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

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!Step 1:    
 
!Step 1:    
 
|-
 
|-
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|First note that <math>f(0)=7</math>.
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|First note that <math style="vertical-align: -3px">f(0)=7</math>.
 
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|-
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|Also, <math>f(-5)=-15-2\sin(-5)+7=-8-2\sin(-5)</math>.
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|Also, <math style="vertical-align: -3px">f(-5)=-15-2\sin(-5)+7=-8-2\sin(-5)</math>.
 
|-
 
|-
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|Since <math>-1\leq \sin(x) \leq 1</math>,
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|Since <math style="vertical-align: -3px">-1\leq \sin(x) \leq 1</math>,
 
|-
 
|-
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|<math>-2\leq -2\sin(x) \leq 2</math>.
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|
 +
::<math>-2\leq -2\sin(x) \leq 2</math>.
 
|-
 
|-
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|Thus, <math>-10\leq f(-5) \leq -6</math> and hence <math>f(-5)<0</math>.
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|Thus, <math style="vertical-align: -5px">-10\leq f(-5) \leq -6</math> and hence <math style="vertical-align: -5px">f(-5)<0</math>.
 
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|}
  
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!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
|-
 
|-
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|Since <math>f(-5)<0</math> and <math>f(0)>0</math>, there exists <math>x</math> with <math>-5<x<0</math> such that  
+
|Since <math style="vertical-align: -5px">f(-5)<0</math> and <math style="vertical-align: -5px">f(0)>0</math>, there exists <math style="vertical-align: -1px">x</math> with <math style="vertical-align: -1px">-5<x<0</math> such that  
 
|-
 
|-
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|<math>f(x)=0</math> by the Intermediate Value Theorem. Hence, <math>f(x)</math> has at least one zero.
+
|<math style="vertical-align: -5px">f(x)=0</math> by the Intermediate Value Theorem. Hence, <math style="vertical-align: -5px">f(x)</math> has at least one zero.
 
|}
 
|}
  
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!Step 1: &nbsp;  
 
!Step 1: &nbsp;  
 
|-
 
|-
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|We have <math>f'(x)=3-2\cos(x)</math>. Since <math>-1\leq \cos(x)\leq 1</math>,
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|We have <math style="vertical-align: -5px">f'(x)=3-2\cos(x)</math>. Since <math style="vertical-align: -5px">-1\leq \cos(x)\leq 1</math>,
 
|-
 
|-
−
|<math>-2 \leq -2\cos(x)\leq 2</math>. So, <math>1\leq f'(x) \leq 5</math>.
+
|<math style="vertical-align: -5px">-2 \leq -2\cos(x)\leq 2</math>. So, <math style="vertical-align: -5px">1\leq f'(x) \leq 5</math>.
 
|-
 
|-
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|Therefore, <math>f'(x)</math> is always positive.
+
|Therefore, <math style="vertical-align: -5px">f'(x)</math> is always positive.
 
|}
 
|}
  
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!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
|-
 
|-
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|Since <math>f'(x)</math> is always positive, <math>f(x)</math> is an increasing function.
+
|Since <math style="vertical-align: -3px">f'(x)</math> is always positive, <math style="vertical-align: -3px">f(x)</math> is an increasing function.
 
|-
 
|-
−
|Thus, <math>f(x)</math> has at most one zero.  
+
|Thus, <math style="vertical-align: -3px">f(x)</math> has at most one zero.  
 
|}
 
|}
  
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!Final Answer: &nbsp;  
 
!Final Answer: &nbsp;  
 
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|-
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|'''(a)''' Since <math>f(-5)<0</math> and <math>f(0)>0</math>, there exists <math>x</math> with <math>-5<x<0</math> such that  
+
|'''(a)''' Since <math style="vertical-align: -5px">f(-5)<0</math> and <math style="vertical-align: -5px">f(0)>0</math>, there exists <math style="vertical-align: -1px">x</math> with <math style="vertical-align: -1px">-5<x<0</math> such that  
 
|-
 
|-
−
|<math>f(x)=0</math> by the Intermediate Value Theorem. Hence, <math>f(x)</math> has at least one zero.
+
|<math style="vertical-align: -5px">f(x)=0</math> by the Intermediate Value Theorem. Hence, <math style="vertical-align: -5px">f(x)</math> has at least one zero.
 
|-
 
|-
−
|'''(b)''' Since <math>f'(x)</math> is always positive, <math>f(x)</math> is an increasing function.
+
|'''(b)''' Since <math style="vertical-align: -3px">f'(x)</math> is always positive, <math style="vertical-align: -3px">f(x)</math> is an increasing function.
 
|-
 
|-
−
|Thus, <math>f(x)</math> has at most one zero.  
+
|Thus, <math style="vertical-align: -3px">f(x)</math> has at most one zero.
 
|}
 
|}
 
[[009A_Sample_Final_1|'''<u>Return to Sample Exam</u>''']]
 
[[009A_Sample_Final_1|'''<u>Return to Sample Exam</u>''']]

Revision as of 15:22, 22 February 2016

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:  

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:  
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} .
Therefore, 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 always positive.
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'(x)} is always positive, 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 an increasing function.
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) 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) 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 always positive, 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 an increasing function.
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.

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