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

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!Step 1:    
 
!Step 1:    
 
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|First, <math>f(x)</math> is continuous on the interval <math>[0,1]</math> since <math>f(x)</math> 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
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&nbsp; &nbsp; &nbsp; &nbsp; <math>f(1)=3-8+2=-3.</math>.
 
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!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
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|Since <math>0</math> is between <math>f(0)=2</math> and <math>f(1)=-3,</math>
 
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|the Intermediate Value Theorem tells us that there is at least one number <math>x</math>
 
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|such that <math>f(x)=0.</math>
 
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|This means that <math>f(x)</math> has a zero in the interval <math>[0,1].</math>
 
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!Final Answer: &nbsp;  
 
!Final Answer: &nbsp;  
 
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|'''(a)'''  
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|&nbsp; &nbsp; '''(a)''' &nbsp; &nbsp; See solution above.
 
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|'''(b)'''  
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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>''']]

Revision as of 13:13, 17 February 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


Foundations:  
?


Solution:

(a)

Step 1:  
Intermediate Value Theorem
        If   is continuous on a closed interval
        and is any number between   and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f(b)} ,

        then there is at least one number Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} in the closed interval such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=c.}

(b)

Step 1:  
First, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} is continuous on the interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [0,1]} since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} is continuous everywhere.
Also,

        Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(0)=2}

and

        Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(1)=3-8+2=-3.} .

Step 2:  
Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0} is between Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(0)=2} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(1)=-3,}
the Intermediate Value Theorem tells us that there is at least one number Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x}
such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=0.}
This means that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} has a zero in the interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [0,1].}


Final Answer:  
    (a)     See solution above.
    (b)     See solution above.

Return to Sample Exam

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