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

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!Foundations:    
 
!Foundations:    
 
|-
 
|-
−
|Recall:
+
|'''1.''' The height of each rectangle in the left-hand Riemann sum is given by choosing
 
|-
 
|-
−
|'''1.''' The height of each rectangle in the lower Riemann sum is given by choosing
+
|        the left endpoints of each interval.
 
|-
 
|-
−
|&nbsp; &nbsp; &nbsp; &nbsp; the minimum &nbsp;<math style="vertical-align: -5px">y</math>&nbsp; value of the left and right endpoints of the rectangle.
+
|'''3.''' The height of each rectangle in the midpoint Riemann sum is given by
 
|-
 
|-
−
|'''2.''' The height of each rectangle in the upper Riemann sum is given by choosing
+
|&nbsp; &nbsp; &nbsp; &nbsp; <math style="vertical-align: -5px">\frac{f(a)+f(b)}{2}</math>&nbsp; where <math>a</math> is the left endpoint of the interval and <math>b</math> is the right endpoint of the interval.
−
|-
 
−
|&nbsp; &nbsp; &nbsp; &nbsp; the maximum &nbsp;<math style="vertical-align: -5px">y</math>&nbsp; value of the left and right endpoints of the rectangle.
 
−
|-
 
−
|'''3.''' The area of the region is given by &nbsp;<math style="vertical-align: -14px">\int_a^b y~dx</math>  
 
−
|-
 
−
|&nbsp; &nbsp; &nbsp; &nbsp; for appropriate values &nbsp;<math style="vertical-align: -4px">a,b</math>.
 
 
|}
 
|}
  
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!Step 1: &nbsp;  
 
!Step 1: &nbsp;  
 
|-
 
|-
−
|We need to set these two equations equal in order to find the intersection points of these functions.
+
|To estimate the distance the bee moved during this experiment,
 
|-
 
|-
−
|So, we let &nbsp;<math style="vertical-align: -5px">2(-x^2+9)=0</math>.&nbsp; Solving for &nbsp;<math style="vertical-align: -4px">x,</math>&nbsp; we get &nbsp;<math style="vertical-align: 0px">x=\pm 3</math>.
+
|we need to calculate the left-hand Riemann sum over the interval &nbsp;<math style="vertical-align: -5px">[0,10].</math>
 
|-
 
|-
−
|This means that we need to calculate the Riemann sums over the interval &nbsp;<math style="vertical-align: -5px">[-3,3]</math>.
+
|Based on the information given in the table, we will have &nbsp;<math style="vertical-align: 0px">5</math>&nbsp; rectangles and
 
|-
 
|-
−
|
+
|each rectangle will have width &nbsp;<math style="vertical-align: 0px">2.</math>
 
|}
 
|}
  
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!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
|-
 
|-
−
|Since the length of our interval is &nbsp;<math style="vertical-align: 0px">6</math>&nbsp; and we are using &nbsp;<math style="vertical-align: 0px">3</math>&nbsp; rectangles,
+
|Let <math>s(t)</math> be the speed of the bee during the experiement.
 
|-
 
|-
−
|each rectangle will have width &nbsp;<math style="vertical-align: 0px">2.</math>
+
|Then, the left-hand Riemann sum is  
−
|-
 
−
|Thus, the lower Riemann sum is
 
 
|-
 
|-
 
|
 
|
−
::<math>2(f(-3)+f(-1)+f(3))\,=\,2(0+16+0)\,=\,32.</math>
+
&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;  
 
|-
 
|-
−
|As in Part (a), the length of our interval is &nbsp;<math style="vertical-align: 0px">6</math>&nbsp; and
+
|To estimate the distance the bee moved during this experiment,
−
|-
 
−
|each rectangle will have width &nbsp;<math style="vertical-align: 0px">2.</math> (See Step 1 and 2 for (a))
 
−
|}
 
−
 
 
−
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
−
!Step 2: &nbsp;
 
−
|-
 
−
|Thus, the upper Riemann sum is
 
 
|-
 
|-
−
|
+
|we need to calculate the Riemann sum using the midpoint rule over the interval &nbsp;<math style="vertical-align: -5px">[0,10].</math>
−
::<math>2(f(-1)+f(-1)+f(1))\,=\,2(16+16+16)\,=\,96.</math>
 
−
|}
 
−
 
 
−
'''(c)'''
 
−
 
 
−
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
−
!Step 1: &nbsp;
 
 
|-
 
|-
−
|To find the actual area of the region, we need to calculate
+
|Based on the information given in the table, we will have &nbsp;<math style="vertical-align: 0px">5</math>&nbsp; rectangles and
 
|-
 
|-
−
|
+
|each rectangle will have width &nbsp;<math style="vertical-align: 0px">2.</math>
−
::<math>\int_{-3}^3 2(-x^2+9)~dx.</math>
 
 
|}
 
|}
  
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!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
|-
 
|-
−
|We integrate to get
+
|Let <math>s(t)</math> be the speed of the bee during the experiement.
 +
|-
 +
|Then, the Riemann sum using the midpoint rule is
 
|-
 
|-
 
|
 
|
−
::<math>\begin{array}{rcl}
+
&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
−
\displaystyle{\int_{-3}^3 2(-x^2+9)~dx} & = & \displaystyle{2\bigg(\frac{-x^3}{3}+9x\bigg)\bigg|_{-3}^3}\\
+
\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}.}
−
&&\\
 
−
& = & \displaystyle{2\bigg(\frac{-3^3}{3}+9\times 3\bigg)-2\bigg(\frac{-(-3)^3}{3}+9(-3)\bigg)}\\
 
−
&&\\
 
−
& = & \displaystyle{2(-9+27)-2(9-27)}\\
 
−
&&\\
 
−
& = & \displaystyle{2(18)-2(-18)}\\
 
−
&&\\
 
−
& = & \displaystyle{72}.\\
 
 
\end{array}</math>
 
\end{array}</math>
 
|}
 
|}
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!Final Answer: &nbsp;  
 
!Final Answer: &nbsp;  
 
|-
 
|-
−
|&nbsp; &nbsp; '''(a)''' &nbsp; &nbsp;<math style="vertical-align: 0px">32</math>
+
|&nbsp; &nbsp; '''(a)''' &nbsp; &nbsp;<math style="vertical-align: 0px">1182\text{ cm}</math>
−
|-
 
−
|&nbsp; &nbsp; '''(b)''' &nbsp; &nbsp;<math style="vertical-align: 0px">96</math>
 
 
|-
 
|-
−
|&nbsp; &nbsp; '''(c)''' &nbsp; &nbsp;<math style="vertical-align: 0px">72</math>
+
|&nbsp; &nbsp; '''(b)''' &nbsp; &nbsp;<math style="vertical-align: 0px">1170\text{ cm}</math>
 
|}
 
|}
 
[[009B_Sample_Final_1|'''<u>Return to Sample Exam</u>''']]
 
[[009B_Sample_Final_1|'''<u>Return to Sample Exam</u>''']]

Revision as of 09:42, 28 February 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 experiement.
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 experiement.
Then, the Riemann sum using the midpoint rule is

       


Final Answer:  
    (a)    
    (b)    

Return to Sample Exam

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