009B Sample Final 3, Problem 1 - Revision history

Kayla Murray at 00:39, 3 December 2017

2017-12-03T00:39:43Z

2017-03-01T00:29:23Z

2017-03-01T00:17:44Z

2017-03-01T00:17:27Z

2017-03-01T00:12:47Z

2017-02-28T20:29:01Z

2017-02-19T02:19:36Z

2017-02-18T17:46:36Z

@@ Line 1: / Line 1: @@
 <span class="exam">Divide the interval &nbsp;<math style="vertical-align: -5px">[-1,1]</math>&nbsp; into four subintervals of equal length &nbsp;<math style="vertical-align: -14px">\frac{1}{2}</math>&nbsp; and compute the left-endpoint Riemann sum of &nbsp;<math style="vertical-align: -5px">y=1-x^2.</math>
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
+<hr>
-!Foundations: &nbsp;
+[[009B Sample Final 3, Problem 1 Solution|'''<u>Solution</u>''']]
-'''Solution:'''
+[[009B Sample Final 3, Problem 1 Detailed Solution|'''<u>Detailed Solution</u>''']]
 [[009B_Sample_Final_3|'''<u>Return to Sample Exam</u>''']]

@@ Line 13: / Line 13: @@
 !Step 1: &nbsp;
 |-
+|Since our interval is &nbsp;<math style="vertical-align: -5px">[-1,1]</math>&nbsp; and we are using &nbsp;<math style="vertical-align: -1px">4</math>&nbsp; rectangles, each rectangle has width &nbsp;<math style="vertical-align: -13px">\frac{1}{2}.</math>
 |-
+|Let &nbsp;<math style="vertical-align: -6px">f(x)=1-x^2.</math>
 |-
+|So, the left-endpoint Riemann sum is
 |-
+| &nbsp;&nbsp; &nbsp; &nbsp; <math style="vertical-align: 0px">S=\frac{1}{2}\bigg(f(-1)+f\bigg(-\frac{1}{2}\bigg)+f(0)+f\bigg(\frac{1}{2}\bigg)\bigg).</math>
 |}
@@ Line 25: / Line 25: @@
 !Step 2: &nbsp;
 |-
+|Thus, the left-endpoint Riemann sum is
 |-
+|
-|-
+&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
+\displaystyle{S} & = & \displaystyle{\frac{1}{2}\bigg(0+\frac{3}{4}+1+\frac{3}{4}\bigg)}\\
-|-
+&&\\
+& = & \displaystyle{\frac{5}{4}.}
 |}
@@ Line 38: / Line 39: @@
 !Final Answer: &nbsp;
 |-
+|&nbsp; &nbsp; &nbsp; &nbsp;<math>\frac{5}{4}</math>
 |}
 [[009B_Sample_Final_3|'''<u>Return to Sample Exam</u>''']]

@@ Line 4: / Line 4: @@
 !Foundations: &nbsp;
 |-
-|The height of each rectangle in the left-hand Riemann sum is given by choosing the left endpoint of the interval.
+|The height of each rectangle in the left-endpoint Riemann sum is given by choosing the left endpoint of the interval.
 |}

@@ Line 4: / Line 4: @@
 !Foundations: &nbsp;
 |-
+|The height of each rectangle in the left-hand Riemann sum is given by choosing the left endpoint of the interval.
 |}

@@ Line 15: / Line 15: @@
 '''Solution:'''
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
@@ Line 36: / Line 34: @@
 |-
+|
 |-
+|
@@ Line 82: / Line 44: @@
 !Final Answer: &nbsp;
 |-
-|'''(a)'''
 |}
 [[009B_Sample_Final_3|'''<u>Return to Sample Exam</u>''']]

← Older revision		Revision as of 20:29, 28 February 2017
Line 1:		Line 1:
−	<span class="exam">Divide the interval <math>[-1,1]</math> into four subintervals of equal length <math>\frac{1}{2}</math> and compute the left-endpoint Riemann sum of <math>y=1-x^2.</math>	+	<span class="exam">Divide the interval  <math style="vertical-align: -5px">[-1,1]</math>  into four subintervals of equal length  <math style="vertical-align: -14px">\frac{1}{2}</math>  and compute the left-endpoint Riemann sum of  <math style="vertical-align: -5px">y=1-x^2.</math>

	{\| class="mw-collapsible mw-collapsed" style = "text-align:left;"		{\| class="mw-collapsible mw-collapsed" style = "text-align:left;"

← Older revision		Revision as of 17:46, 18 February 2017
Line 1:		Line 1:
−	<span class="exam">~~Consider~~ the ~~region bounded by the following two functions:~~	+	<span class="exam">Divide the interval <math>[-1,1]</math> into four subintervals of equal length <math>\frac{1}{2}</math> and compute the left-endpoint Riemann sum of <math>y=1-x^2.</math>
−	~~::::::::<span class="exam">~~ <math ~~style="vertical-align: -5px"~~>~~y=2(~~-~~x^2+9)~~</math> ~~and~~ <math ~~style="vertical-align: -4px"~~>~~y=0~~</math>.
−
−	~~<span class="exam">a) Using~~ the ~~lower~~ sum ~~with three rectangles having equal width, approximate the area.~~
−
−	<~~span class~~=~~"exam">b) Using the upper sum with three rectangles having equal width, approximate the area~~.
−
−	<~~span class="exam"~~>~~c) Find the actual area of the region.~~

	{\| class="mw-collapsible mw-collapsed" style = "text-align:left;"		{\| class="mw-collapsible mw-collapsed" style = "text-align:left;"