009A Sample Final 2, Problem 7 - Revision history

Kayla Murray: Replaced content with " Show that the equation $x^3+2x-2=0$ has exactly one real root.
009A Sample Final 2, Problem..."

2017-12-01T17:21:00Z

Replaced content with " Show that the equation <math style="vertical-align: -2px">x^3+2x-2=0</math> has exactly one real root. <hr> 009A Sample Final 2, Problem..."

Kayla Murray at 23:28, 7 March 2017

2017-03-07T23:28:26Z

Kayla Murray at 21:56, 6 March 2017

2017-03-06T21:56:02Z

Kayla Murray at 21:49, 6 March 2017

2017-03-06T21:49:31Z

Kayla Murray at 01:55, 19 February 2017

2017-02-19T01:55:55Z

Kayla Murray at 02:29, 18 February 2017

2017-02-18T02:29:17Z

Kayla Murray: Created page with "Compute ::a) $\lim_{x\rightarrow 4} \frac{\sqrt{x+5}-3}{x-4}$ ::b)
2017-02-18T02:14:09Z

Created page with "Compute ::a) <math style="vertical-align: -14px">\lim_{x\rightarrow 4} \frac{\sqrt{x+5}-3}{x-4}</math> ::b) <math st..."

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Compute

::a) <math style="vertical-align: -14px">\lim_{x\rightarrow 4} \frac{\sqrt{x+5}-3}{x-4}</math>

::b) <math style="vertical-align: -14px">\lim_{x\rightarrow 0} \frac{\sin^2x}{3x}</math>

::c) <math style="vertical-align: -14px">\lim_{x\rightarrow -\infty} \frac{\sqrt{x^2+2}}{2x-1}</math>

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'''(b)'''

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[[009A_Sample_Final_2|'''Return to Sample Exam''']]

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 <span class="exam"> Show that the equation &nbsp;<math style="vertical-align: -2px">x^3+2x-2=0</math>&nbsp; has exactly one real root.
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-!Foundations: &nbsp;
+[[009A Sample Final 2, Problem 7 Solution|'''<u>Solution</u>''']]
-'''Solution:'''
+[[009A Sample Final 2, Problem 7 Detailed Solution|'''<u>Detailed Solution</u>''']]
 [[009A_Sample_Final_2|'''<u>Return to Sample Exam</u>''']]

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 [[009A_Sample_Final_2|'''<u>Return to Sample Exam</u>''']]

← Older revision		Revision as of 21:49, 6 March 2017
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−	<span class="exam"> Show that the equation <math>x^3+2x-2=0</math> has exactly one real root.	+	<span class="exam"> Show that the equation  <math style="vertical-align: -2px">x^3+2x-2=0</math>  has exactly one real root.

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← Older revision		Revision as of 02:29, 18 February 2017
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−	<span class="exam">~~Compute~~	+	<span class="exam"> Show that the equation <math>x^3+2x-2=0</math> has exactly one real root.
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−	~~::<span class="exam">a)~~ <math ~~style="vertical-align: -14px"~~>~~\lim_{x\rightarrow 4} \frac{\sqrt{~~x+5}-~~3}{x-4}</math>~~
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−	~~::<span class~~=~~"exam">b) <math style="vertical-align: -14px">\lim_{x\rightarrow~~ 0~~} \frac{\sin^2x}{3x}</math>~~
−
−	~~::<span class="exam">c) <math style="vertical-align: -14px">\lim_{x\rightarrow -\infty} \frac{\sqrt{x^2+2}}{2x-1}~~</math>

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+|First, we note that
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+|&nbsp; &nbsp; &nbsp; &nbsp;<math style="vertical-align: -5px">f(x)=0</math>&nbsp;
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+|by the Intermediate Value Theorem.
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+|Hence, &nbsp;<math style="vertical-align: -5px">f(x)</math>&nbsp; has at least one zero.
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+|Suppose that &nbsp;<math style="vertical-align: -5px">f(x)</math>&nbsp; has more than one zero.
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+|Therefore, it is impossible for &nbsp;<math style="vertical-align: -5px">f'(c)=0.</math>&nbsp; Hence, &nbsp;<math style="vertical-align: -5px">f(x)</math>&nbsp; has at most one zero.
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 !Final Answer: &nbsp;
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+|&nbsp; &nbsp; &nbsp; &nbsp; See solution above.
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 [[009A_Sample_Final_2|'''<u>Return to Sample Exam</u>''']]