Difference between revisions of "009A Sample Final 2, Problem 9"

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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 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.
  
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!Foundations: &nbsp;
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[[009A Sample Final 2, Problem 9 Solution|'''<u>Solution</u>''']]
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|'''Mean Value Theorem'''
 
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|&nbsp; &nbsp; &nbsp; &nbsp; Suppose &nbsp;<math style="vertical-align: -5px">f(x)</math>&nbsp; is a function that satisfies the following:
 
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&nbsp; &nbsp; &nbsp; &nbsp;<math style="vertical-align: -5px">f(x)</math>&nbsp; is continuous on the closed interval &nbsp;<math style="vertical-align: -5px">[a,b].</math>
 
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&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>
 
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&nbsp; &nbsp; &nbsp; &nbsp;Then, there is a number &nbsp;<math style="vertical-align: 0px">c</math>&nbsp; such that &nbsp;<math style="vertical-align: 0px">a<c<b</math>&nbsp; and &nbsp;<math style="vertical-align: -14px">f'(c)=\frac{f(b)-f(a)}{b-a}.</math>
 
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'''Solution:'''
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[[009A Sample Final 2, Problem 9 Detailed Solution|'''<u>Detailed Solution</u>''']]
  
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!Step 1: &nbsp;
 
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|On average the plane flew
 
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|&nbsp; &nbsp; &nbsp; &nbsp;<math>\frac{2500 \text{ miles}}{5.5 \text{ hrs}}\approx 454.5 \text{ miles/hr}.</math>
 
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!Step 2: &nbsp;
 
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|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.
 
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|This means that there will be at least two times where the plane of the speed is 400mph.
 
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!Final Answer: &nbsp;
 
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|&nbsp; &nbsp; &nbsp; &nbsp;See above.
 
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[[009A_Sample_Final_2|'''<u>Return to Sample Exam</u>''']]
 
[[009A_Sample_Final_2|'''<u>Return to Sample Exam</u>''']]

Latest revision as of 10:25, 1 December 2017

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.

Solution


Detailed Solution


Return to Sample Exam

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