Vipul: /* Definition */

2011-09-22T17:33:41Z

Definition

← Older revision		Revision as of 17:33, 22 September 2011
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	The function also has the following description using a [[piecewise definition of function\|piecewise definition]]:		The function also has the following description using a [[piecewise definition of function\|piecewise definition]]:

	<math>x \mapsto \left\lbrace \begin{array}{rl} \sin x, & 2n\pi \le x \le (2n + 1)\pi, n \in \mathbb{Z} \\ 0, & (2n - 1)\pi \le x \le 2n\pi, n \in mathbb{Z} \end{array}\right.</math>		<math>x \mapsto \left\lbrace \begin{array}{rl} \sin x, & 2n\pi \le x \le (2n + 1)\pi, n \in \mathbb{Z} \\ 0, & (2n - 1)\pi \le x \le 2n\pi, n \in \mathbb{Z} \end{array}\right.</math>

	==Key data==		==Key data==

Vipul: Created page with "{{particular function}} ==Definition== This function of one variable, denoted $\sin^+$ is defined as the [[defining ingredient::composite of two functions|compos..."

2011-09-22T17:33:25Z

Created page with "{{particular function}} ==Definition== This function of one variable, denoted <math>\sin^+</math> is defined as the [[defining ingredient::composite of two functions|compos..."

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{{particular function}}

==Definition==

This [[function]] of one variable, denoted <math>\sin^+</math> is defined as the [[defining ingredient::composite of two functions|composite]] of the [[defining ingredient::positive part function]] and the [[defining ingredient::sine function]].

Alternatively, it can be defined as the function <math>\max \{ \sin x, 0 \}</math>

The function also has the following description using a [[piecewise definition of function|piecewise definition]]:

<math>x \mapsto \left\lbrace \begin{array}{rl} \sin x, & 2n\pi \le x \le (2n + 1)\pi, n \in \mathbb{Z} \\ 0, & (2n - 1)\pi \le x \le 2n\pi, n \in mathbb{Z} \end{array}\right.</math>

==Key data==

{| class="sortable" border="1"
! Item !! Value
|-
| [[domain]] || all [[real number]]s, i.e., all of <matH>\R</math>
|-
| [[range]] || the [[closed interval]] <math>[0,1]</math><br>[[absolute maximum value]]: 1, [[absolute minimum value]]: 0
|-
| [[period]] || <math>2\pi</math>
|-
| [[local maximum value]]s and points of attainment || 1, at number of the form <math>2n\pi + \pi/2, n \in \mathbb{Z}</math>
|-
| [[local minimum value]]s and points of attainment || 0, at all points <math>x</math> in intervals of the form <math>[(2n - 1)\pi,2n\pi]</math>, with <math>n \in \mathbb{Z}</math>.
|-
| [[first derivative]] || <math>x \mapsto \left\lbrace \begin{array}{rl} \cos x, & 2n\pi < x < (2n + 1)\pi\\ 0, & (2n - 1)\pi < x < 2n\pi\end{array}\right.</math><br>The function is not differentiable at integer multiples of <math>\pi</math>. At even multiples of <math>\pi</math>, the [[left hand derivative]] is 0 and the [[right hand derivative]] is 1. At odd multiples of <math>\pi</math>, the [[left hand derivative]] is -1 and the [[right hand derivative]] is 0.
|-
| [[second derivative]] || <math>-\sin^+</math>, except at integer multiples of <math>\pi</math>, where it is undefined.
|-
| mean value over a period || <math>1/\pi</math>
|}

Positive part of sine function - Revision history

Vipul: /* Definition */

Vipul: Created page with "{{particular function}} ==Definition== This function of one variable, denoted \sin^+ is defined as the [[defining ingredient::composite of two functions|compos..."

Vipul: Created page with "{{particular function}} ==Definition== This function of one variable, denoted $\sin^+$ is defined as the [[defining ingredient::composite of two functions|compos..."