009A Sample Midterm 2, Problem 2 - Revision history

MathAdmin at 22:19, 9 November 2017

2017-11-09T22:19:41Z

MathAdmin at 15:49, 7 November 2017

2017-11-07T15:49:48Z

MathAdmin at 02:33, 14 April 2017

2017-04-14T02:33:25Z

MathAdmin: Created page with "The function $f(x)=3x^7-8x+2$ is a polynomial and therefore continuous everywhere. (..."

2017-04-09T18:51:34Z

Created page with "The function <math style="vertical-align: -5px">f(x)=3x^7-8x+2</math> is a polynomial and therefore continuous everywhere. (..."

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The function  <math style="vertical-align: -5px">f(x)=3x^7-8x+2</math>  is a polynomial and therefore continuous everywhere.

(a) State the Intermediate Value Theorem.

(b) Use the Intermediate Value Theorem to show that  <math style="vertical-align: -5px">f(x)</math>  has a zero in the interval  <math style="vertical-align: -5px">[0,1].</math>

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Foundations:  
|-
|What is a zero of the function  <math style="vertical-align: -5px">f(x)?</math>
|-
|        A zero is a value  <math style="vertical-align: -1px">c</math>  such that  <math style="vertical-align: -5px">f(c)=0.</math>
|}

'''Solution:'''

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!(a)  
|-
|'''Intermediate Value Theorem'''
|-
|        If  <math style="vertical-align: -5px">f(x)</math>  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 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: 0px">x</math>  in the closed interval such that  <math style="vertical-align: -5px">f(x)=c.</math>
|}

'''(b)'''
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 1:  
|-
|First,  <math style="vertical-align: -5px">f(x)</math>  is continuous on the interval  <math style="vertical-align: -5px">[0,1]</math>  since  <math style="vertical-align: -5px">f(x)</math>  is continuous everywhere.
|-
|Also,
|-
|
        <math>f(0)=2</math>
|-
|and
        <math>f(1)=3-8+2=-3.</math>.
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 2:  
|-
|Since  <math style="vertical-align: -1px">0</math>  is between  <math style="vertical-align: -5px">f(0)=2</math>  and  <math style="vertical-align: -5px">f(1)=-3,</math>
|-
|the Intermediate Value Theorem tells us that there is at least one number  <math style="vertical-align: -1px">x</math>
|-
|such that  <math style="vertical-align: -5px">f(x)=0.</math>
|-
|This means that  <math style="vertical-align: -5px">f(x)</math>  has a zero in the interval  <math style="vertical-align: -5px">[0,1].</math>
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Final Answer:  
|-
|    '''(a)'''     See solution above.
|-
|    '''(b)'''     See solution above.
|}
[[009A_Sample_Midterm_2|'''Return to Sample Exam''']]

@@ Line 5: / Line 5: @@
 <span class="exam">(b) Use the Intermediate Value Theorem to show that &nbsp;<math style="vertical-align: -5px">f(x)</math>&nbsp; has a zero in the interval &nbsp;<math style="vertical-align: -5px">[0,1].</math>
 <hr>
-[[File:9ASM2P2.jpg|600px|thumb|center]]
+[[009A Sample Midterm 2, Problem 2 Detailed Solution|'''<u>Detailed Solution</u>''']]
 [[009A_Sample_Midterm_2|'''<u>Return to Sample Exam</u>''']]

@@ Line 4: / Line 4: @@
 <span class="exam">(b) Use the Intermediate Value Theorem to show that &nbsp;<math style="vertical-align: -5px">f(x)</math>&nbsp; has a zero in the interval &nbsp;<math style="vertical-align: -5px">[0,1].</math>
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
+[[File:9ASM2P2.jpg|600px|thumb|center]]
 [[009A_Sample_Midterm_2|'''<u>Return to Sample Exam</u>''']]

@@ Line 48: / Line 48: @@
 !Step 2: &nbsp;
 |-
-|Since &nbsp;<math style="vertical-align: -1px">0</math>&nbsp; is between &nbsp;<math style="vertical-align: -5px">f(0)=2</math>&nbsp; and &nbsp;<math style="vertical-align: -5px">f(1)=-3,</math>
+|Since &nbsp;<math style="vertical-align: -5px">y=0</math>&nbsp; is between &nbsp;<math style="vertical-align: -5px">f(0)=2</math>&nbsp; and &nbsp;<math style="vertical-align: -5px">f(1)=-3,</math>
 |-
 |the Intermediate Value Theorem tells us that there is at least one number &nbsp;<math style="vertical-align: -1px">x</math>

009A Sample Midterm 2, Problem 2 - Revision history

MathAdmin at 22:19, 9 November 2017

MathAdmin at 15:49, 7 November 2017

MathAdmin at 02:33, 14 April 2017

MathAdmin: Created page with "The function f(x)=3x^7-8x+2 is a polynomial and therefore continuous everywhere. (..."

MathAdmin: Created page with "The function $f(x)=3x^7-8x+2$ is a polynomial and therefore continuous everywhere. (..."