Difference between revisions of "009A Sample Midterm 2, Problem 2"

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<span class="exam">(b) Use the Intermediate Value Theorem to show that &nbsp;<math style="vertical-align: -5px">f(x)</math>&nbsp; has a zero in the interval &nbsp;<math style="vertical-align: -5px">[0,1].</math>
 
<span class="exam">(b) Use the Intermediate Value Theorem to show that &nbsp;<math style="vertical-align: -5px">f(x)</math>&nbsp; has a zero in the interval &nbsp;<math style="vertical-align: -5px">[0,1].</math>
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[[009A Sample Midterm 2, Problem 2 Solution|'''<u>Solution</u>''']]
  
  
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[[009A Sample Midterm 2, Problem 2 Detailed Solution|'''<u>Detailed Solution</u>''']]
!Foundations: &nbsp;
 
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|What is a zero of the function &nbsp;<math style="vertical-align: -5px">f(x)?</math>
 
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|&nbsp; &nbsp; &nbsp; &nbsp; A zero is a value &nbsp;<math style="vertical-align: -1px">c</math>&nbsp; such that &nbsp;<math style="vertical-align: -5px">f(c)=0.</math>
 
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'''Solution:'''
 
 
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!(a) &nbsp;
 
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|'''Intermediate Value Theorem'''
 
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|&nbsp; &nbsp; &nbsp; &nbsp; If &nbsp;<math style="vertical-align: -5px">f(x)</math>&nbsp; is continuous on a closed interval &nbsp;<math style="vertical-align: -5px">[a,b]</math>
 
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|&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp;<math style="vertical-align: 0px">c</math>&nbsp; is any number between &nbsp;<math style="vertical-align: -5px">f(a)</math>&nbsp; and &nbsp;<math style="vertical-align: -5px">f(b),</math>
 
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&nbsp; &nbsp; &nbsp; &nbsp; then there is at least one number &nbsp;<math style="vertical-align: 0px">x</math>&nbsp; in the closed interval such that &nbsp;<math style="vertical-align: -5px">f(x)=c.</math>
 
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'''(b)'''
 
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!Step 1: &nbsp;
 
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|First, &nbsp;<math style="vertical-align: -5px">f(x)</math>&nbsp; is continuous on the interval &nbsp;<math style="vertical-align: -5px">[0,1]</math>&nbsp; since &nbsp;<math style="vertical-align: -5px">f(x)</math>&nbsp; is continuous everywhere.
 
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|Also,
 
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&nbsp; &nbsp; &nbsp; &nbsp; <math>f(0)=2</math>
 
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|and
 
&nbsp; &nbsp; &nbsp; &nbsp; <math>f(1)=3-8+2=-3.</math>.
 
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!Step 2: &nbsp;
 
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|Since &nbsp;<math style="vertical-align: -5px">y=0</math>&nbsp; is between &nbsp;<math style="vertical-align: -5px">f(0)=2</math>&nbsp; and &nbsp;<math style="vertical-align: -5px">f(1)=-3,</math>
 
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|the Intermediate Value Theorem tells us that there is at least one number &nbsp;<math style="vertical-align: -1px">x</math>
 
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|such that &nbsp;<math style="vertical-align: -5px">f(x)=0.</math>
 
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|This means that &nbsp;<math style="vertical-align: -5px">f(x)</math>&nbsp; has a zero in the interval &nbsp;<math style="vertical-align: -5px">[0,1].</math>
 
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!Final Answer: &nbsp;
 
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|&nbsp; &nbsp; '''(a)''' &nbsp; &nbsp; See solution above.
 
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|&nbsp; &nbsp; '''(b)''' &nbsp; &nbsp; See solution above.
 
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[[009A_Sample_Midterm_2|'''<u>Return to Sample Exam</u>''']]
 
[[009A_Sample_Midterm_2|'''<u>Return to Sample Exam</u>''']]

Latest revision as of 15:19, 9 November 2017

The function    is a polynomial and therefore continuous everywhere.

(a) State the Intermediate Value Theorem.

(b) Use the Intermediate Value Theorem to show that    has a zero in the interval  


Solution


Detailed Solution


Return to Sample Exam