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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| + | |Also, <math>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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| − | | | + | |<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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!Step 2: | !Step 2: | ||
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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 |
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| − | | | + | |<math>f(x)=0</math> by the Intermediate Value Theorem. Hence, <math>f(x)</math> has at least one zero. |
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!Final Answer: | !Final Answer: | ||
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| − | |'''(a)''' | + | |'''(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 |
| + | |- | ||
| + | |<math>f(x)=0</math> by the Intermediate Value Theorem. Hence, <math>f(x)</math> has at least one zero. | ||
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|'''(b)''' | |'''(b)''' | ||
|} | |} | ||
[[009A_Sample_Final_1|'''<u>Return to Sample Exam</u>''']] | [[009A_Sample_Final_1|'''<u>Return to Sample Exam</u>''']] | ||
Revision as of 09:29, 15 February 2016
Consider the following function:
a) Use the Intermediate Value Theorem to show that Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f(x)} has at least one zero.
b) Use the Mean Value Theorem to show that Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f(x)} has at most one zero.
| Foundations: |
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Solution:
(a)
| Step 1: |
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| First note that . |
| Also, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f(-5)=-15-2\sin(-5)+7=-8-2\sin(-5)} . |
| Since Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle -1\leq \sin(x)\leq 1} , |
| Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle -2\leq -2\sin(x)\leq 2} . |
| Thus, and hence Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f(-5)<0} . |
| Step 2: |
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| Since Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f(-5)<0} and , 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: |
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| Step 2: |
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| Step 3: |
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| Final Answer: |
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| (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) |