Difference between revisions of "009B Sample Midterm 3, Problem 1"

From Grad Wiki
Jump to navigation Jump to search
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>
 
<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 Detailed Solution|'''<u>Detailed Solution for this Problem</u>''']]
  
−
{| 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; &nbsp; &nbsp; 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: &nbsp;
 
−
|-
 
−
|Let &nbsp;<math style="vertical-align: -5px">f(x)=\sin(x).</math>
 
−
|-
 
−
|Each interval has length &nbsp;<math>\frac{\pi}{4}.</math>
 
−
|-
 
−
|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
 
−
|-
 
−
|
 
−
&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>
 
−
|}
 
−
 
−
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
−
!Step 2: &nbsp;
 
−
|-
 
−
|Thus, the right-endpoint Riemann sum is
 
−
|-
 
−
|
 
−
&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)}\\
 
−
&&\\
 
−
& = & \displaystyle{\frac{\pi}{4}(\sqrt{2}+1).}\\
 
−
\end{array}</math>
 
−
|}
 
−
 
−
 
−
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
−
!Final Answer: &nbsp;
 
−
|-
 
−
|&nbsp;&nbsp; &nbsp; &nbsp; <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>''']]

Revision as of 17:22, 4 November 2017

Divide the interval  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [0,\pi]}   into four subintervals of equal length   Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {\pi }{4}}}   and compute the right-endpoint Riemann sum of  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=\sin (x).}

(insert picture of handwritten solution)

Detailed Solution for this Problem

Return to Sample Exam

Retrieved from "https://gradwiki.math.ucr.edu/index.php?title=009B_Sample_Midterm_3,_Problem_1&oldid=3858"