009A Sample Final 2, Problem 9 - Revision history

MathAdmin at 18:33, 1 June 2017

2017-06-01T18:33:50Z

MathAdmin at 15:06, 1 June 2017

2017-06-01T15:06:51Z

MathAdmin: Created page with "A plane begins its takeoff at 2:00pm on a 2500-mile flight. After 5.5 hours, the plane arrives at its destination. Give a precise mathematical reason using..."

2017-04-10T16:25:59Z

Created page with "<span class="exam">A plane begins its takeoff at 2:00pm on a 2500-mile flight. After 5.5 hours, the plane arrives at its destination. Give a precise mathematical reason using..."

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<span class="exam">A plane begins its takeoff at 2:00pm on a 2500-mile flight. After 5.5 hours, the plane arrives at its destination. Give a precise mathematical reason using the mean value theorem to explain why there are at least two times during the flight when the speed of the plane is 400 miles per hour.

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Foundations:  
|-
|'''Mean Value Theorem'''
|-
|        Suppose  <math style="vertical-align: -5px">f(x)</math>  is a function that satisfies the following:
|-
|
       <math style="vertical-align: -5px">f(x)</math>  is continuous on the closed interval  <math style="vertical-align: -5px">[a,b].</math>
|-
|
       <math style="vertical-align: -5px">f(x)</math>  is differentiable on the open interval  <math style="vertical-align: -5px">(a,b).</math>
|-
|
       Then, there is a number  <math style="vertical-align: 0px">c</math>  such that  <math style="vertical-align: 0px">a<c<b</math>  and  <math style="vertical-align: -14px">f'(c)=\frac{f(b)-f(a)}{b-a}.</math>
|}

'''Solution:'''

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 1:  
|-
|On average the plane flew
|-
|       <math>\frac{2500 \text{ miles}}{5.5 \text{ hrs}}\approx 454.5 \text{ miles/hr}.</math>
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 2:  
|-
|In order to average this speed, the plane had to go from 0mph, up to full speed, past 454.5mph, and then it had to go back down to 0mph to land.
|-
|This means that there will be at least two times where the plane of the speed is 400mph.
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Final Answer:  
|-
|       See above.
|}
[[009A_Sample_Final_2|'''<u>Return to Sample Exam</u>''']]

← Older revision		Revision as of 18:33, 1 June 2017
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−	<span class="exam">A plane begins its takeoff at 2:00pm on a 2500-mile flight. After 5.5 hours, the plane arrives at its destination. Give a precise mathematical reason ~~using the mean value theorem~~ to explain why there are at least two times during the flight when the speed of the plane is 400 miles per hour.	+	<span class="exam">A plane begins its takeoff at 2:00pm on a 2500-mile flight. After 5.5 hours, the plane arrives at its destination. Give a precise mathematical reason to explain why there are at least two times during the flight when the speed of the plane is 400 miles per hour.

	{\| class="mw-collapsible mw-collapsed" style = "text-align:left;"		{\| class="mw-collapsible mw-collapsed" style = "text-align:left;"

@@ Line 4: / Line 4: @@
 !Foundations: &nbsp;
 |-
-|'''Mean Value Theorem'''
+|'''Intermediate Value Theorem'''
 |-
-|&nbsp; &nbsp; &nbsp; &nbsp; Suppose &nbsp;<math style="vertical-align: -5px">f(x)</math>&nbsp; is a function that satisfies the following:
+|&nbsp; &nbsp; &nbsp; &nbsp; Let &nbsp;<math style="vertical-align: -5px">f(x)</math>&nbsp; be a continuous function on the interval &nbsp;<math style="vertical-align: -5px">[a,b]</math>&nbsp; and
 |-
+|&nbsp; &nbsp; &nbsp; &nbsp; without loss of generality, let &nbsp;<math style="vertical-align: -5px">f(a)<f(b).</math>
 |-
+|
-&nbsp; &nbsp; &nbsp; &nbsp;<math style="vertical-align: -5px">f(x)</math>&nbsp; is differentiable on the open interval &nbsp;<math style="vertical-align: -5px">(a,b).</math>
+&nbsp; &nbsp; &nbsp; &nbsp;Then, for every value &nbsp;<math style="vertical-align: -5px">y,</math>&nbsp; where <math style="vertical-align: -5px">f(a)<y<f(b),</math>&nbsp;
 |-
+|&nbsp; &nbsp; &nbsp; &nbsp;there is a value &nbsp;<math style="vertical-align: 0px">c</math> &nbsp; in &nbsp;<math style="vertical-align: -5px">[a,b]</math>&nbsp; such that &nbsp;<math style="vertical-align: -5px">f(c)=y.</math>
 |}
@@ Line 34: / Line 32: @@
 |In order to average this speed, the plane had to go from 0mph, up to full speed, past 454.5mph, and then it had to go back down to 0mph to land.
 |-
-|This means that there will be at least two times where the plane of the speed is 400mph.
+|This means that there will be at least two times where the plane of the speed is 400mph by the Intermediate Value Theorem.
 |}