Cosine-squared function - Revision history

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2011-12-19T01:47:03Z

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← Older revision		Revision as of 01:47, 19 December 2011
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	\| [[derivative]] \|\| <math>x \mapsto -\sin (2x)</math> i.e., negative of the [[double-angle sine function]].		\| [[derivative]] \|\| <math>x \mapsto -\sin (2x)</math> i.e., negative of the [[double-angle sine function]].
	\|-		\|-
	\| [[second derivative]] \|\| <math>x \mapsto 2 \cos(2x)</math>		\| [[second derivative]] \|\| <math>x \mapsto -2 \cos(2x)</math>
	\|-		\|-
	\| higher derivatives \|\| <math>2^{~~n−1~~}</math> times an expression that is <math>\pm \sin</math> or <math>\pm \cos</math> of <math>2x</math>, depending on the remainder of <math>n</math> mod 4.		\| higher derivatives \|\| <math>2^{n - 1}</math> times an expression that is <math>\pm \sin</math> or <math>\pm \cos</math> of <math>2x</math>, depending on the remainder of <math>n</math> mod 4.
	\|-		\|-
	\| [[antiderivative]] \|\| <math>\frac{x}{2} + \frac{\sin(2x)}{4} + C</math>		\| [[antiderivative]] \|\| <math>\frac{x}{2} + \frac{\sin(2x)}{4} + C</math>

Vipul: /* Key data */

2011-08-26T17:50:47Z

Key data

← Older revision		Revision as of 17:50, 26 August 2011
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	\| expression as a [[sinusoidal function]] plus a constant function \|\| <math>(1/2) + (\cos(2x))/2</math>		\| expression as a [[sinusoidal function]] plus a constant function \|\| <math>(1/2) + (\cos(2x))/2</math>
	\|-		\|-
	\| important symmetries \|\| [[even function]]<br>More generally, has [[mirror symmetry]] about all vertical lines <math>x = n\pi/2</math>, <math>n</math> an integer.<br>Also, has half turn symmetry about all points of the form <math>(n \pi + \pi/4,1/2)</math>, i.e., all points of inflection.		\| important symmetries \|\| [[even function]]<br>More generally, has [[mirror symmetry]] about all vertical lines <math>x = n\pi/2</math>, <math>n</math> an integer.<br>Also, has half turn symmetry about all points of the form <math>(n \pi/2 + \pi/4,1/2)</math>, i.e., all points of inflection.
	\|-		\|-
	\| interval description based on increase/decrease and concave up/down \|\| For each integer <math>n</math>, the interval from <math>n\pi</math> to <math>(n+1)\pi</math> is subdivided into four parts:<br><math>(n\pi,n\pi + \pi/4)</math>: decreasing and concave down<br><math>(n\pi + \pi/4,n\pi + \pi/2)</math>: decreasing and concave up<br><math>(n\pi + \pi/2,n\pi + 3\pi/4)</math>: increasing and concave up<br><math>(n\pi + 3\pi/4, (n+1)\pi)</math>: increasing and concave down.		\| interval description based on increase/decrease and concave up/down \|\| For each integer <math>n</math>, the interval from <math>n\pi</math> to <math>(n+1)\pi</math> is subdivided into four parts:<br><math>(n\pi,n\pi + \pi/4)</math>: decreasing and concave down<br><math>(n\pi + \pi/4,n\pi + \pi/2)</math>: decreasing and concave up<br><math>(n\pi + \pi/2,n\pi + 3\pi/4)</math>: increasing and concave up<br><math>(n\pi + 3\pi/4, (n+1)\pi)</math>: increasing and concave down.
	\|}		\|}

Vipul: /* Key data */

2011-08-26T17:50:19Z

Key data

← Older revision		Revision as of 17:50, 26 August 2011
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	\| expression as a [[sinusoidal function]] plus a constant function \|\| <math>(1/2) + (\cos(2x))/2</math>		\| expression as a [[sinusoidal function]] plus a constant function \|\| <math>(1/2) + (\cos(2x))/2</math>
	\|-		\|-
	\| important symmetries \|\| [[even function]]<br>More generally, has [[mirror symmetry]] about all vertical lines <math>x = n\pi</math>, <math>n</math> an integer.<br>Also, has half turn symmetry about all points of the form <math>(n \pi + \pi/2,0)</math>, i.e., all points of inflection.		\| important symmetries \|\| [[even function]]<br>More generally, has [[mirror symmetry]] about all vertical lines <math>x = n\pi/2</math>, <math>n</math> an integer.<br>Also, has half turn symmetry about all points of the form <math>(n \pi + \pi/4,1/2)</math>, i.e., all points of inflection.
	\|-		\|-
	\| interval description based on increase/decrease and concave up/down \|\| For each integer <math>n</math>, the interval from <math>n\pi</math> to <math>(n+1)\pi</math> is subdivided into four parts:<br><math>(n\pi,n\pi + \pi/4)</math>: decreasing and concave down<br><math>(n\pi + \pi/4,n\pi + \pi/2)</math>: decreasing and concave up<br><math>(n\pi + \pi/2,n\pi + 3\pi/4)</math>: increasing and concave up<br><math>(n\pi + 3\pi/4, (n+1)\pi)</math>: increasing and concave down.		\| interval description based on increase/decrease and concave up/down \|\| For each integer <math>n</math>, the interval from <math>n\pi</math> to <math>(n+1)\pi</math> is subdivided into four parts:<br><math>(n\pi,n\pi + \pi/4)</math>: decreasing and concave down<br><math>(n\pi + \pi/4,n\pi + \pi/2)</math>: decreasing and concave up<br><math>(n\pi + \pi/2,n\pi + 3\pi/4)</math>: increasing and concave up<br><math>(n\pi + 3\pi/4, (n+1)\pi)</math>: increasing and concave down.
	\|}		\|}

Vipul: /* Key data */

2011-08-26T17:49:21Z

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← Older revision		Revision as of 17:49, 26 August 2011
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	\|-		\|-
	\| expression as a [[sinusoidal function]] plus a constant function \|\| <math>(1/2) + (\cos(2x))/2</math>		\| expression as a [[sinusoidal function]] plus a constant function \|\| <math>(1/2) + (\cos(2x))/2</math>
			\|-
			\| important symmetries \|\| [[even function]]<br>More generally, has [[mirror symmetry]] about all vertical lines <math>x = n\pi</math>, <math>n</math> an integer.<br>Also, has half turn symmetry about all points of the form <math>(n \pi + \pi/2,0)</math>, i.e., all points of inflection.
	\|-		\|-
	\| interval description based on increase/decrease and concave up/down \|\| For each integer <math>n</math>, the interval from <math>n\pi</math> to <math>(n+1)\pi</math> is subdivided into four parts:<br><math>(n\pi,n\pi + \pi/4)</math>: decreasing and concave down<br><math>(n\pi + \pi/4,n\pi + \pi/2)</math>: decreasing and concave up<br><math>(n\pi + \pi/2,n\pi + 3\pi/4)</math>: increasing and concave up<br><math>(n\pi + 3\pi/4, (n+1)\pi)</math>: increasing and concave down.		\| interval description based on increase/decrease and concave up/down \|\| For each integer <math>n</math>, the interval from <math>n\pi</math> to <math>(n+1)\pi</math> is subdivided into four parts:<br><math>(n\pi,n\pi + \pi/4)</math>: decreasing and concave down<br><math>(n\pi + \pi/4,n\pi + \pi/2)</math>: decreasing and concave up<br><math>(n\pi + \pi/2,n\pi + 3\pi/4)</math>: increasing and concave up<br><math>(n\pi + 3\pi/4, (n+1)\pi)</math>: increasing and concave down.
	\|}		\|}

Vipul: /* Key data */

2011-08-26T11:13:07Z

Key data

← Older revision		Revision as of 11:13, 26 August 2011
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	\| [[second derivative]] \|\| <math>x \mapsto 2 \cos(2x)</math>		\| [[second derivative]] \|\| <math>x \mapsto 2 \cos(2x)</math>
	\|-		\|-
	\| higher derivatives \|\| <math>2^{~~n − 1~~}</math> times an expression that is <math>\pm \sin</math> or <math>\pm \cos</math> of <math>2x</math>, depending on the remainder of <math>n</math> mod 4.		\| higher derivatives \|\| <math>2^{n−1}</math> times an expression that is <math>\pm \sin</math> or <math>\pm \cos</math> of <math>2x</math>, depending on the remainder of <math>n</math> mod 4.
	\|-		\|-
	\| [[antiderivative]] \|\| <math>\frac{x}{2} + \frac{\sin(2x)}{4} + C</math>		\| [[antiderivative]] \|\| <math>\frac{x}{2} + \frac{\sin(2x)}{4} + C</math>

Vipul: /* Key data */

2011-08-26T11:12:21Z

Key data

← Older revision		Revision as of 11:12, 26 August 2011
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	\| [[range]] \|\| <math>[0,1]</math>, i.e., <math>\{ y \mid 0 \le y \le 1 \}</math>.		\| [[range]] \|\| <math>[0,1]</math>, i.e., <math>\{ y \mid 0 \le y \le 1 \}</math>.
	\|-		\|-
	\| [[period]] <math>\pi</math>, i.e., <math>180\,^\circ</math>.		\| [[period]] \|\| <math>\pi</math>, i.e., <math>180\,^\circ</math>.
	\|-		\|-
	\| [[local maximum value]]s and points of attainment \|\| All local maximum values are equal to 1, and they are attained at integer multiples of <math>\pi</math>.		\| [[local maximum value]]s and points of attainment \|\| All local maximum values are equal to 1, and they are attained at integer multiples of <math>\pi</math>.

Vipul: Created page with "{{particular function}} {{angular function radian convention}} ==Definition== This function, denoted $\cos^2$ , is defined as the [[defining ingredient::composite of..."

2011-08-26T11:12:01Z

Created page with "{{particular function}} {{angular function radian convention}} ==Definition== This function, denoted <math>\cos^2</math>, is defined as the [[defining ingredient::composite of..."

New page

{{particular function}}

{{angular function radian convention}}

==Definition==

This function, denoted <math>\cos^2</math>, is defined as the [[defining ingredient::composite of two functions|composite]] of the [[defining ingredient::square function]] and the [[defining ingredient::cosine function]]. Explicitly, it is the map:

<math>x \mapsto (\cos x)^2</math>

For brevity, we write <math>(\cos x)^2</math> as <math>\cos^2x</math>.

==Key data==

{| class="sortable" border="1"
! Item !! Value
|-
| Default [[domain]] || all [[real number]]s, i.e., all of <math>\R</math>.
|-
| [[range]] || <math>[0,1]</math>, i.e., <math>\{ y \mid 0 \le y \le 1 \}</math>.
|-
| [[period]] <math>\pi</math>, i.e., <math>180\,^\circ</math>.
|-
| [[local maximum value]]s and points of attainment || All local maximum values are equal to 1, and they are attained at integer multiples of <math>\pi</math>.
|-
| [[local minimum value]]s and points of attainment || All local minimum values are equal to 0, and they are attained at odd integer multiples of <math>\pi/2</math>.
|-
| [[point of inflection|points of inflection]] (both coordinates) || odd multiples of <math>\pi/4</math>, with value <math>1/2</math> at each point
|-
| [[derivative]] || <math>x \mapsto -\sin (2x)</math> i.e., negative of the [[double-angle sine function]].
|-
| [[second derivative]] || <math>x \mapsto 2 \cos(2x)</math>
|-
| higher derivatives || <math>2^{n − 1}</math> times an expression that is <math>\pm \sin</math> or <math>\pm \cos</math> of <math>2x</math>, depending on the remainder of <math>n</math> mod 4.
|-
| [[antiderivative]] || <math>\frac{x}{2} + \frac{\sin(2x)}{4} + C</math>
|-
| [[mean value]] over a period || <math>1/2</math>
|-
| expression as a [[sinusoidal function]] plus a constant function || <math>(1/2) + (\cos(2x))/2</math>
|-
| interval description based on increase/decrease and concave up/down || For each integer <math>n</math>, the interval from <math>n\pi</math> to <math>(n+1)\pi</math> is subdivided into four parts:<br><math>(n\pi,n\pi + \pi/4)</math>: decreasing and concave down<br><math>(n\pi + \pi/4,n\pi + \pi/2)</math>: decreasing and concave up<br><math>(n\pi + \pi/2,n\pi + 3\pi/4)</math>: increasing and concave up<br><math>(n\pi + 3\pi/4, (n+1)\pi)</math>: increasing and concave down.
|}

Cosine-squared function - Revision history

Vipul: /* Key data */

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Vipul: Created page with "{{particular function}} {{angular function radian convention}} ==Definition== This function, denoted \cos^2, is defined as the [[defining ingredient::composite of..."

Vipul: Created page with "{{particular function}} {{angular function radian convention}} ==Definition== This function, denoted $\cos^2$ , is defined as the [[defining ingredient::composite of..."