Worksheets/Simpsons Rule - Revision history

Admin: /* Question 4 */

2016-05-21T20:46:46Z

Question 4

← Older revision		Revision as of 20:46, 21 May 2016
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	<math>		<math>
	\epsilon \sim ~~e^{~~\Delta x}		\epsilon \sim \Delta x^n
	</math>		</math>

	<math>		<math>
	\epsilon \sim \Delta x^n		\epsilon \sim e^{\Delta x}
	</math>		</math>

Admin at 20:45, 21 May 2016

2016-05-21T20:45:19Z

← Older revision		Revision as of 20:45, 21 May 2016
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	This worksheet is intended to answer the following questions: How do we evaluate integrals with a computer? When do we evaluate integrals with computers?		This worksheet is intended to answer the following questions: How do we evaluate integrals with a computer? When do we evaluate integrals with computers?

	~~Expectations:~~		==Student expectations==
	* You will submit a one-page report that answers each of the questions.
	* You will need to do some analytical work, and some numerical work. You will need to show your work for both kinds of questions.
	* If a question asks for a plot or a chart, attach it to your report and refer to it in your answer.
	* Answer questions with complete sentences.

	~~=PowerPoint Problem Setup=~~		You will submit a one-page report that answers each of the questions.

	~~Explanation~~ of ~~numerical integration background, equation derivation~~		You will need to do some analytical work, and some numerical work. You will need to show your work for both kinds of questions.

	~~Cover the questions~~		If a question asks for a plot or a chart, attach it to your report and refer to it in your answer.

	~~Cover strategy, writing out what to do before implementing, how to implement in a spreadsheet~~		Answer questions with complete sentences.

	Spend a day working in a computer lab		==Problem setup==

			(Essential information is on handout, but go through it with board/slides/etc)

			Numerical integration: background

			Equation derivation: referring back to introduction to integrals, rectangle approximations, then through linear and quadratic approximations

			Cover each question, cover strategy and writing out before implementing, how to implement in spreadsheet

			Spend one day working in computer lab

	=Worksheet Content=		=Worksheet Content=
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	Read through each section of the worksheet carefully. It contains the information you need to successfully implement Simpson's Rule in a spreadsheet.		Read through each section of the worksheet carefully. It contains the information you need to successfully implement Simpson's Rule in a spreadsheet.

	~~Answer~~ each question at the end of the worksheet with a complete ~~sentence on a separate sheet of paper~~. This assignment is due in class on ~~XYZ~~.		You will submit a one-page report that answers each question at the end of the worksheet. Answer each question with complete sentences. This assignment is due in class on (date).

	==Numerical Integration==		==Numerical Integration==
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	[[Image:SimpsonsRule5.jpg\|500px]]		[[Image:SimpsonsRule5.jpg\|500px]]

	=~~Calc II~~ Questions=

			=Worksheet Questions for Calculus 2=

	On a separate sheet of paper, answer each of the following questions with a complete sentence. Please be considerate of the environment – do not print out your spreadsheets.		On a separate sheet of paper, answer each of the following questions with a complete sentence. Please be considerate of the environment – do not print out your spreadsheets.
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	==Question 1==		==Question 1==

	~~Verify,~~ using the ~~substitution <math>u^2 = 4x</math>, that~~ the ~~indefinite~~ integral ~~is given by:~~		Create a new worksheet using a spreadsheet program. Use it to approximate the value of the integral I using Simpson's Rule with N = 10 subintervals. Report the absolute error in scientific notation. Report the relative error as a percentage.

	~~<math>~~
	~~\int 2 + \cos{(2 \sqrt{x})} dx~~ = ~~2x + \sqrt{x} \sin{(2 \sqrt{x})} - \frac{ \cos{(2 \sqrt{x})} }{ 2 } + C~~
	~~</math>~~

	==Question 2==		==Question 2==

	~~Create~~ a new worksheet ~~using a spreadsheet program. Use it to~~ approximate the value of the integral I using Simpson's Rule with N = 10 subintervals. Report the absolute error in scientific notation. Report the relative error as a percentage.		In a new worksheet, approximate the value of the integral I using Simpson's Rule with N = 50 subintervals. Report the absolute error in scientific notation. Report the relative error as a percentage.

	==Question 3==		==Question 3==

	In a new worksheet, approximate the value of the integral I using Simpson's Rule with N = 50 subintervals. Report the absolute error in scientific notation. Report the relative error as a percentage.		In a new worksheet, approximate the value of the integral I using Simpson's Rule with N = 100 subintervals. Report the absolute error in scientific notation. Report the relative error as a percentage.

	==Question 4==		==Question 4==

	In a new worksheet, approximate the value of the integral I using Simpson's Rule with N = 100 subintervals. Report the absolute error in scientific notation. Report the relative error as a percentage.

	~~==Question 5==~~

	We would expect that the error would be a function of the step size , . Determine what the form of the functional relationship is by plotting vs for the cases above. What is the functional form? Some examples to consider:		We would expect that the error would be a function of the step size , . Determine what the form of the functional relationship is by plotting vs for the cases above. What is the functional form? Some examples to consider:
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	<math>		<math>
	\epsilon \sim \ln{(\Delta x)}		\epsilon \sim \ln{(\Delta x)}
			</math>

			==Extra Credit==

			Verify, using the substitution <math>u^2 = 4x</math>, that the indefinite integral is given by:

			<math>
			\int 2 + \cos{(2 \sqrt{x})} dx = 2x + \sqrt{x} \sin{(2 \sqrt{x})} - \frac{ \cos{(2 \sqrt{x})} }{ 2 } + C
	</math>		</math>

Admin: /* Background */

2016-05-21T20:40:22Z

Background

← Older revision		Revision as of 20:40, 21 May 2016
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	=~~Background~~=		=Read Before You Begin=

	~~When might we need to evaluate an integral~~ with ~~data?~~		Before you begin with this worksheet, let's talk about what it will cover.

	~~Numerical routine~~ to ~~integrate two functions and find~~ a ~~centroid - Magrab p. 154 (uses quadc() routine)~~		This worksheet is intended to answer the following questions: How do we evaluate integrals with a computer? When do we evaluate integrals with computers?

	Expectations:		Expectations:
	* ~~Answers with complete sentences~~		* You will submit a one-page report that answers each of the questions.
	* ~~Some~~ analytical work, some numerical work		* You will need to do some analytical work, and some numerical work. You will need to show your work for both kinds of questions.
	* ~~Final answers~~ in ~~report!~~		* If a question asks for a plot or a chart, attach it to your report and refer to it in your answer.
	* ~~Plot/chart to answer question~~		* Answer questions with complete sentences.

	=PowerPoint Problem Setup=		=PowerPoint Problem Setup=

Admin: /* Simpson's Rule */

2016-05-08T20:29:15Z

Simpson's Rule

← Older revision		Revision as of 20:29, 8 May 2016
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	To approximate the total area, simply sum all of the <math>A_i</math> terms.		To approximate the total area, simply sum all of the <math>A_i</math> terms.

	Simpson's Rule fits a polynomial of degree 2 every three data points (two sub-intervals) and approximates the real function with a set of polynomials . Therefore, Simpson's Rule is exact when is a polynomial of degree 2 or less.		Simpson's Rule fits a polynomial of degree 2 every three data points (two sub-intervals) and approximates the real function with a set of polynomials . Therefore, Simpson's Rule is exact when is a polynomial of degree 2 or less.

			[[Image:SimpsonsRule5.jpg\|500px]]

	=Calc II Questions=		=Calc II Questions=

Admin: /* Left Hand, Right Hand, and Midpoint Rules */

2016-05-08T20:27:49Z

Left Hand, Right Hand, and Midpoint Rules

← Older revision		Revision as of 20:27, 8 May 2016
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	The left-hand, right-hand, and midpoint rules approximate the behavior of the function as a constant over the interval . Recall that we covered these when we were first introduced to the topic of integration. The constant function has one unknown parameter (y=c), and therefore requires one function evaluation.		The left-hand, right-hand, and midpoint rules approximate the behavior of the function as a constant over the interval . Recall that we covered these when we were first introduced to the topic of integration. The constant function has one unknown parameter (y=c), and therefore requires one function evaluation.

	[[~~Images~~:SimpsonsRule2.jpg\|500px]]		[[Image:SimpsonsRule2.jpg\|500px]]

	[[Image:SimpsonsRule3.jpg\|500px]]		[[Image:SimpsonsRule3.jpg\|500px]]

Admin: /* Trapezoid Rule */

2016-05-08T20:27:37Z

Trapezoid Rule

← Older revision		Revision as of 20:27, 8 May 2016
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	[[Images:SimpsonsRule2.jpg\|500px]]		[[Images:SimpsonsRule2.jpg\|500px]]

			[[Image:SimpsonsRule3.jpg\|500px]]

	==Trapezoid Rule==		==Trapezoid Rule==
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	To approximate the total area, simply sum all of the <math>A_i</math> terms.		To approximate the total area, simply sum all of the <math>A_i</math> terms.

			[[Image:SimpsonsRule4.jpg\|500px]]

	==Simpson's Rule==		==Simpson's Rule==

Admin: /* Left Hand, Right Hand, and Midpoint Rules */

2016-05-08T20:27:05Z

Left Hand, Right Hand, and Midpoint Rules

← Older revision		Revision as of 20:27, 8 May 2016
Line 54:		Line 54:

	The left-hand, right-hand, and midpoint rules approximate the behavior of the function as a constant over the interval . Recall that we covered these when we were first introduced to the topic of integration. The constant function has one unknown parameter (y=c), and therefore requires one function evaluation.		The left-hand, right-hand, and midpoint rules approximate the behavior of the function as a constant over the interval . Recall that we covered these when we were first introduced to the topic of integration. The constant function has one unknown parameter (y=c), and therefore requires one function evaluation.

			[[Images:SimpsonsRule2.jpg\|500px]]

	==Trapezoid Rule==		==Trapezoid Rule==

Admin: /* Numerical Integration */

2016-05-08T20:26:21Z

Numerical Integration

← Older revision		Revision as of 20:26, 8 May 2016
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	This partitioning, as applied to a function, is shown in the figure on the right. There are N sub-intervals. The behavior of the function between the points <math>f(x_i)</math> depends on the numerical integration technique being used.		This partitioning, as applied to a function, is shown in the figure on the right. There are N sub-intervals. The behavior of the function between the points <math>f(x_i)</math> depends on the numerical integration technique being used.

	[[Image:SimpsonsRule1.jpg]]		[[Image:SimpsonsRule1.jpg\|500px]]

	==Left Hand, Right Hand, and Midpoint Rules==		==Left Hand, Right Hand, and Midpoint Rules==

Admin: /* Numerical Integration */

2016-05-08T20:25:40Z

Numerical Integration

← Older revision		Revision as of 20:25, 8 May 2016
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	This partitioning, as applied to a function, is shown in the figure on the right. There are N sub-intervals. The behavior of the function between the points <math>f(x_i)</math> depends on the numerical integration technique being used.		This partitioning, as applied to a function, is shown in the figure on the right. There are N sub-intervals. The behavior of the function between the points <math>f(x_i)</math> depends on the numerical integration technique being used.

			[[Image:SimpsonsRule1.jpg]]

	==Left Hand, Right Hand, and Midpoint Rules==		==Left Hand, Right Hand, and Midpoint Rules==

Admin: /* Flags */

2016-05-08T10:44:03Z

Flags

← Older revision		Revision as of 10:44, 8 May 2016
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	=Flags=		=Flags=

	{{~~WorksheetsFlag~~}}		{{WorksheetFlag}}