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

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!Foundations:    
 
!Foundations:    
 
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|What is a zero of the function <math>f(x)?</math>
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|What is a zero of the function <math style="vertical-align: -5px">f(x)?</math>
 
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|-
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|&nbsp; &nbsp; &nbsp; &nbsp; A zero is a value <math>c</math> such that <math>f(c)=0.</math>
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|&nbsp; &nbsp; &nbsp; &nbsp; A zero is a value <math style="vertical-align: -1px">c</math> such that <math style="vertical-align: -5px">f(c)=0.</math>
 
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|&nbsp; &nbsp; &nbsp; &nbsp; If <math style="vertical-align: -5px">f(x)</math>&thinsp; is continuous on a closed interval <math style="vertical-align: -5px">[a,b]</math>  
 
|&nbsp; &nbsp; &nbsp; &nbsp; If <math style="vertical-align: -5px">f(x)</math>&thinsp; is continuous on a closed interval <math style="vertical-align: -5px">[a,b]</math>  
 
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|&nbsp; &nbsp; &nbsp; &nbsp; and <math style="vertical-align: 0px">c</math> is any number between <math style="vertical-align: -5px">f(a)</math>&thinsp; and <math style="vertical-align: -5px">f(b)</math>,
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|&nbsp; &nbsp; &nbsp; &nbsp; and <math style="vertical-align: 0px">c</math> is any number between <math style="vertical-align: -5px">f(a)</math>&thinsp; and <math style="vertical-align: -5px">f(b),</math>
 
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!Step 1: &nbsp;  
 
!Step 1: &nbsp;  
 
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|-
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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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|First, <math style="vertical-align: -5px">f(x)</math> is continuous on the interval <math style="vertical-align: -5px">[0,1]</math> since <math style="vertical-align: -5px">f(x)</math> is continuous everywhere.
 
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|Also,
 
|Also,
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!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
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|-
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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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|Since <math style="vertical-align: -1px">0</math> is between <math style="vertical-align: -5px">f(0)=2</math>&nbsp; and <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 <math>x</math>
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|the Intermediate Value Theorem tells us that there is at least one number <math style="vertical-align: -1px">x</math>
 
|-
 
|-
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|such that <math>f(x)=0.</math>
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|such that <math style="vertical-align: -5px">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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|This means that <math style="vertical-align: -5px">f(x)</math> has a zero in the interval <math style="vertical-align: -5px">[0,1].</math>
 
|}
 
|}
  

Revision as of 15:15, 18 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 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].}


Foundations:  
What is a zero of the function 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)?}
        A zero is a value 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 c} 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(c)=0.}


Solution:

(a)

Step 1:  
Intermediate Value Theorem
        If 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 a closed 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 [a,b]}
        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 c} is any number 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(a)}   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(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.

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