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

From Grad Wiki
Jump to navigation Jump to search
 
(21 intermediate revisions by 2 users not shown)
Line 1: Line 1:
−
<span class="exam">Consider the region bounded by the following two functions:
+
<span class="exam">Suppose the speed of a bee is given in the table.
−
::::::::<span class="exam"> <math>y=2(-x^2+9)</math> and <math>y=0</math>
 
  
−
<span class="exam">a) Using the lower sum with three rectangles having equal width , approximate the area.
+
<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>
  
−
<span class="exam">b) Using the upper sum with three rectangles having equal width, approximate the area.
+
</table>
  
−
<span class="exam">c) Find the actual area of the region.
+
<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;  
 
|-
 
|-
−
|Link to Riemann sums page
+
|'''1.''' The height of each rectangle in the left-hand Riemann sum is given by choosing
 +
|-
 +
|&nbsp; &nbsp; &nbsp; &nbsp; the left endpoints of each interval.
 +
|-
 +
|'''3.''' The height of each rectangle in the midpoint Riemann sum is given by
 +
|-
 +
|&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:'''
Line 21: Line 57:
 
!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 <math>2(-x^2+9)=0</math>. Solving for <math>x</math>, we get <math>x=-3,3</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 <math>[-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>
 
|}
 
|}
  
Line 33: Line 69:
 
!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
|-
 
|-
−
|Since the length of our interval is <math>6</math> and we are using <math>3</math> rectangles,
+
|Let &nbsp;<math style="vertical-align: -5px">s(t)</math>&nbsp; be the speed of the bee during the experiment.
 
|-
 
|-
−
|each rectangle will have width <math>2</math>.
+
|Then, the left-hand Riemann sum is
 
|-
 
|-
−
|Thus, the lower Riemann sum is
+
|
−
|-
+
&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
−
|<math>2(f(-3)+f(-1)+f(3))=2(0+16+0)=32</math>.
+
\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>
 
|}
 
|}
  
Line 47: Line 86:
 
!Step 1: &nbsp;  
 
!Step 1: &nbsp;  
 
|-
 
|-
−
|As in Part (a), the length of our inteval is <math>6</math> and
+
|To estimate the distance the bee moved during this experiment,
−
|-
 
−
|each rectangle will have width <math>2</math>. (See Step 1 and 2 for part (a))
 
−
|}
 
−
 
 
−
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
−
!Step 2: &nbsp;
 
−
|-
 
−
|Thus, the upper Riemann sum is
 
 
|-
 
|-
−
|<math>2(f(-1)+f(-1)+f(1))=2(16+16+16)=96</math>
+
|we need to calculate the Riemann sum using the midpoint rule over the interval &nbsp;<math style="vertical-align: -5px">[0,10].</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
 
|-
 
|-
−
|<math>\int_{-3}^3 2(-x^2+9)~dx</math>
+
|each rectangle will have width &nbsp;<math style="vertical-align: 0px">2.</math>
 
|}
 
|}
  
Line 73: Line 98:
 
!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
|-
 
|-
−
|We integrate to get
+
|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
 
|-
 
|-
 
|
 
|
−
::<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>
 
|}
 
|}
 +
  
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Final Answer: &nbsp;  
 
!Final Answer: &nbsp;  
 
|-
 
|-
−
|'''(a)''' <math>64</math>
+
|&nbsp; &nbsp; '''(a)''' &nbsp; &nbsp;<math style="vertical-align: 0px">1182\text{ cm}</math>
−
|-
 
−
|'''(b)''' <math>64</math>
 
 
|-
 
|-
−
|'''(c)''' <math>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>''']]

Latest revision as of 09:45, 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 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

Retrieved from "https://gradwiki.math.ucr.edu/index.php?title=009B_Sample_Final_1,_Problem_1&oldid=2886"