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

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!Foundations:    
 
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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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!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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Revision as of 20:30, 7 March 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.

Foundations:  
Mean Value Theorem
        Suppose    is a function that satisfies the following:

         is continuous on the closed interval  

         is differentiable on the open interval  

       Then, there is a number    such that    and  


Solution:

Step 1:  
On average the plane flew
       
Step 2:  


Final Answer:  

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