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

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(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.
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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 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;"
 
!Foundations: &nbsp;  
 
!Foundations: &nbsp;  
 
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|'''Mean Value Theorem'''  
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|'''Intermediate 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; 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
 
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|&nbsp; &nbsp; &nbsp; &nbsp; without loss of generality, let &nbsp;<math style="vertical-align: -5px">f(a)<f(b).</math>
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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, 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;
 
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|&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>
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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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|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.
 
|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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|This means that there will be at least two times where the plane of the speed is 400mph by the Intermediate Value Theorem.
 
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Latest revision as of 10:33, 1 June 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 to explain why there are at least two times during the flight when the speed of the plane is 400 miles per hour.

Foundations:  
Intermediate Value Theorem
        Let    be a continuous function on the interval    and
        without loss of generality, let  

       Then, for every value    where  

       there is a value     in    such that  


Solution:

Step 1:  
On average the plane flew
       
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 by the Intermediate Value Theorem.


Final Answer:  
       See above.

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