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

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::between <math style="vertical-align: -5px">f(a)</math> and <math style="vertical-align: -5px">f(b)</math>, then there is at least one number <math style="vertical-align: -1px">x</math> in the closed interval such that <math style="vertical-align: -5px">f(x)=c</math>.
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::between <math style="vertical-align: -5px">f(a)</math> and <math style="vertical-align: -5px">f(b)</math>, then there is at least one number <math style="vertical-align: -1px">x</math> in the closed interval such that <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> is a function that satisfies the following:
 
|'''2. Mean Value Theorem''' Suppose <math style="vertical-align: -5px">f(x)</math> is a function that satisfies the following:
 
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::<math style="vertical-align: -5px">f(x)</math> is continuous on the closed interval <math style="vertical-align: -5px">[a,b]</math>.
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::<math style="vertical-align: -5px">f(x)</math> is continuous on the closed interval <math style="vertical-align: -5px">[a,b].</math>
 
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::<math style="vertical-align: -5px">f(x)</math> is differentiable on the open interval <math style="vertical-align: -5px">(a,b)</math>.
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::<math style="vertical-align: -5px">f(x)</math> is differentiable on the open interval <math style="vertical-align: -5px">(a,b).</math>
 
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::Then, there is a number <math style="vertical-align: -1px">c</math> such that <math style="vertical-align: -1px">a<c<b</math> and <math style="vertical-align: -14px">f'(c)=\frac{f(b)-f(a)}{b-a}</math>.
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::Then, there is a number <math style="vertical-align: -1px">c</math> such that <math style="vertical-align: -1px">a<c<b</math> and <math style="vertical-align: -14px">f'(c)=\frac{f(b)-f(a)}{b-a}.</math>
 
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!Step 1: &nbsp;  
 
!Step 1: &nbsp;  
 
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|First note that <math style="vertical-align: -3px">f(0)=7</math>.
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|First note that <math style="vertical-align: -3px">f(0)=7.</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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|Also, <math style="vertical-align: -3px">f(-5)=-15-2\sin(-5)+7=-8-2\sin(-5).</math>
 
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|Since <math style="vertical-align: -3px">-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 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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|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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!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
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|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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|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 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.
 
|<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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|Suppose that <math style="vertical-align: -5px">f(x)</math> has more than one zero. So, there exists <math style="vertical-align: -5px">a,b</math> such that <math style="vertical-align: -5px">f(a)=f(b)=0</math>.
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|Suppose that <math style="vertical-align: -5px">f(x)</math> has more than one zero. So, there exists <math style="vertical-align: -5px">a,b</math> such that <math style="vertical-align: -5px">f(a)=f(b)=0.</math>
 
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|Then, by the Mean Value Theorem, there exists <math style="vertical-align: -1px">c</math> with <math style="vertical-align: -1px">a<c<b</math> such that <math style="vertical-align: -5px">f'(c)=0</math>.
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|Then, by the Mean Value Theorem, there exists <math style="vertical-align: -1px">c</math> with <math style="vertical-align: -1px">a<c<b</math> such that <math style="vertical-align: -5px">f'(c)=0.</math>
 
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!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
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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>,
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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>
 
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|<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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|<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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|which contradicts <math style="vertical-align: -5px">f'(c)=0</math>. Thus, <math style="vertical-align: -5px">f(x)</math> has at most one zero.
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|which contradicts <math style="vertical-align: -5px">f'(c)=0.</math> Thus, <math style="vertical-align: -5px">f(x)</math> has at most one zero.
 
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!Final Answer: &nbsp;  
 
!Final Answer: &nbsp;  
 
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|'''(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  
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|'''(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  
 
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|<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.
 
|<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.

Revision as of 11:35, 29 February 2016

Consider the following function:

a) Use the Intermediate Value Theorem to show that has at least one zero.

b) Use the Mean Value Theorem to show that has at most one zero.

Foundations:  
Recall:
1. Intermediate Value Theorem If is continuous on a closed interval and is any number
between 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 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 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 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.}
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) 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) See Step 1 and Step 2 above.

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