Sine-cubed function - Revision history

Vipul: /* Computation of power series */

2011-09-03T21:26:00Z

Computation of power series

← Older revision		Revision as of 21:26, 3 September 2011
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	We have the power series:		We have the power series:

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

	We thus get the power series:		We thus get the power series:

Vipul: /* Repeated antidifferentiation */

2011-09-03T21:25:36Z

Repeated antidifferentiation

← Older revision		Revision as of 21:25, 3 September 2011
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	The antiderivative of <math>\sin^3</math> involves [[cos^3]] and [[cos]], both of which can be antidifferentiated, and this now involves [[sin^3]] and [[sin]]. We can thus antidifferentiate (i.e., integrate) the function any number of times, with the antiderivative expression alternating between a cubic function of sine and a cubic function of cosine.		The antiderivative of <math>\sin^3</math> involves [[cos^3]] and [[cos]], both of which can be antidifferentiated, and this now involves [[sin^3]] and [[sin]]. We can thus antidifferentiate (i.e., integrate) the function any number of times, with the antiderivative expression alternating between a cubic function of sine and a cubic function of cosine.

			==Power series and Taylor series==

			===Computation of power series===

			We can use the identity:

			<math>\! \sin^3x = \frac{3\sin x - \sin(3x)}{4}</math>

			We have the power series:

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

			We thus get the power series:

			<math>\! \sin(3x) = (3x) - \frac{(3x)^3}{3!} + \frac{(3x)^5}{5!} - \dots = \sum_{k=0}^\infty \frac{(-1)^k3^{2k+1}x^{2k+1}}{(2k+1)!}</math>

			Plugging into the formula, we get:

			<math>\! \sin^3x = \sum_{k=0}^\infty \frac{(-1)^k(3 - 3^{2k+1})}{4 \cdot (2k+1)!}</math>

			The first few terms are as follows:

			<math>\! \sin^3x = x^3 - \frac{x^5}{2} + \dots</math>

Vipul: /* Graph */

2011-09-03T20:34:45Z

Graph

← Older revision		Revision as of 20:34, 3 September 2011
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	==Graph==		==Graph==

	Here is the basic graph:		Here is the basic graph, drawn on the interval <math>[-2\pi,2\pi]</math>:

	[[File:Sinecubedbasic.png\|600px]]		[[File:Sinecubedbasic.png\|600px]]

			Here is a more close-up graph, drawn on the interval <math>[-\pi,\pi]</math>. The thick black dots correspond to local extreme values, and the thick red dots correspond to points of inflection.

			[[File:Sinecubedcloseup.png\|800px]]

	==Differentiation==		==Differentiation==

Vipul: /* Differentiation */

2011-09-03T20:27:07Z

Differentiation

← Older revision		Revision as of 20:27, 3 September 2011
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	<math>\! \sin^3 x = \frac{3\sin x - \sin(3x)}{4}</math>		<math>\! \sin^3 x = \frac{3\sin x - \sin(3x)}{4}</math>

			==Graph==

			Here is the basic graph:

			[[File:Sinecubedbasic.png\|600px]]

	==Differentiation==		==Differentiation==

Vipul: /* Key data */

2011-08-28T12:57:40Z

Key data

← Older revision		Revision as of 12:57, 28 August 2011
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	\| [[antiderivative]] \|\| <math>\frac{\cos^3x}{3} - \cos x + C</math>		\| [[antiderivative]] \|\| <math>\frac{\cos^3x}{3} - \cos x + C</math>
	\|-		\|-
	\| important symmetries \|\| [[odd function]]<br>[[half turn symmetry]] about all points of the form <math>(n\pi,0)</math><br>[[mirror symmetry]] about all lines <math>x = n\pi + \pi/2</math>.		\| important symmetries \|\| [[odd function]] (follows from [[composite of odd functions is odd]], and the fact that the [[cube function]] and [[sine function]] are both odd)<br>[[half turn symmetry]] about all points of the form <math>(n\pi,0)</math><br>[[mirror symmetry]] about all lines <math>x = n\pi + \pi/2</math>.
	\|}		\|}

Vipul at 02:27, 27 August 2011

2011-08-27T02:27:30Z

@@ Line 33: / Line 33: @@
 |-
 | [[antiderivative]] || <math>\frac{\cos^3x}{3} - \cos x + C</math>
 |}

Vipul: /* Key data */

2011-08-26T17:44:25Z

Key data

← Older revision		Revision as of 17:44, 26 August 2011
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	\| [[period]] \|\| <math>2\pi</math>, i.e., <math>360\,^\circ</math>		\| [[period]] \|\| <math>2\pi</math>, i.e., <math>360\,^\circ</math>
	\|-		\|-
	\| [[local maximum value]]s and points of attainment \|\| All local maximum values are equal to 1, and they are attained at all points of the form <math>2n\pi + \pi/2</math>		\| [[local maximum value]]s and points of attainment \|\| All local maximum values are equal to 1, and they are attained at all points of the form <math>2n\pi + \pi/2</math> where <math>n</math> varies over integers.
	\|-		\|-
	\| [[local minimum value]]s and points of attainment \|\| All local minimum values are equal to -1, and they are attained at all points of the form <math>2n\pi - \pi/2</math>.		\| [[local minimum value]]s and points of attainment \|\| All local minimum values are equal to -1, and they are attained at all points of the form <math>2n\pi - \pi/2</math> where <math>n</math> varies over integers.
	\|-		\|-
	\| [[point of inflection\|points of inflection]] (both coordinates) \|\| All points of the form <math>(n\pi,0)</math>, as well as points of the form <math>(n\pi + \alpha, (-1)^n2\sqrt{2}/3\sqrt{3})</math> and <math>(n\pi + \pi - \alpha, (-1)^n2\sqrt{2}/3\sqrt{3})</math> where <math>\alpha = \arcsin(\sqrt{2/3})</math>.		\| [[point of inflection\|points of inflection]] (both coordinates) \|\| All points of the form <math>(n\pi,0)</math>, as well as points of the form <math>(n\pi + \alpha, (-1)^n2\sqrt{2}/3\sqrt{3})</math> and <math>(n\pi + \pi - \alpha, (-1)^n2\sqrt{2}/3\sqrt{3})</math> where <math>\alpha = \arcsin(\sqrt{2/3})</math> where <math>n</math> varies over integers.
	\|-		\|-
	\| [[derivative]] \|\| <math>x \mapsto 3\sin^2x \cos x</math>		\| [[derivative]] \|\| <math>x \mapsto 3\sin^2x \cos x</math>

Vipul: Created page with "{{particular function}} {{angular function radian convention}} ==Definition== This function, denoted $\sin^3$ , is defined as the [[defining ingredient::composite of..."

2011-08-26T17:43:11Z

Created page with "{{particular function}} {{angular function radian convention}} ==Definition== This function, denoted <math>\sin^3</math>, is defined as the [[defining ingredient::composite of..."

New page

{{particular function}}

{{angular function radian convention}}

==Definition==

This function, denoted <math>\sin^3</math>, is defined as the [[defining ingredient::composite of two functions|composite]] of the [[defining ingredient::cube function]] and the [[defining ingredient::sine function]]. Explicitly, it is the map:

<math>\! x \mapsto (\sin x)^3</math>

For brevity, we write <math>(\sin x)^3</math> or <math>\sin^3x</math>.

==Key data==

{| class="sortable" border="1"
! Item !! Value
|-
| Default [[domain]] || all [[real number]]s, i.e., all of <math>\R</math>
|-
| [[range]] || the [[closed interval]] <math>[-1,1]</math>, i.e., <math>\{ y \mid -1 \le y \le 1 \}</math><br>[[absolute maximum value]]: 1, [[absolute minimum value]]: -1
|-
| [[period]] || <math>2\pi</math>, i.e., <math>360\,^\circ</math>
|-
| [[local maximum value]]s and points of attainment || All local maximum values are equal to 1, and they are attained at all points of the form <math>2n\pi + \pi/2</math>
|-
| [[local minimum value]]s and points of attainment || All local minimum values are equal to -1, and they are attained at all points of the form <math>2n\pi - \pi/2</math>.
|-
| [[point of inflection|points of inflection]] (both coordinates) || All points of the form <math>(n\pi,0)</math>, as well as points of the form <math>(n\pi + \alpha, (-1)^n2\sqrt{2}/3\sqrt{3})</math> and <math>(n\pi + \pi - \alpha, (-1)^n2\sqrt{2}/3\sqrt{3})</math> where <math>\alpha = \arcsin(\sqrt{2/3})</math>.
|-
| [[derivative]] || <math>x \mapsto 3\sin^2x \cos x</math>
|-
| [[second derivative]] || <math>x \mapsto 6 \sin x \cos^2x - 3\sin^3x = 6 \sin x - 9 \sin^3x</math>
|-
| [[antiderivative]] || <math>\frac{\cos^3x}{3} - \cos x + C</math>
|}

Sine-cubed function - Revision history

Vipul: /* Computation of power series */

Vipul: /* Repeated antidifferentiation */

Vipul: /* Graph */

Vipul: /* Differentiation */

Vipul: /* Key data */

Vipul at 02:27, 27 August 2011

Vipul: /* Key data */

Vipul: Created page with "{{particular function}} {{angular function radian convention}} ==Definition== This function, denoted \sin^3, is defined as the [[defining ingredient::composite of..."

Vipul: Created page with "{{particular function}} {{angular function radian convention}} ==Definition== This function, denoted $\sin^3$ , is defined as the [[defining ingredient::composite of..."