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		p style="MARGIN-TOP: 0px; MARGIN-BOTTOM: 0px" align="left">b>
		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>
	/font>
  		font face="Verdana" size="6" Language="VbScript">
		p align="left">b>Runge-Kutta 4sup>th/sup> Order Method/b>/p>
		/font>
	p>&nbsp;/div>
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		p class="MsoNormal">font size="7" face="Georgia">b>Q1. /b>To solve the ordinary 
		differential equation/font>/p>
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/v:shape>![endif]-->![if !vml]>img border=0 width=293 height=69
src="runge_4th_method_mobile_files/image002.gif" v:shapes="_x0000_s1161">![endif]>/font>/span>!--[if gte mso 9]>![endif]-->/p>
		p class="MsoNormal">font size="7" face="Georgia">by Runge-Kutta 4sup>th/sup> 
		order method, you need to rewrite the equation as/font>/p>
		p style="word-spacing: 0; text-indent: 0; margin: 0" align="left">
		font size="7" face="Georgia">
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src="runge_4th_method_mobile_files/image004.gif" v:shapes="_x0000_s1160">![endif]>/font>/span>/span>font size="7" face="Georgia">br>
		/font>font size="7" face="Georgia">
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src="runge_4th_method_mobile_files/image008.gif" v:shapes="_x0000_s1158">![endif]>/font>/span>!--[if gte mso 9]>![endif]-->/span>p style="word-spacing: 0; text-indent: 0; margin: 0" align="left">
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		/font>/span>span style="font-family: Georgia">
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		hr>font size="7" face="Georgia">/center>/center>/center>
		/font>
		p style="margin-top: 0; margin-bottom: 0">
		font face="Georgia" size="7">
		b>Q2. /b>Given /font>/p>
		p style="margin-top: 0; margin-bottom: 0">font face="Georgia">
		span style="position: relative; top: 12.0pt">font size="7">!--[if gte vml 1]>v:shape
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/v:shape>![endif]-->![if !vml]>img border=0 width=294 height=65
src="runge_4th_method_mobile_files/image012.gif" v:shapes="_x0000_s1156">![endif]>/font>/span>/font>span style="font-size: 12.0pt; font-family: Times New Roman">!--[if gte mso 9]>![endif]-->/span>font face="Georgia">font size="7">&nbsp;/font>/font>/p>
		p style="margin-top: 0; margin-bottom: 0">
		font size="7" face="Georgia">and 
		using a step size of i>h=0.3/i>/font>!--[if gte mso 9]>![endif]-->span style="font-size: 12.0pt; font-family: Times New Roman">span style="font-family: Georgia">font size="7">, 
		the value of i>y/i>(i>0.9/i>)/font>!--[if gte mso 9]>![endif]-->/span>font size="7">span style="font-family: Georgia">&nbsp;using 
		Runge-Kutta 4sup>th/sup> order method is most nearly /span>/font>
		/span>/p>
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		font face="Georgia">
		font size="7">
		input type="radio" style="width: 60; height: 60" style="width: 60; height: 60" name="Q2" value="V1" Language="VbScript">- 
		0.25011 /font>font size="6">x /font>font size="7">10sup>40/sup>/font>/font>!--[if gte mso 9]>![endif]-->/p>
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		font size="7" face="Georgia">
		input type="radio" style="width: 60; height: 60" style="width: 60; height: 60" name="Q2" value="V1" Language="VbScript"> 
		-4297.4/font>/p>
		p style="word-spacing: 0; text-indent: 0; margin: 0" align="left">
		font size="7" face="Georgia">
		input type="radio" style="width: 60; height: 60" style="width: 60; height: 60" name="Q2" value="V1" Language="VbScript">-1261.5br>
		/font>font size="7" face="Georgia">
		input type="radio" style="width: 60; height: 60" style="width: 60; height: 60" name="Q2" value="V1" Language="VbScript">0.88498br>
		/font>i>font size="7" face="Georgia">Note: At the end of this 
		document, see formulas used to answer this question as there are a few 
		different versions of the Runge-Kutta 4sup>th/sup> order method./font>/i>/p>
		p style="word-spacing: 0; text-indent: 0; margin: 0" align="left">
		&nbsp;/p>hr>
		font size="7" face="Georgia">/center>
		/center>/center>/center>/center>
		/font>
		p>font face="Georgia" size="7">b>Q3. /b>Given /font>/p>
		p>font face="Georgia">
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src="runge_4th_method_mobile_files/image014.gif" v:shapes="_x0000_s1155">![endif]>/font>/span>/font>span style="font-size: 12.0pt; font-family: Times New Roman">!--[if gte mso 9]>![endif]-->/span>font size="7" face="Georgia"> /font>/p>
		p>font face="Georgia">font size="7">and using a step size of /font>/font>
		font size="7" face="Georgia">i>h=0.3/i>/font>!--[if gte mso 9]>![endif]-->span style="font-size: 12.0pt; font-family: Times New Roman">span style="font-family: Georgia">font size="7">, 
		the best estimate of /font>
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src="runge_4th_method_mobile_files/image016.gif" v:shapes="_x0000_s1154">![endif]>/font>/span>!--[if gte mso 9]>![endif]-->/span>font size="7">span style="font-family: Georgia">&nbsp;/span>/font>/span>/p>
		p>span style="font-size: 12.0pt; font-family: Times New Roman">
		font size="7">span style="font-family: Georgia">Runge-Kutta 
		4th order method is most nearly/span>/font>/span>/p>
		p class="MsoNormal" align="left" style="margin-top: 0; margin-bottom: 0">
		font size="7" face="Georgia">
		input type="radio" style="width: 60; height: 60" style="width: 60; height: 60" name="Q3" value="V1" Language="VbScript">-1.6604br>
		/font>font size="7" face="Georgia">
		input type="radio" style="width: 60; height: 60" style="width: 60; height: 60" name="Q3" value="V1" Language="VbScript">-1.1785br>
		/font>font size="7" face="Georgia">
		input type="radio" style="width: 60; height: 60" style="width: 60; height: 60" name="Q3" value="V1" Language="VbScript">-0.45831br>
		/font>font size="7" face="Georgia">
		input type="radio" style="width: 60; height: 60" style="width: 60; height: 60" name="Q3" value="V1" Language="VbScript">2.7270/font>/p>
		p class="MsoNormal">i>font size="7" face="Georgia">Note: At the end 
		of this document, see formulas used to answer this question as there are 
		a few different versions of the Runge-Kutta 4sup>th/sup> order method./font>/i>/p>
		hr>font size="7" face="Georgia">/center>
		/center>/center>/center>/center>
		/font>
		p class="MsoNormal">font size="7" face="Georgia">b>Q4. /b>The velocity (m/s) of 
		a parachutist is given as a function of time (seconds) by /font>/p>
		p class="MsoNormal" style="margin-left: .75in">
		span style="position: relative; top: 5.0pt">
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 id="_x0000_s1153" type="#_x0000_t75" style='width:253.5pt;height:32.25pt'>
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/v:shape>![endif]-->![if !vml]>img border=0 width=338 height=43
src="runge_4th_method_mobile_files/image018.gif" v:shapes="_x0000_s1153">![endif]>/font>/span>!--[if gte mso 9]>![endif]-->/p>
		p class="MsoNormal">font face="Georgia">font size="7">Using 
		Runge-Kutta 4sup>th/sup> order method with a step size of 5 seconds, 
		the distance traveled by the body from i>t=2/i>/font>/font>!--[if gte mso 9]>![endif]-->font face="Georgia">font size="7">&nbsp;to
		i>t=12/i>/font>/font>!--[if gte mso 9]>![endif]-->font size="7" face="Georgia">&nbsp;seconds 
		is estimated most nearly as/font>/p>
		
		
		p align="left" style="word-spacing: 0; text-indent: 0; margin: 0">
		font size="7" face="Georgia">
		input type="radio" style="width: 60; height: 60" style="width: 60; height: 60" name="Q4" value="V1" Language="VbScript">341.43 
		m&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; !--[if gte mso 9]>
		![endif]-->br>
		/font>font size="7" face="Georgia">
		input type="radio" style="width: 60; height: 60" style="width: 60; height: 60" name="Q4" value="V1" Language="VbScript">428.97 
		mbr>/font>font size="7" face="Georgia">
		input type="radio" style="width: 60; height: 60" style="width: 60; height: 60" name="Q4" value="V1" Language="VbScript">429.05 
		mbr>/font>font size="7" face="Georgia">
		input type="radio" style="width: 60; height: 60" style="width: 60; height: 60" name="Q4" value="V1" Language="VbScript">703.50 
		m/font>p class="MsoNormal">i>font size="7" face="Georgia">
		Note: At the end of this document, see formulas used to answer this 
		question as there are a few different versions of the Runge-Kutta 4sup>th/sup> 
		order method./font>/i>/p>
		hr>
		p class="MsoNormal">font face="Georgia">font size="7">b>Q5/b>.&nbsp; Runge-Kutta 
		method can be derived from using first three terms of Taylor series of 
		writing the value of /font>
		/font>
		font face="Georgia" size="7"> i>ysub>i/sub>/i>sub>+1/sub>/font>!--[if gte mso 9]>![endif]-->!--[if gte mso 9]>![endif]-->font face="Georgia">font size="7">, 
		that is the value of /font>
		/font>font face="Georgia" size="7">i>y/i>/font>!--[if gte mso 9]>![endif]-->font face="Georgia">font size="7"> at
		/font>/font>
		font face="Georgia" size="7">
		i>xsub>i/sub>/i>sub>+1/sub>/font>!--[if gte mso 9]>![endif]-->font face="Georgia">font size="7">, 
		&nbsp;in terms of /font>/font>
		font face="Georgia" size="7"> i>ysub>i/sub>/i>/font>!--[if gte mso 9]>![endif]-->font face="Georgia">font size="7">&nbsp;and 
		all the derivatives of /font>
		/font>font face="Georgia" size="7">i>y /i> /font>!--[if gte mso 9]>
		![endif]-->font face="Georgia">font size="7">at
		/font>/font>
		font face="Georgia" size="7">
		i>xsub>i/sub>/i>/font>!--[if gte mso 9]>![endif]-->font face="Georgia">font size="7">.&nbsp; 
		If /font>/font>font face="Georgia" size="7"> 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">font size="7">, 
		the explicit expression for /font>
		/font>
		font face="Georgia" size="7"> i>ysub>i/sub>/i>sub>+1 /sub>/font>!--[if gte mso 9]>
		![endif]-->font size="7" face="Georgia">if 
		the first five terms of the Taylor series are chosen for the ordinary 
		differential equation/font>/p>
		p class="MsoNormal" style="text-indent: .5in">
		span style="position: relative; top: 12.0pt">
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src="runge_4th_method_mobile_files/image020.gif" v:shapes="_x0000_s1152">![endif]>/font>/span>!--[if gte mso 9]>![endif]-->/p>
		p class="MsoNormal">font size="7" face="Georgia">would be/font>/p>
		p class="MsoNormal">font size="7" face="Georgia">
		input type="radio" style="width: 60; height: 60" 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_4th_method_mobile_files/image022.gif" v:shapes="_x0000_s1151">![endif]>/font>/span>/p>
		p align="left" style="word-spacing: 0; text-indent: 0; margin: 0">
		font size="7" face="Georgia">
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		&nbsp;hr>
		p class="MsoNormal">font face="Georgia">font size="7">b>Q6. /b>A hot solid 
		cylinder is immersed in an cool oil bath as part of a quenching process. 
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		change with time.&nbsp; If the initial temperature of the bar and the oil 
		bath is given as 600/font>span style="font-family: Georgia">°/span>font face="Georgia"> 
		C and 27/font>span style="font-family: Georgia">°/span>font face="Georgia">C, 
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table cellpadding="0" cellspacing="0" width="100%" id="table7">
	tr>
		td>![endif]>
		div>
			p class="MsoNormal">Oil/div>
		![if !mso]>/td>
	/tr>
/table>
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table cellpadding="0" cellspacing="0" width="100%" id="table6">
	tr>
		td>![endif]>
		div>
			p class="MsoNormal">Cylinder/div>
		![if !mso]>/td>
	/tr>
/table>
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font size="7">Length of cylinder = 30 cm/font>/p>
		font face="Georgia" size="6">Radius of cylinder = 3 cm/font>/p>
		font face="Georgia" size="6">Density of cylinder = 2700 kg/m^3/font>/p>
		font face="Georgia" size="6">Specific heat of cylinder = 895 J/kg-K/font>/p>
		font face="Georgia" size="6">Convection heat transfer coefficient = 100 W/(m^2-K)/font>/p>
		font face="Georgia" size="6">Specific heat of oil = 1910 J/(kg-K)/font>/p>
		font face="Georgia" size="6">Mass of oil = 2 kg/font>/p>
		font face="Georgia" size="7">The coupled ordinary differential equations governing the heat transfer are given by/font>/p>
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		p style="word-spacing: 0; text-indent: 0; margin: 0" align="left">
		font face="Georgia">font size="7">&nbsp;&nbsp;&nbsp;&nbsp; /font>
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		p style="word-spacing: 0; text-indent: 0; margin: 0" align="left">
		font size="7" face="Georgia">
		input type="radio" style="width: 60; height: 60" style="width: 60; height: 60" name="Q6" 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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		p style="word-spacing: 0; text-indent: 0; margin: 0" align="left">
		font face="Georgia">font size="7">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
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		/font>font size="7" face="Georgia">
		input type="radio" style="width: 60; height: 60" style="width: 60; height: 60" name="Q6" value="V1" Language="VbScript">/font>!--[if gte mso 9]>![endif]-->span style="position: relative; top: 12.0pt">font size="7" face="Georgia">!--[if gte vml 1]>v:shape
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		font face="Georgia">font size="7">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
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		font size="7" face="Georgia">
		input type="radio" style="width: 60; height: 60" style="width: 60; height: 60" name="Q6" value="V1" Language="VbScript">/font>!--[if gte mso 9]>![endif]-->span style="position: relative; top: 12.0pt">font size="7" face="Georgia">!--[if gte vml 1]>v:shape
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		p style="word-spacing: 0; text-indent: 0; margin: 0" align="left">
		font face="Georgia">font size="7">&nbsp;&nbsp;&nbsp;&nbsp; /font>
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font size="7" face="Georgia" Language="VbScript">
		
		p class="MsoNormal">The following 
		equations are used to answer questions#2, 3, and 4/p>
		
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		input type="submit" value="Check my answers" name="btncheck" style="font-size: 36">/font>/font>font size="7">/font>/font>font size="7" face="Georgia" Language="VbScript">br>
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