Difference between revisions of "009A Sample Final 1, Problem 6"

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(Created page with "<span class="exam"> Consider the following function: ::::::<math>f(x)=3x-2\sin x+7</math> <span class="exam">a) Use the Intermediate Value Theorem to show that <math style="...")
 
 
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<span class="exam"> Consider the following function:
 
<span class="exam"> Consider the following function:
  
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::::::<math>f(x)=3x-2\sin x+7</math>
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::<math>f(x)=3x-2\sin x+7</math>
  
−
<span class="exam">a) Use the Intermediate Value Theorem to show that <math style="vertical-align: -5px">f(x)</math>&thinsp; has at least one zero.
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<span class="exam">(a) Use the Intermediate Value Theorem to show that &nbsp;<math style="vertical-align: -5px">f(x)</math>&nbsp; has at least one zero.
  
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<span class="exam">b) Use the Mean Value Theorem to show that <math style="vertical-align: -5px">f(x)</math>&thinsp; has at most one zero.
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<span class="exam">(b) Use the Mean Value Theorem to show that &nbsp;<math style="vertical-align: -5px">f(x)</math>&nbsp; has at most one zero.
  
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Foundations: &nbsp;  
 
!Foundations: &nbsp;  
 
|-
 
|-
−
|Recall:
+
|'''1.''' '''Intermediate Value Theorem'''
 
|-
 
|-
−
|'''1. Intermediate Value Theorem:''' If <math style="vertical-align: -5px">f(x)</math>&thinsp; is continuous on a closed interval <math style="vertical-align: -5px">[a,b]</math> and <math style="vertical-align: 0px">c</math> is any number
+
|&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>&nbsp; and &nbsp;<math style="vertical-align: 0px">c</math>&nbsp; is any number
 
|-
 
|-
 
|
 
|
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::between <math style="vertical-align: -5px">f(a)</math>&thinsp; and <math style="vertical-align: -5px">f(b)</math>, then there is at least one number <math style="vertical-align: 0px">x</math> in the closed interval such that <math style="vertical-align: -5px">f(x)=c.</math>
+
&nbsp; &nbsp; &nbsp; &nbsp;between &nbsp;<math style="vertical-align: -5px">f(a)</math>&nbsp; and &nbsp;<math style="vertical-align: -5px">f(b),</math>&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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|'''2. Mean Value Theorem:''' Suppose <math style="vertical-align: -5px">f(x)</math>&thinsp; is a function that satisfies the following:
+
|'''2.'''  '''Mean Value Theorem'''  
 +
|-
 +
|&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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::<math style="vertical-align: -5px">f(x)</math>&thinsp; is continuous on the closed interval &thinsp;<math style="vertical-align: -5px">[a,b].</math>
+
&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>
 
|-
 
|-
 
|
 
|
−
::<math style="vertical-align: -5px">f(x)</math>&thinsp; is differentiable on the open interval <math style="vertical-align: -5px">(a,b).</math>
+
&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>
 
|-
 
|-
 
|
 
|
−
::Then, there is a number <math style="vertical-align: 0px">c</math> such that &thinsp;<math style="vertical-align: 0px">a<c<b</math>&thinsp; and <math style="vertical-align: -14px">f'(c)=\frac{f(b)-f(a)}{b-a}.</math>
+
&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>
 
|}
 
|}
 +
  
 
'''Solution:'''
 
'''Solution:'''
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!Step 1: &nbsp;  
 
!Step 1: &nbsp;  
 
|-
 
|-
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|First note that&thinsp; <math style="vertical-align: -5px">f(0)=7.</math>
+
|First note that &nbsp; <math style="vertical-align: -5px">f(0)=7.</math>
 
|-
 
|-
−
|Also,&thinsp; <math style="vertical-align: -5px">f(-5)=-15-2\sin(-5)+7=-8-2\sin(-5).</math>
+
|Also, &nbsp;<math style="vertical-align: -5px">f(-5)=-15-2\sin(-5)+7=-8-2\sin(-5).</math>
 
|-
 
|-
−
|Since&thinsp; <math style="vertical-align: -5px">-1\leq \sin(x) \leq 1,</math>
+
|Since &nbsp;<math style="vertical-align: -5px">-1\leq \sin(x) \leq 1,</math>
 
|-
 
|-
 
|
 
|
−
::<math>-2\leq -2\sin(x) \leq 2.</math>
+
&nbsp; &nbsp; &nbsp; &nbsp;<math>-2\leq -2\sin(x) \leq 2.</math>
 
|-
 
|-
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|Thus,&thinsp; <math style="vertical-align: -5px">-10\leq f(-5) \leq -6</math>&thinsp; and hence &thinsp;<math style="vertical-align: -5px">f(-5)<0.</math>
+
|Thus, &nbsp;<math style="vertical-align: -5px">-10\leq f(-5) \leq -6</math>&nbsp; and hence &nbsp;<math style="vertical-align: -5px">f(-5)<0.</math>
 
|}
 
|}
  
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!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
|-
 
|-
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|Since <math style="vertical-align: -5px">f(-5)<0</math>&thinsp; and &thinsp;<math style="vertical-align: -5px">f(0)>0,</math>&thinsp; there exists <math style="vertical-align: 0px">x</math> with &thinsp;<math style="vertical-align: 0px">-5<x<0</math>&thinsp; such that  
+
|Since &nbsp;<math style="vertical-align: -5px">f(-5)<0</math>&nbsp; and &nbsp;<math style="vertical-align: -5px">f(0)>0,</math>&nbsp; there exists &nbsp;<math style="vertical-align: 0px">x</math>&nbsp; with &nbsp;<math style="vertical-align: 0px">-5<x<0</math>&nbsp; such that  
 
|-
 
|-
−
|<math style="vertical-align: -5px">f(x)=0</math>&thinsp; by the Intermediate Value Theorem. Hence, <math style="vertical-align: -5px">f(x)</math>&thinsp; has at least one zero.
+
|<math style="vertical-align: -5px">f(x)=0</math>&nbsp; by the Intermediate Value Theorem. Hence, &nbsp;<math style="vertical-align: -5px">f(x)</math>&nbsp; has at least one zero.
 
|}
 
|}
  
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!Step 1: &nbsp;  
 
!Step 1: &nbsp;  
 
|-
 
|-
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|Suppose that <math style="vertical-align: -5px">f(x)</math> has more than one zero. So, there exist <math style="vertical-align: -4px">a,b</math> such that &thinsp;<math style="vertical-align: -5px">f(a)=f(b)=0.</math>
+
|Suppose that &nbsp;<math style="vertical-align: -5px">f(x)</math>&nbsp; has more than one zero. So, there exist &nbsp;<math style="vertical-align: -4px">a,b</math>&nbsp; with &nbsp;<math style="vertical-align: 0px">a<b</math>&nbsp; such that &nbsp;<math style="vertical-align: -5px">f(a)=f(b)=0.</math>
 
|-
 
|-
−
|Then, by the Mean Value Theorem, there exists <math style="vertical-align: 0px">c</math> with &thinsp;<math style="vertical-align: 0px">a<c<b</math> such that &thinsp;<math style="vertical-align: -5px">f'(c)=0.</math>
+
|Then, by the Mean Value Theorem, there exists &nbsp;<math style="vertical-align: 0px">c</math>&nbsp; with &nbsp;<math style="vertical-align: 0px">a<c<b</math>&nbsp; such that &nbsp;<math style="vertical-align: -5px">f'(c)=0.</math>
 
|}
 
|}
  
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!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
|-
 
|-
−
|We have <math style="vertical-align: -5px">f'(x)=3-2\cos(x).</math>&thinsp; Since &thinsp;<math style="vertical-align: -5px">-1\leq \cos(x)\leq 1,</math>
+
|We have &nbsp;<math style="vertical-align: -5px">f'(x)=3-2\cos(x).</math>&nbsp;  
 +
|-
 +
|Since &nbsp;<math style="vertical-align: -5px">-1\leq \cos(x)\leq 1,</math>
 +
|-
 +
|&nbsp; &nbsp; &nbsp; &nbsp;<math style="vertical-align: -5px">-2 \leq -2\cos(x)\leq 2.</math>&nbsp;
 
|-
 
|-
−
|<math style="vertical-align: -5px">-2 \leq -2\cos(x)\leq 2.</math>&thinsp; So, <math style="vertical-align: -5px">1\leq f'(x) \leq 5,</math>
+
|So, &nbsp;<math style="vertical-align: -5px">1\leq f'(x) \leq 5,</math>
 
|-
 
|-
−
|which contradicts <math style="vertical-align: -5px">f'(c)=0.</math> Thus, <math style="vertical-align: -5px">f(x)</math>&thinsp; has at most one zero.
+
|which contradicts &nbsp;<math style="vertical-align: -5px">f'(c)=0.</math>&nbsp; Thus, &nbsp;<math style="vertical-align: -5px">f(x)</math>&nbsp; has at most one zero.
 
|}
 
|}
 +
  
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Final Answer: &nbsp;  
 
!Final Answer: &nbsp;  
 
|-
 
|-
−
|'''(a)''' Since <math style="vertical-align: -5px">f(-5)<0</math>&thinsp; and &thinsp;<math style="vertical-align: -5px">f(0)>0,</math>&thinsp; there exists <math style="vertical-align: 0px">x</math> with &thinsp;<math style="vertical-align: 0px">-5<x<0</math>&thinsp; such that
+
|&nbsp; &nbsp; '''(a)''' &nbsp; &nbsp; See solution above.
−
|-
 
−
|<math style="vertical-align: -5px">f(x)=0</math>&thinsp; by the Intermediate Value Theorem. Hence, <math style="vertical-align: -5px">f(x)</math>&thinsp; has at least one zero.
 
 
|-
 
|-
−
|'''(b)''' See '''Step 1''' and '''Step 2''' above.
+
|&nbsp; &nbsp; '''(b)''' &nbsp; &nbsp; See solution above.
 
|}
 
|}
 
[[009A_Sample_Final_1|'''<u>Return to Sample Exam</u>''']]
 
[[009A_Sample_Final_1|'''<u>Return to Sample Exam</u>''']]

Latest revision as of 08:12, 10 April 2017

Consider the following 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)=3x-2\sin x+7}

(a) 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 at least one zero.

(b) Use the Mean 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 at most one zero.

Foundations:  
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.}

2. Mean Value Theorem
        Suppose  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 a function that satisfies the following:

       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 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].}

       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 differentiable on the open 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).}

       Then, there is a 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 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 a<c<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 f'(c)=\frac{f(b)-f(a)}{b-a}.}


Solution:

(a)

Step 1:  
First note 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(0)=7.}
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(-5)=-15-2\sin(-5)+7=-8-2\sin(-5).}
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 -1\leq \sin(x) \leq 1,}

       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 -2\leq -2\sin(x) \leq 2.}

Thus,  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 -10\leq f(-5) \leq -6}   and hence  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(-5)<0.}
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 f(-5)<0}   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(0)>0,}   there exists  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}   with  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 -5<x<0}   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}   by the Intermediate Value Theorem. Hence,  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 at least one zero.

(b)

Step 1:  
Suppose 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 more than one zero. So, there exist  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}   with  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}   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(a)=f(b)=0.}
Then, by the Mean Value Theorem, there exists  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}   with  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<c<b}   such that  
Step 2:  
We have  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)=3-2\cos(x).}  
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 -1\leq \cos(x)\leq 1,}
       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 -2 \leq -2\cos(x)\leq 2.}  
So,  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 1\leq f'(x) \leq 5,}
which contradicts  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.}   Thus,  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 at most one zero.


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

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