Arc tangent function - Revision history

Vipul: /* Second derivative */

2011-12-19T16:34:38Z

Second derivative

← Older revision		Revision as of 16:34, 19 December 2011
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	===Second derivative===		===Second derivative===

	{{quotation\|~~We use here~~ [[chain rule for differentiation]] and [[differentiation rule for power functions]]}}		{{quotation\|'''WHAT WE USE''': [[chain rule for differentiation]] and [[differentiation rule for power functions]]}}

	The second derivative is given as:		The second derivative is given as:

2011-12-19T16:26:24Z

First derivative

← Older revision		Revision as of 16:26, 19 December 2011
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	===First derivative===		===First derivative===

			{{quotation\|'''WHAT WE USE''': [[inverse function theorem]] and [[tangent function#First derivative]], which in turn relies on [[sine function#First derivative]], [[cosine function#First derivative]], and [[quotient rule for differentiation]]}}

	We use the [[inverse function theorem]], ''and'' the fact that the derivative of <math>\tan</math> is <math>\sec^2</math>.		We use the [[inverse function theorem]], ''and'' the fact that the derivative of <math>\tan</math> is <math>\sec^2</math>.

2011-12-19T16:25:16Z

Second derivative

← Older revision		Revision as of 16:25, 19 December 2011
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	===Second derivative===		===Second derivative===

	{{quotation\|We use here [[chain rule for differentiation]] and [[~~differentation~~ rule for power functions]]}}		{{quotation\|We use here [[chain rule for differentiation]] and [[differentiation rule for power functions]]}}

	The second derivative is given as:		The second derivative is given as:

2011-12-19T16:24:45Z

Second derivative

← Older revision		Revision as of 16:24, 19 December 2011
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	===Second derivative===		===Second derivative===

			{{quotation\|We use here [[chain rule for differentiation]] and [[differentation rule for power functions]]}}

	The second derivative is given as:		The second derivative is given as:

2011-12-19T16:23:36Z

Higher antiderivatives

← Older revision		Revision as of 16:23, 19 December 2011
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	===Higher antiderivatives===		===Higher antiderivatives===

	The function can be antidifferentiated any number of times using integration by parts.		The function can be antidifferentiated any number of times using integration by parts. The reason for this is that the derivative of the function is a rational function, and [[rational functions can be repeatedly integrated within elementarily expressible functions]].

			All the antiderivatives can be expressed in the form:

			<math>p_0(x) + p_1(x)\arctan x + p_2(x)\ln(1 + x^2)</math>

			where <math>p_0,p_1,p_2</math> are polynomial. Note that <math>p_0</math> is ambiguous up to addition of polynomials of degree <math>n - 1</math> if we are integrating <math>n</math> times.

	===Higher antiderivatives===		===Higher antiderivatives===

2011-12-19T16:21:34Z

Key data

← Older revision		Revision as of 16:21, 19 December 2011
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	\| [[horizontal asymptote]]s \|\| The line <math>y = \pi/2</math> as <math>x \to \infty</math><br>The line <math>y = -\pi/2</math> as <math>x \to -\infty</math>		\| [[horizontal asymptote]]s \|\| The line <math>y = \pi/2</math> as <math>x \to \infty</math><br>The line <math>y = -\pi/2</math> as <math>x \to -\infty</math>
	\|-		\|-
	\| [[derivative]] \|\| <math>x \mapsto \frac{1}{1 + x^2}</math>		\| [[derivative]] \|\| <math>x \mapsto \frac{1}{1 + x^2}</math> (see [[#First derivative]])
	\|-		\|-
	\| [[second derivative]] \|\| <math>x \mapsto \frac{-2x}{(1 + x^2)^2}</math>		\| [[second derivative]] \|\| <math>x \mapsto \frac{-2x}{(1 + x^2)^2}</math> (see [[#Second derivative]])
	\|-		\|-
	\| higher derivatives \|\| The <math>n^{th}</math> derivative is a rational function of <math>x</math> of degree <math>-n-1</math>.		\| higher derivatives \|\| The <math>n^{th}</math> derivative is a rational function of <math>x</math> where the denominator degree is <math>n + 1</math> more than the numerator degree.
	\|-		\|-
	\| [[antiderivative]] \|\| <math>x \arctan x - \frac{1}{2}\ln(1 + x^2) + C</math>. We use [[integration by parts]] (see [[#Integration]]).		\| [[antiderivative]] \|\| <math>x \arctan x - \frac{1}{2}\ln(1 + x^2) + C</math>. We use [[integration by parts]] (see [[#Integration]]).

2011-12-19T05:20:54Z

Key data

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	\| default [[domain]] \|\| all [[real number]]s, i.e., all of <math>\R</math>		\| default [[domain]] \|\| all [[real number]]s, i.e., all of <math>\R</math>
	\|-		\|-
	\| [[range]] \|\| the [[open interval]] <math>(-\pi/2,\pi/2)</math>, i.e., the set <math>\{y \mid -\pi/2 ~~\le~~ y ~~\le~~ \pi/2 \}</math><br>no absolute minimum or absolute maximum, because the extremes are attained asymptotically at infinity.		\| [[range]] \|\| the [[open interval]] <math>(-\pi/2,\pi/2)</math>, i.e., the set <math>\{y \mid -\pi/2 < y< \pi/2 \}</math><br>no absolute minimum or absolute maximum, because the extremes are attained asymptotically at infinity.
	\|-		\|-
	\| [[local minimum value]]s and points of attainment \|\| no local minimum values		\| [[local minimum value]]s and points of attainment \|\| no local minimum values

2011-09-04T01:02:27Z

Key data

@@ Line 36: / Line 36: @@
 | higher antiderivatives || The function can be antidifferentiated any number of times in terms of [[elementary function]]s.
 |}
 ==Differentiation==

2011-09-04T01:00:09Z

Power series and Taylor series

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	==Power series and Taylor series==		==Power series and Taylor series==

			===Computation of power series===

	The [[power series]] for the function about 0 can be obtained as follows.		The [[power series]] for the function about 0 can be obtained as follows.

2011-08-27T14:15:12Z

Power series and Taylor series

← Older revision		Revision as of 14:15, 27 August 2011
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	The left side is <math>[\arctan t]_0^x = \arctan x - \arctan 0 = \arctan x</math>, so we get:		The left side is <math>[\arctan t]_0^x = \arctan x - \arctan 0 = \arctan x</math>, so we get:

	<math>\arctan x = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \dots = \sum_{k=1}^\infty \frac{(-1)^kx^{2k+1}}{2k + 1}, \qquad -1 < x < 1</math>		<math>\arctan x = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \dots = \sum_{k=0}^\infty \frac{(-1)^kx^{2k+1}}{2k + 1}, \qquad -1 < x < 1</math>

	By the [[alternating series theorem]], we note that the power series on the right converges for <math>x = -1</math> ''and'' for <math>x = 1</math>, so by [[Abel's theorem]], we conclude that it converges ''to'' <math>\arctan</math> of the respective inputs. We thus get:		By the [[alternating series theorem]], we note that the power series on the right converges for <math>x = -1</math> ''and'' for <math>x = 1</math>, so by [[Abel's theorem]], we conclude that it converges ''to'' <math>\arctan</math> of the respective inputs. We thus get:

	<math>\arctan x = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \dots = \sum_{k=0}^\infty \frac{(-1)^kx^{2k+1}}{2k + 1}, \qquad -1 \le x \le 1</math>		<math>\arctan x = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \dots = \sum_{k=0}^\infty \frac{(-1)^kx^{2k+1}}{2k + 1}, \qquad -1 \le x \le 1</math>