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		font face="Verdana" size="6">Holistic Numerical Methods Institute/font>/b>/p>
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		font face="Verdana" size="6">committed to bringing numerical methods to 
		undergraduates/font>/p>
		p align="left">b>font face="Verdana" size="6">Multiple Choice Test/font>/b>/p>
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  		font face="Verdana" size="6" Language="VbScript">
		p align="left">b>Runge-Kutta 2sup>nd/sup> Order Method/b>/p>
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		font face="Georgia" size="7">b>Q1./b>i> /i>To solve the ordinary differential 
		equation /font>/p>
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		&nbsp;/p>
		p class="MsoNormal" style="  margin-top: 0; margin-bottom: 0" align="left">
		font face="Georgia" size="7">by the Runge-Kutta 2sup>nd/sup> order 
		method, you need to rewrite the equation as/font>/p>
		p style="word-spacing: 0; text-indent: 0; margin: 0; line-height:120%" align="left">
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		&nbsp;/p>
		center>center>center>hr align="left">/center>/center>/center>
		p class="MsoNormal" style="  margin-top: 0; margin-bottom: 0" align="left">
		font face="Georgia" size="7">b>Q2. /b>Given /font>/p>
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		font face="Georgia" size="7">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
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		font face="Georgia" size="7">and 
		using a step size of i>h/i>=0.3/font>!--[if gte mso 9]>![endif]-->font face="Georgia" size="7">, 
		the value of i>y/i>(0.9)/font>!--[if gte mso 9]>![endif]-->font face="Georgia" size="7">using 
		the Runge-Kutta 2sup>nd/sup> order Heun's method is most nearly /font>
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		font size="7" face="Georgia">
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		font size="7" face="Georgia">
		input type="radio" style="width: 60; height: 60" name="Q2" value="V1" Language="VbScript">-0.21336/font>font size="6" face="Georgia"> 
		x
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		10sup>14/sup>br>
		/font>font size="7" face="Georgia">
		input type="radio" style="width: 60; height: 60" name="Q2" value="V1" Language="VbScript">-0.24489/font>font size="6" face="Georgia"> 
		x
		/font>font size="7" face="Georgia">
		10sup>14/sup>br>
		/font>/p>center>center>center>center>center>hr align="left">/center>
		/center>/center>/center>/center>
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		font face="Georgia" size="7">b>Q3. /b>Given /font>/p>
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		font face="Georgia" size="7">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
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		/font>/p>
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		font face="Georgia" size="7">and using a step size of i>h/i>=0.3/font>!--[if gte mso 9]>![endif]-->font face="Georgia" size="7">, 
		the best estimate ofi> dy/dx/i>(0.9) using the Runge-Kutta 2nd order midpoint-method most nearly is/font>font size="7">/font>/font>/p>
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		/font>font size="7" face="Georgia">
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		/font>font face="Times New Roman">font size="7" face="Georgia">
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		/font>font size="7" face="Georgia">
		input type="radio" style="width: 60; height: 60" name="Q3" value="V1" Language="VbScript">-3.2045/font>/font>/p>
		center>center>center>center>center>center>hr align="left">/center>
		/center>/center>/center>/center>
		p class="MsoNormal" style="  margin-top: 0; margin-bottom: 0" align="left">
		font face="Georgia" size="7">b>Q4/b>. The velocity (m/s) of a body is given as a 
		function of time (seconds) by /font>/p>
		center>
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		font face="Georgia" size="7">&nbsp;span style="position: relative; top: 5.0pt">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; !--[if gte vml 1]>v:shape
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		center>
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		font face="Georgia" size="7">Using the Runge-Kutta 2sup>nd/sup> order 
		Ralston method with a step size of 5 seconds, the distance in meters traveled by 
		the body from i>t/i>=2/font>!--[if gte mso 9]>![endif]-->font face="Georgia" size="7">&nbsp;to
		i>t/i>=12/font>!--[if gte mso 9]>![endif]-->font face="Georgia" size="7">&nbsp;seconds 
		is estimated most nearly is/font>/p>center>
		center>
		center>
		center>
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		font size="7" face="Georgia">
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		font size="7" face="Georgia"> !--[if gte mso 9]>xml>
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		font face="Georgia" size="7">b>Q5. /b>Theb> /b>Runge-Kutta 2sup>nd/sup> order method 
		can be derived by using the first three terms of the Taylor series of 
		writing the value of i>ysub>i/sub>/i>sub>+1/sub>/font>!--[if gte mso 9]>![endif]-->font face="Georgia" size="7">&nbsp;(that 
		is the value of i>y/i> /font>!--[if gte mso 9]>![endif]-->
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		i>xsub>i/sub>/i>sub>+1/sub>/font>!--[if gte mso 9]>![endif]-->font face="Georgia" size="7">&nbsp;/font>!--[if gte mso 9]>![endif]-->font face="Georgia" size="7">) 
		in terms of i>ysub>i/sub>/i>/font>!--[if gte mso 9]>![endif]-->font face="Georgia" size="7">&nbsp;/font>!--[if gte mso 9]>![endif]-->font face="Georgia" size="7">&nbsp;(that 
		is the value of span style="position: relative; top: 5.0pt">i>y/i> /span>at
		span style="position: relative; top: 6.0pt">i>xsub>i/sub>/i>/span>/font>!--[if gte mso 9]>![endif]-->font face="Georgia" size="7">) 
		and all the derivatives of i>y /i>/font>!--[if gte mso 9]>![endif]-->
		font face="Georgia" size="7">at
		i>xsub>i/sub>/i>/font>!--[if gte mso 9]>![endif]-->font face="Georgia" size="7">&nbsp;/font>!--[if gte mso 9]>![endif]-->font face="Georgia" size="7">.&nbsp; 
		If i>h=xsub>i/sub>/i>sub>+1/sub>/font>!--[if gte mso 9]>![endif]-->font face="Georgia" size="7">-i>xsub>i/sub>/i>/font>!--[if gte mso 9]>![endif]-->font face="Georgia" size="7">, 
		the explicit expression for i>ysub>i/sub>/i>sub>+1/sub>/font>!--[if gte mso 9]>![endif]-->font face="Georgia" size="7">&nbsp;/font>!--[if gte mso 9]>![endif]-->font face="Georgia" size="7">if 
		the first three terms of the Taylor series are chosen for solving the 
		ordinary differential equation /font>/p>
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		font face="Georgia" size="7">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
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		font face="Georgia" size="7">would be/font>/p>
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		font size="7" face="Georgia">
		input type="radio" style="width: 60; height: 60" name="Q5" value="V1" Language="VbScript">/font>span style="position: relative; top: 12.0pt">font size="7" face="Georgia">!--[if gte vml 1]>v:shape
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		font size="7" face="Georgia">
		input type="radio" style="width: 60; height: 60" name="Q5" value="V1" Language="VbScript">/font>span style="position: relative; top: 12.0pt">font size="7" face="Georgia">!--[if gte vml 1]>v:shape
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src="runge_2nd_method_mobile_files/image022.gif" v:shapes="_x0000_s1033">![endif]>/font>/span>p align="left" style="word-spacing: 0; text-indent: 0; margin: 0; line-height:120%">
		font size="7" face="Georgia">
		input type="radio" style="width: 60; height: 60" name="Q5" value="V1" Language="VbScript">/font>span style="position: relative; top: 12.0pt">font size="7" face="Georgia">!--[if gte vml 1]>v:shape
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 v:imagedata src="runge_2nd_method_mobile_files/image023.wmz" o:title=""/>
/v:shape>![endif]-->![if !vml]>img border=0 width=408 height=69
src="runge_2nd_method_mobile_files/image024.gif" v:shapes="_x0000_s1032">![endif]>/font>/span>p align="left" style="word-spacing: 0; text-indent: 0; margin: 0; line-height:120%">
		font size="7" face="Georgia">
		input type="radio" style="width: 60; height: 60" name="Q5" value="V1" Language="VbScript">/font>span style="font-family: Georgia">span style="position: relative; top: 12.0pt">font size="7">!--[if gte vml 1]>v:shape
 id="_x0000_s1031" type="#_x0000_t75" style='width:307.5pt;height:48pt'>
 v:imagedata src="runge_2nd_method_mobile_files/image025.wmz" o:title=""/>
/v:shape>![endif]-->![if !vml]>img border=0 width=410 height=64
src="runge_2nd_method_mobile_files/image026.gif" v:shapes="_x0000_s1031">![endif]>/font>/span>/span>p align="left" style="word-spacing: 0; text-indent: 0; margin: 0; line-height:120%">
		&nbsp;hr align="left">
		p class="MsoNormal" style="  margin-top: 0; margin-bottom: 0" align="left">
		font face="Georgia" size="7">b>Q6. /b>A spherical ball is taken out of a furnace 
		at 1200K and is allowed to cool in air. Given the following,/font>/p>
		p class="MsoNormal" style="  margin-top: 0; margin-bottom: 0" align="left">
		font face="Georgia" size="7">radius of&nbsp; ball = 2 cm/font>/p>
		p class="MsoNormal" style="  margin-top: 0; margin-bottom: 0" align="left">
		font face="Georgia" size="7">specific heat of ball = 420
		J/(kg-K)/font>/p>
		p class="MsoNormal" style="  margin-top: 0; margin-bottom: 0" align="left">
		font face="Georgia" size="7">density of ball = 7800
		!--[if gte mso 9]>kg/m^3![endif]-->/font>/p>
		p class="MsoNormal" style="  margin-top: 0; margin-bottom: 0" align="left">
		font face="Georgia" size="7">convection coefficient = 350
		J/s-m^2-K/font>!--[if gte mso 9]>![endif]-->/p>
		p class="MsoNormal" style="  margin-top: 0; margin-bottom: 0" align="left">
		font size="7" face="Georgia">The ordinary differential equation is 
		given for the temperature, /font>
		span style="font-family: Georgia">
		span style="position: relative; top: 3.0pt">font size="7">!--[if gte vml 1]>v:shape
 id="_x0000_s1030" type="#_x0000_t75" style='width:21pt;height:30.75pt'>
 v:imagedata src="runge_2nd_method_mobile_files/image027.wmz" o:title=""/>
/v:shape>![endif]-->![if !vml]>img border=0 width=28 height=41
src="runge_2nd_method_mobile_files/image028.gif" v:shapes="_x0000_s1030">![endif]>/font>/span>/span>font size="7" face="Georgia"> 
		of the ball/font>/p>
		p class="MsoNormal" style="text-indent: .5in;   margin-top: 0; margin-bottom: 0" align="left">
		font face="Georgia" size="7">
		span style="position: relative; top: 12.0pt">!--[if gte vml 1]>v:shape
 id="_x0000_s1029" type="#_x0000_t75" style='width:296.25pt;height:51pt'>
 v:imagedata src="runge_2nd_method_mobile_files/image029.wmz" o:title=""/>
/v:shape>![endif]-->![if !vml]>img border=0 width=395 height=68
src="runge_2nd_method_mobile_files/image030.gif" v:shapes="_x0000_s1029">![endif]>/span>/font>!--[if gte mso 9]>![endif]-->/p>
		p class="MsoNormal" style="text-indent: .5in;   margin-top: 0; margin-bottom: 0" align="left">
		!--[if gte mso 9]>![endif]-->/p>
		p class="MsoNormal" style="  margin-top: 0; margin-bottom: 0" align="left">
		font face="Georgia" size="7">if only radiation is accounted for.&nbsp; The 
		ordinary differential equation if convection is accounted for in 
		addition to radiation is&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; /font>/p>
		p style="word-spacing: 0; text-indent: 0; margin: 0; line-height:120%" align="left">
		font size="7" face="Georgia">br>/font>font size="7" face="Georgia">
		input type="radio" style="width: 60; height: 60" name="Q6" value="V1" Language="VbScript">/font>span style="font-family: Georgia">span style="position: relative; top: 12.0pt">font size="7"> !--[if gte vml 1]>v:shape
 id="_x0000_s1028" type="#_x0000_t75" style='width:376.5pt;height:40.5pt'>
 v:imagedata src="runge_2nd_method_mobile_files/image031.wmz" o:title=""/>
/v:shape>![endif]-->![if !vml]>img border=0 width=502 height=54
src="runge_2nd_method_mobile_files/image032.gif" v:shapes="_x0000_s1028">![endif]>/font>/span>!--[if gte mso 9]>![endif]-->/span>/p>
		p style="word-spacing: 0; text-indent: 0; margin: 0; line-height:120%" align="left">
		font size="7" face="Georgia">
		input type="radio" style="width: 60; height: 60" name="Q6" value="V1" Language="VbScript">/font>span style="font-family: Georgia">span style="position: relative; top: 12.0pt">font size="7"> !--[if gte vml 1]>v:shape
 id="_x0000_s1027" type="#_x0000_t75" style='width:378.75pt;height:40.5pt'>
 v:imagedata src="runge_2nd_method_mobile_files/image033.wmz" o:title=""/>
/v:shape>![endif]-->![if !vml]>img border=0 width=505 height=54
src="runge_2nd_method_mobile_files/image034.gif" v:shapes="_x0000_s1027">![endif]>/font>/span>!--[if gte mso 9]>![endif]-->/span>/p>
		p style="word-spacing: 0; text-indent: 0; margin: 0; line-height:120%" align="left">
		font size="7" face="Georgia">
		input type="radio" style="width: 60; height: 60" name="Q6" value="V1" Language="VbScript"> /font>span style="font-family: Georgia">
		span style="position: relative; top: 12.0pt">font size="7">!--[if gte vml 1]>v:shape
 id="_x0000_s1026" type="#_x0000_t75" style='width:222.75pt;height:48.75pt'>
 v:imagedata src="runge_2nd_method_mobile_files/image035.wmz" o:title=""/>
/v:shape>![endif]-->![if !vml]>img border=0 width=297 height=65
src="runge_2nd_method_mobile_files/image036.gif" v:shapes="_x0000_s1026">![endif]>/font>/span>!--[if gte mso 9]>![endif]-->/span>/p>
		p style="word-spacing: 0; text-indent: 0; margin: 0; line-height:120%" align="left">
		font size="7" face="Georgia">
		input type="radio" style="width: 60; height: 60" name="Q6" value="V1" Language="VbScript">/font>span style="font-family: Georgia">span style="position: relative; top: 12.0pt">font size="7"> !--[if gte vml 1]>v:shape
 id="_x0000_s1025" type="#_x0000_t75" style='width:223.5pt;height:48pt'>
 v:imagedata src="runge_2nd_method_mobile_files/image037.wmz" o:title=""/>
/v:shape>![endif]-->![if !vml]>img border=0 width=298 height=64
src="runge_2nd_method_mobile_files/image038.gif" v:shapes="_x0000_s1025">![endif]>/font>/span>!--[if gte mso 9]>![endif]-->/span>/p>
		p style="word-spacing: 0; text-indent: 0; margin: 0; line-height:120%" align="left">
		&nbsp;/p>
		p style="word-spacing: 0; text-indent: 0; margin: 0; line-height:120%" align="left">
		!--[if gte mso 9]>![endif]-->/p>
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