Taylor Series - Revision history

Carver at 16:54, 1 December 2011

2011-12-01T16:54:46Z

2011-03-15T13:04:53Z

‎New Concept: Taylor Series

2011-03-15T12:25:14Z

‎New Concept: Taylor Series

2011-03-15T12:24:25Z

‎New Concept: Taylor Series

2011-03-15T12:14:42Z

‎New Concept: Taylor Series

2011-03-15T11:11:05Z

‎Try it yourself

2011-03-15T07:58:30Z

‎Questions for Exploration and Discussion in Groups of Three (With Computer)

2011-03-15T07:56:49Z

‎Questions for Exploration and Discussion in Groups of Three (With Computer)

2011-03-15T02:27:40Z

‎Looking Ahead

2011-03-15T02:26:39Z

‎Looking Ahead

← Older revision		Revision as of 16:54, 1 December 2011
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		+	This lecture was delivered in February 2011 to the Department of Mathematics and Statistics search committee at the University of Maryland, Baltimore County. I taught the class to illustrate the teaching methods I developed for the Modeling and Identifying Neurosystems class that I taught at the Johns Hopkins university in Spring of 2009. The topic ''Taylor Series'' was assigned to me the night before.
		+
	== Review Concepts ==		== Review Concepts ==

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 Definition copied, verified, and adapted from Wikipedia, [http://en.wikipedia.org/w/index.php?title=Taylor_series&oldid=418169005 this page (permanent link)].  See [http://en.wikipedia.org/wiki/Wikipedia:Text_of_the_GNU_Free_Documentation_License license to copy, modify, distribute.]  Also adapted somewhat based on textbook Finney & Thomas, Calculus, 2nd edition.
-Let f be a function with derivatives of all orders throughout some interval containing a as an interior point.  The Taylor series of a function ''ƒ''(''x'') at ''a'' is the power series
+Let f be a function with derivatives of all orders throughout some interval containing a as an interior point.  The Taylor series of the function ''ƒ''(''x'') at ''a'' is the power series
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 * The Taylor Series is derived from the function.
-Definition copied, verified, and adapted from Wikipedia, [http://en.wikipedia.org/w/index.php?title=Taylor_series&oldid=418169005 this page (permanent link)].  See [http://en.wikipedia.org/wiki/Wikipedia:Text_of_the_GNU_Free_Documentation_License license to copy, modify, distribute.]  Also adapted somewhat based on textbook Finney & Thomas.
+Definition copied, verified, and adapted from Wikipedia, [http://en.wikipedia.org/w/index.php?title=Taylor_series&oldid=418169005 this page (permanent link)].  See [http://en.wikipedia.org/wiki/Wikipedia:Text_of_the_GNU_Free_Documentation_License license to copy, modify, distribute.]  Also adapted somewhat based on textbook Finney & Thomas, Calculus, 2nd edition.
 Let f be a function with derivatives of all orders throughout some interval containing a as an interior point.  The Taylor series of a function ''ƒ''(''x'') at ''a'' is the power series

@@ Line 12: / Line 12: @@
 * The Taylor Series is derived from the function.
-Definition copied, verified, and adapted from Wikipedia, [http://en.wikipedia.org/w/index.php?title=Taylor_series&oldid=418169005 this page (permanent link)].  See [http://en.wikipedia.org/wiki/Wikipedia:Text_of_the_GNU_Free_Documentation_License license to copy, modify, distribute.]
+Definition copied, verified, and adapted from Wikipedia, [http://en.wikipedia.org/w/index.php?title=Taylor_series&oldid=418169005 this page (permanent link)].  See [http://en.wikipedia.org/wiki/Wikipedia:Text_of_the_GNU_Free_Documentation_License license to copy, modify, distribute.]  Also adapted somewhat based on textbook Finney & Thomas.
 Let f be a function with derivatives of all orders throughout some interval containing a as an interior point.  The Taylor series of a function ''ƒ''(''x'') at ''a'' is the power series

@@ Line 14: / Line 14: @@
 Definition copied, verified, and adapted from Wikipedia, [http://en.wikipedia.org/w/index.php?title=Taylor_series&oldid=418169005 this page (permanent link)].  See [http://en.wikipedia.org/wiki/Wikipedia:Text_of_the_GNU_Free_Documentation_License license to copy, modify, distribute.]
-The Taylor series of a function ''ƒ''(''x'') at ''a'' is the power series
+Let f be a function with derivatives of all orders throughout some interval containing a as an interior point.  The Taylor series of a function ''ƒ''(''x'') at ''a'' is the power series
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 After you have typed in the above you can change the parameters a, n1, n2, M by scrolling up and changing them, then pressing return.
-* <math> a </math> is where the derivatives are taken
+* <math> a </math> is where the derivatives are taken (center)
 * n1 and n2 are the orders for the two taylor series plots
 * [-M, M] is the range of the plot

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 * Is there an interval where the Taylor series seems to converge?  What is it?
-* Use the formula above to compute the Taylor series to order 4 (i.e. n=4) of  <math> 4 x^2 + 2 x + 5 </math> at the point a=1.
+* Use the formula above to compute (by hand) the Taylor series to order 4 (i.e. n=4) of  <math> 4 x^2 + 2 x + 5 </math> at the point a=1.
 == Looking Ahead ==
 * Taylor's theorem -- a precise statement about the error in a Taylor approximation of f(x) (ie. a partial sum of the Taylor series of f(x)).

← Older revision		Revision as of 02:27, 15 March 2011
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	== Looking Ahead ==		== Looking Ahead ==

−	* Taylor's theorem -- a precise statement about the error in a Taylor approximation of f(x) (ie. a partial sum of the Taylor series of f(x) ~~at a~~).	+	* Taylor's theorem -- a precise statement about the error in a Taylor approximation of f(x) (ie. a partial sum of the Taylor series of f(x)).

← Older revision		Revision as of 02:26, 15 March 2011
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	== Looking Ahead ==		== Looking Ahead ==

−	* Taylor's theorem -- a precise statement about the error in a Taylor approximation of f(x) (ie. a partial sum of Taylor series).	+	* Taylor's theorem -- a precise statement about the error in a Taylor approximation of f(x) (ie. a partial sum of the Taylor series of f(x) at a).