See above
 |-
-|&nbsp;&nbsp; '''(b)'''
+|&nbsp; &nbsp; '''(b)'''&nbsp; &nbsp;  There is no tangent line at the origin.
 |}
 [[009C_Sample_Final_3|'''<u>Return to Sample Exam</u>''']]