Difference between revisions of "009B Sample Final 1, Problem 1"

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<span class="exam">Consider the region bounded by the following two functions:
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<span class="exam">Suppose the speed of a bee is given in the table.
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::::::::<span class="exam"> <math style="vertical-align: -5px">y=2(-x^2+9)</math> and <math style="vertical-align: -4px">y=0</math>.
 
  
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<span class="exam">a) Using the lower sum with three rectangles having equal width, approximate the area.
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<table border="1" cellspacing="0" cellpadding="6" align = "center">
 +
  <tr>
 +
    <td align = "center">Time (s)</td>
 +
    <td align = "center">Speed (cm/s)</td>
 +
  </tr>
 +
  <tr>
 +
    <td align = "center"><math>0.0</math></td>
 +
    <td align = "center"><math> 125.0  </math></td>
 +
  </tr>
 +
<tr>
 +
    <td align = "center"><math>2.0</math></td>
 +
    <td align = "center"><math>  118.0</math></td>
 +
  </tr>
 +
<tr>
 +
    <td align = "center"><math>4.0</math></td>
 +
    <td align = "center"><math> 116.0 </math></td>
 +
  </tr>
 +
<tr>
 +
    <td align = "center"><math>6.0</math></td>
 +
    <td align = "center"><math> 112.0 </math></td>
 +
  </tr>
 +
<tr>
 +
    <td align = "center"><math>8.0</math></td>
 +
    <td align = "center"><math> 120.0  </math></td>
 +
  </tr>
 +
<tr>
 +
    <td align = "center"><math>10.0</math></td>
 +
    <td align = "center"><math> 113.0  </math></td>
 +
  </tr>
  
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<span class="exam">b) Using the upper sum with three rectangles having equal width, approximate the area.
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</table>
  
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<span class="exam">c) Find the actual area of the region.
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<span class="exam">(a) Using the given measurements, find the left-hand estimate for the distance the bee moved during this experiment.
 +
 
 +
<span class="exam">(b) Using the given measurements, find the midpoint estimate for the distance the bee moved during this experiment.
  
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Foundations: &nbsp;  
 
!Foundations: &nbsp;  
 
|-
 
|-
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|Recall:
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|'''1.''' The height of each rectangle in the left-hand Riemann sum is given by choosing
 
|-
 
|-
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|'''1.''' The height of each rectangle in the lower Riemann sum is given by choosing the minimum <math style="vertical-align: -5px">y</math> value of the left and right endpoints of the rectangle.
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|&nbsp; &nbsp; &nbsp; &nbsp; the left endpoints of each interval.
 
|-
 
|-
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|'''2.''' The height of each rectangle in the upper Riemann sum is given by choosing the maximum <math style="vertical-align: -5px">y</math> value of the left and right endpoints of the rectangle.
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|'''3.''' The height of each rectangle in the midpoint Riemann sum is given by  
 
|-
 
|-
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|'''3.''' The area of the region is given by <math style="vertical-align: -14px">\int_a^b y~dx</math> for appropriate values <math style="vertical-align: -4px">a,b</math>.
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|&nbsp; &nbsp; &nbsp; &nbsp; <math style="vertical-align: -14px">\frac{f(a)+f(b)}{2}</math>&nbsp; where &nbsp;<math>a</math>&nbsp; is the left endpoint of the interval and &nbsp;<math style="vertical-align: -1px">b</math>&nbsp; is the right endpoint of the interval.
 
|}
 
|}
 +
  
 
'''Solution:'''
 
'''Solution:'''
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!Step 1: &nbsp;  
 
!Step 1: &nbsp;  
 
|-
 
|-
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|We need to set these two equations equal in order to find the intersection points of these functions.
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|To estimate the distance the bee moved during this experiment,
 
|-
 
|-
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|So, we let <math style="vertical-align: -5px">2(-x^2+9)=0</math>. Solving for <math style="vertical-align: 0px">x</math>, we get <math style="vertical-align: 0px">x=\pm 3</math>.
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|we need to calculate the left-hand Riemann sum over the interval &nbsp;<math style="vertical-align: -5px">[0,10].</math>
 
|-
 
|-
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|This means that we need to calculate the Riemann sums over the interval <math style="vertical-align: -5px">[-3,3]</math>.
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|Based on the information given in the table, we will have &nbsp;<math style="vertical-align: 0px">5</math>&nbsp; rectangles and
 
|-
 
|-
−
|
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|each rectangle will have width &nbsp;<math style="vertical-align: 0px">2.</math>
 
|}
 
|}
  
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!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
|-
 
|-
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|Since the length of our interval is <math style="vertical-align: 0px">6</math> and we are using <math style="vertical-align: 0px">3</math> rectangles,
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|Let &nbsp;<math style="vertical-align: -5px">s(t)</math>&nbsp; be the speed of the bee during the experiment.
 
|-
 
|-
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|each rectangle will have width <math style="vertical-align: 0px">2</math>&thinsp;.
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|Then, the left-hand Riemann sum is  
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|-
 
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|Thus, the lower Riemann sum is
 
 
|-
 
|-
 
|
 
|
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::<math>2(f(-3)+f(-1)+f(3))\,=\,2(0+16+0)\,=\,32.</math>
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&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
 +
\displaystyle{2(s(0)+s(2)+s(4)+s(6)+s(8))} & = & \displaystyle{2(125+118+116+112+120)}\\
 +
&&\\
 +
& = & \displaystyle{1182\text{ cm}.}
 +
\end{array}</math>
 
|}
 
|}
  
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!Step 1: &nbsp;  
 
!Step 1: &nbsp;  
 
|-
 
|-
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|As in Part (a), the length of our inteval is <math style="vertical-align: 0px">6</math> and
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|To estimate the distance the bee moved during this experiment,
 
|-
 
|-
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|each rectangle will have width <math style="vertical-align: 0px">2</math>. (See Step 1 and 2 for '''(a)''')
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|we need to calculate the Riemann sum using the midpoint rule over the interval &nbsp;<math style="vertical-align: -5px">[0,10].</math>
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|}
 
−
 
 
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
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!Step 2: &nbsp;
 
 
|-
 
|-
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|Thus, the upper Riemann sum is
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|Based on the information given in the table, we will have &nbsp;<math style="vertical-align: 0px">5</math>&nbsp; rectangles and
 
|-
 
|-
−
|
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|each rectangle will have width &nbsp;<math style="vertical-align: 0px">2.</math>
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::<math>2(f(-1)+f(-1)+f(1))\,=\,2(16+16+16)\,=\,96.</math>
 
 
|}
 
|}
−
 
−
'''(c)'''
 
  
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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!Step 1: &nbsp;  
+
!Step 2: &nbsp;
 
|-
 
|-
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|To find the actual area of the region, we need to calculate
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|Let &nbsp;<math style="vertical-align: -5px">s(t)</math>&nbsp; be the speed of the bee during the experiment.
 
|-
 
|-
−
|
+
|Then, the Riemann sum using the midpoint rule is
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::<math>\int_{-3}^3 2(-x^2+9)~dx.</math>
 
−
|}
 
−
 
 
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
−
!Step 2: &nbsp;
 
−
|-
 
−
|We integrate to get
 
 
|-
 
|-
 
|
 
|
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::<math>\begin{array}{rcl}
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&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
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\displaystyle{\int_{-3}^3 2(-x^2+9)~dx} & = & \displaystyle{2\bigg(\frac{-x^3}{3}+9x\bigg)\bigg|_{-3}^3}\\
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\displaystyle{2\bigg(\frac{s(0)+s(2)}{2}+\frac{s(2)+s(4)}{2}+\frac{s(4)+s(6)}{2}+\frac{s(6)+s(8)}{2}+\frac{s(8)+s(10)}{2}\bigg)} & = & \displaystyle{1170\text{ cm}.}
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&&\\
 
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& = & \displaystyle{2\bigg(\frac{-3^3}{3}+9\times 3\bigg)-2\bigg(\frac{-(-3)^3}{3}+9(-3)\bigg)}\\
 
−
&&\\
 
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& = & \displaystyle{2(-9+27)-2(9-27)}\\
 
−
&&\\
 
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& = & \displaystyle{2(18)-2(-18)}\\
 
−
&&\\
 
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& = & \displaystyle{72}.\\
 
 
\end{array}</math>
 
\end{array}</math>
 
|}
 
|}
 +
  
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Final Answer: &nbsp;  
 
!Final Answer: &nbsp;  
 
|-
 
|-
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|'''(a)''' &nbsp;<math style="vertical-align: 0px">32</math>
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|&nbsp; &nbsp; '''(a)''' &nbsp; &nbsp;<math style="vertical-align: 0px">1182\text{ cm}</math>
 
|-
 
|-
−
|'''(b)''' &nbsp;<math style="vertical-align: 0px">96</math>
+
|&nbsp; &nbsp; '''(b)''' &nbsp; &nbsp;<math style="vertical-align: 0px">1170\text{ cm}</math>
−
|-
 
−
|'''(c)''' &nbsp;<math style="vertical-align: 0px">72</math>
 
 
|}
 
|}
 
[[009B_Sample_Final_1|'''<u>Return to Sample Exam</u>''']]
 
[[009B_Sample_Final_1|'''<u>Return to Sample Exam</u>''']]
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−
[[File:9BF1 1 GP.png|center|1000px]]
 

Latest revision as of 08:40, 10 April 2017

Suppose the speed of a bee is given in the table.

Time (s) Speed (cm/s)

(a) Using the given measurements, find the left-hand estimate for the distance the bee moved during this experiment.

(b) Using the given measurements, find the midpoint estimate for the distance the bee moved during this experiment.

Foundations:  
1. The height of each rectangle in the left-hand Riemann sum is given by choosing
        the left endpoints of each interval.
3. The height of each rectangle in the midpoint Riemann sum is given by
          where    is the left endpoint of the interval and    is the right endpoint of the interval.


Solution:

(a)

Step 1:  
To estimate the distance the bee moved during this experiment,
we need to calculate the left-hand Riemann sum over the interval  
Based on the information given in the table, we will have    rectangles and
each rectangle will have width  
Step 2:  
Let    be the speed of the bee during the experiment.
Then, the left-hand Riemann sum is

       

(b)

Step 1:  
To estimate the distance the bee moved during this experiment,
we need to calculate the Riemann sum using the midpoint rule over the interval  
Based on the information given in the table, we will have    rectangles and
each rectangle will have width  
Step 2:  
Let    be the speed of the bee during the experiment.
Then, the Riemann sum using the midpoint rule is

       


Final Answer:  
    (a)    
    (b)    

Return to Sample Exam

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