# Difference between revisions of "Sine-squared function"

From Calculus

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| [[derivative]] || <math>x \mapsto \sin(2x) = 2\sin x \cos x</math>, i.e., [[double-angle sine function]]. | | [[derivative]] || <math>x \mapsto \sin(2x) = 2\sin x \cos x</math>, i.e., [[double-angle sine function]]. | ||

+ | |- | ||

+ | | [[second derivative]] || <math>x \mapsto -2\cos(2x)</math> | ||

+ | |- | ||

+ | | <math>n^{th}</math> derivative || <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 <math>4</math> | ||

|- | |- | ||

| [[antiderivative]] || <math>x \mapsto \frac{x}{2} - \frac{\sin(2x)}{4} + C</math> | | [[antiderivative]] || <math>x \mapsto \frac{x}{2} - \frac{\sin(2x)}{4} + C</math> |

## Revision as of 10:47, 26 August 2011

## Definition

This function, denoted , is defined as the composite of the square function and the sine function. Explicitly, it is the map:

For brevity, we write as .

## Key data

Item | Value |
---|---|

Default domain | all real numbers, i.e., all of |

range | , i.e., absolute maximum value: 1, absolute minimum value: 0 |

period | , i.e., |

local maximum value and points of attainment | All local maximum values are equal to 1, and are attained at odd integer multiples of . |

local minimum value and points of attainment | All local minimum values are equal to 0, and are attained at integer multiples of . |

points of inflection (both coordinates) | odd multiples of , with value 1/2 at each point. |

derivative | , i.e., double-angle sine function. |

second derivative | |

derivative | times an expression that is or of , depending on the remainder of mod |

antiderivative | |

mean value over a period | 1/2 |

expression as a sinusoidal function plus a constant function |