009B Sample Midterm 3, Problem 1 - Revision history

Kayla Murray at 01:27, 13 November 2017

2017-11-13T01:27:49Z

Kayla Murray at 01:22, 5 November 2017

2017-11-05T01:22:12Z

Kayla Murray at 01:33, 27 February 2017

2017-02-27T01:33:56Z

Kayla Murray at 00:13, 19 February 2017

2017-02-19T00:13:56Z

Kayla Murray at 01:29, 8 February 2017

2017-02-08T01:29:49Z

Kayla Murray at 17:09, 7 February 2017

2017-02-07T17:09:18Z

Kayla Murray at 16:45, 6 February 2017

2017-02-06T16:45:33Z

Kayla Murray at 16:45, 6 February 2017

2017-02-06T16:45:09Z

Kayla Murray at 16:44, 6 February 2017

2017-02-06T16:44:34Z

Kayla Murray at 00:54, 19 April 2016

2016-04-19T00:54:23Z

@@ 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>
 <hr>
-(insert picture of handwritten solution)
+[[009B Sample Midterm 3, Problem 1 Solution|'''<u>Solution</u>''']]
 [[009B_Sample_Midterm_3|'''<u>Return to Sample Exam</u>''']]

← Older revision		Revision as of 01:22, 5 November 2017
Line 1:		Line 1:
	<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>
		+	<hr>
		+	(insert picture of handwritten solution)

		+	[[009B Sample Midterm 3, Problem 1 Detailed Solution\|'''<u>Detailed Solution for this Problem</u>''']]

−	~~{\| class="mw-collapsible mw-collapsed" style = "text-align:left;"~~
−	~~!Foundations:  ~~
−	\|-
−	\|
−	~~'''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>~~
−	\|-
−	~~\|Therefore, 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>''']]		[[009B_Sample_Midterm_3\|'''<u>Return to Sample Exam</u>''']]

@@ Line 1: / Line 1: @@
-<span class="exam">Divide the interval <math style="vertical-align: -5px">[0,\pi]</math> into four subintervals of equal length &nbsp; <math>\frac{\pi}{4}</math> &nbsp; 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 19: / Line 19: @@
 !Step 1: &nbsp;
 |-
-|Let <math style="vertical-align: -5px">f(x)=\sin(x).</math>
+|Let &nbsp;<math style="vertical-align: -5px">f(x)=\sin(x).</math>
 |-
-|Each interval has length <math>\frac{\pi}{4}.</math>
+|Each interval has length &nbsp;<math>\frac{\pi}{4}.</math>
 |-
-|Therefore, 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
+|Therefore, the right-endpoint Riemann sum of &nbsp;<math style="vertical-align: -5px">f(x)</math>&nbsp; on the interval &nbsp;<math style="vertical-align: -5px">[0,\pi]</math>&nbsp; is
 |-
+|

← Older revision		Revision as of 00:13, 19 February 2017
Line 1:		Line 1:
−	<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 4: / Line 4: @@
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 !Foundations: &nbsp;
 |-
+|
 '''1.''' The height of each rectangle in the right-hand Riemann sum
 |-
-|&nbsp; &nbsp; is given by choosing the right endpoint of the interval.
+|&nbsp; &nbsp; &nbsp; &nbsp; is given by choosing the right endpoint of the interval.
 |-
+|
@@ Line 28: / Line 26: @@
 |-
+|
-&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>
+&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 37: / Line 35: @@
 |-
+|
-&nbsp; &nbsp;<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 48: / Line 46: @@
 !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>''']]

@@ Line 8: / Line 8: @@
 |-
+|
-'''1.''' The height of each rectangle in the right-hand Riemann sum is given by choosing the right endpoint of the interval.
+'''1.''' The height of each rectangle in the right-hand Riemann sum
 |-
+|
@@ Line 19: / 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 <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
+|Therefore, 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
 |-
+|

@@ Line 8: / Line 8: @@
 |-
+|
-::'''1.''' The height of each rectangle in the right-hand Riemann sum is given by choosing the right endpoint of the interval.
+'''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.
+'''2.''' See the Riemann sums (insert link) for more information.
 |}

@@ Line 13: / Line 13: @@
 ::'''2.''' See the Riemann sums (insert link) for more information.
 |}
 '''Solution:'''
@@ Line 23: / Line 24: @@
 |-
+|
-::<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;<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 38: / Line 39: @@
 \end{array}</math>
 |}
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"

@@ Line 7: / Line 7: @@
 ||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.
 |}
@@ Line 16: / Line 18: @@
 !Step 1: &nbsp;
 |-
-|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
+|Let <math style="vertical-align: -5px">f(x)=\sin(x).</math> Each interval has length <math>\frac{\pi}{4}.</math>
 |-
+|
@@ Line 28: / Line 32: @@
 |-
+|
-::<math>\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)=\frac{\pi}{4}\bigg(\frac{\sqrt{2}}{2}+1+\frac{\sqrt{2}}{2}+0\bigg)=\frac{\pi}{4}(\sqrt{2}+1).</math>
+::<math>\begin{array}{rcl}
 |}
@@ Line 34: / Line 42: @@
 !Final Answer: &nbsp;
 |-
-|<math>\frac{\pi}{4}(\sqrt{2}+1)</math>
+|&nbsp;&nbsp; <math>\frac{\pi}{4}(\sqrt{2}+1)</math>
 |}
 [[009B_Sample_Midterm_3|'''<u>Return to Sample Exam</u>''']]