009B Sample Midterm 3, Problem 1 - Revision history

MathAdmin at 01:32, 13 November 2017

2017-11-13T01:32:50Z

MathAdmin at 19:18, 9 April 2017

2017-04-09T19:18:46Z

MathAdmin at 22:27, 12 May 2016

2016-05-12T22:27:29Z

MathAdmin: Created page with "Divide the interval $[0,\pi]$ into four subintervals of equal length $\frac{\pi}{4}$ and compute the righ..."

2016-04-19T00:42:45Z

Created page with "<span class="exam">Divide the interval <math style="vertical-align: -5px">[0,\pi]</math> into four subintervals of equal length <math>\frac{\pi}{4}</math> and compute the righ..."

New page

<span class="exam">Divide the interval <math style="vertical-align: -5px">[0,\pi]</math> into four subintervals of equal length <math>\frac{\pi}{4}</math> and compute the right-endpoint Riemann sum of <math style="vertical-align: -5px">y=\sin (x).</math>

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Foundations:  
|-
||Recall:
|-
|
::'''1.''' The height of each rectangle in the right-hand Riemann sum is given by choosing the right endpoint of the interval.
|-
|
::'''2.''' See the Riemann sums (insert link) for more information.
|}

'''Solution:'''
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 1:  
|-
|Let <math style="vertical-align: -5px">f(x)=\sin(x).</math> Each interval has length <math>\frac{\pi}{4}.</math>
|-
|So, the right-endpoint Riemann sum of <math style="vertical-align: -5px">f(x)</math> on the interval <math style="vertical-align: -5px">[0,\pi]</math> is
|-
|
::<math>\frac{\pi}{4}\bigg(f\bigg(\frac{\pi}{4}\bigg)+f\bigg(\frac{\pi}{2}\bigg)+f\bigg(\frac{3\pi}{4}\bigg)+f(\pi)\bigg).</math>
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 2:  
|-
|Thus, the right-endpoint Riemann sum is
|-
|
::<math>\begin{array}{rcl}
\displaystyle{\frac{\pi}{4}\bigg(\sin\bigg(\frac{\pi}{4}\bigg)+\sin\bigg(\frac{\pi}{2}\bigg)+\sin\bigg(\frac{3\pi}{4}\bigg)+\sin(\pi)\bigg)} & = & \displaystyle{\frac{\pi}{4}\bigg(\frac{\sqrt{2}}{2}+1+\frac{\sqrt{2}}{2}+0\bigg)}\\
&&\\
& = & \displaystyle{\frac{\pi}{4}(\sqrt{2}+1).}\\
\end{array}</math>
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Final Answer:  
|-
|   <math>\frac{\pi}{4}(\sqrt{2}+1)</math>
|}
[[009B_Sample_Midterm_3|'''<u>Return to Sample Exam</u>''']]

@@ Line 1: / Line 1: @@
 <span class="exam">Divide the interval &nbsp;<math style="vertical-align: -5px">[0,\pi]</math>&nbsp; into four subintervals of equal length &nbsp; <math>\frac{\pi}{4}</math> &nbsp; and compute the right-endpoint Riemann sum of &nbsp;<math style="vertical-align: -5px">y=\sin (x).</math>
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
+[[009B Sample Midterm 3, Problem 1 Detailed Solution|'''<u>Detailed Solution</u>''']]
 [[009B_Sample_Midterm_3|'''<u>Return to Sample Exam</u>''']]

← Older revision		Revision as of 22:27, 12 May 2016
Line 1:		Line 1:
		+	[[File:9B_SM3_1_GP.png\|right\|375px]]
		+
	<span class="exam">Divide the interval <math style="vertical-align: -5px">[0,\pi]</math> into four subintervals of equal length <math>\frac{\pi}{4}</math> and compute the right-endpoint Riemann sum of <math style="vertical-align: -5px">y=\sin (x).</math>		<span class="exam">Divide the interval <math style="vertical-align: -5px">[0,\pi]</math> into four subintervals of equal length <math>\frac{\pi}{4}</math> and compute the right-endpoint Riemann sum of <math style="vertical-align: -5px">y=\sin (x).</math>

@@ Line 1: / Line 1: @@
 [[File:9B_SM3_1_GP.png|right|375px]]
-<span class="exam">Divide the interval <math style="vertical-align: -5px">[0,\pi]</math> into four subintervals of equal length <math>\frac{\pi}{4}</math> and compute the right-endpoint Riemann sum of <math style="vertical-align: -5px">y=\sin (x).</math>
+<span class="exam">Divide the interval &nbsp;<math style="vertical-align: -5px">[0,\pi]</math>&nbsp; into four subintervals of equal length &nbsp; <math>\frac{\pi}{4}</math> &nbsp; and compute the right-endpoint Riemann sum of &nbsp;<math style="vertical-align: -5px">y=\sin (x).</math>
@@ Line 7: / Line 7: @@
 !Foundations: &nbsp;
 |-
-||Recall:
 |-
+|&nbsp; &nbsp; &nbsp; &nbsp; is given by choosing the right endpoint of the interval.
 |-
+|
-::'''2.''' See the Riemann sums (insert link) for more information.
+'''2.''' See the Riemann sums (insert link) for more information.
 |}
 '''Solution:'''
@@ Line 20: / Line 21: @@
 !Step 1: &nbsp;
 |-
-|Let <math style="vertical-align: -5px">f(x)=\sin(x).</math> Each interval has length <math>\frac{\pi}{4}.</math>
+|Let &nbsp;<math style="vertical-align: -5px">f(x)=\sin(x).</math>
 |-
-|So, the right-endpoint Riemann sum of <math style="vertical-align: -5px">f(x)</math> on the interval <math style="vertical-align: -5px">[0,\pi]</math> is
+|Each interval has length &nbsp;<math>\frac{\pi}{4}.</math>
 |-
+|
-::<math>\frac{\pi}{4}\bigg(f\bigg(\frac{\pi}{4}\bigg)+f\bigg(\frac{\pi}{2}\bigg)+f\bigg(\frac{3\pi}{4}\bigg)+f(\pi)\bigg).</math>
+&nbsp; &nbsp; &nbsp; &nbsp; <math>\frac{\pi}{4}\bigg(f\bigg(\frac{\pi}{4}\bigg)+f\bigg(\frac{\pi}{2}\bigg)+f\bigg(\frac{3\pi}{4}\bigg)+f(\pi)\bigg).</math>
 |}
@@ Line 34: / Line 37: @@
 |-
+|
-::<math>\begin{array}{rcl}
+&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
 \displaystyle{\frac{\pi}{4}\bigg(\sin\bigg(\frac{\pi}{4}\bigg)+\sin\bigg(\frac{\pi}{2}\bigg)+\sin\bigg(\frac{3\pi}{4}\bigg)+\sin(\pi)\bigg)} & = & \displaystyle{\frac{\pi}{4}\bigg(\frac{\sqrt{2}}{2}+1+\frac{\sqrt{2}}{2}+0\bigg)}\\
 &&\\
@@ Line 40: / Line 43: @@
 \end{array}</math>
 |}
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 !Final Answer: &nbsp;
 |-
-|&nbsp;&nbsp; <math>\frac{\pi}{4}(\sqrt{2}+1)</math>
+|&nbsp;&nbsp; &nbsp; &nbsp; <math>\frac{\pi}{4}(\sqrt{2}+1)</math>
 |}
 [[009B_Sample_Midterm_3|'''<u>Return to Sample Exam</u>''']]

009B Sample Midterm 3, Problem 1 - Revision history

MathAdmin at 01:32, 13 November 2017

MathAdmin at 19:18, 9 April 2017

MathAdmin at 22:27, 12 May 2016

MathAdmin: Created page with "Divide the interval [0,\pi] into four subintervals of equal length \frac{\pi}{4} and compute the righ..."

MathAdmin: Created page with "Divide the interval $[0,\pi]$ into four subintervals of equal length $\frac{\pi}{4}$ and compute the righ..."