Tool to compute the period of a function: the value t such that the function repeats itself: f(x+t)=f(x-t)=f(x), that is the case for trigo functions (cos, sin, etc.)

Period of a Function - dCode

Tag(s) : Functions

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The period $ t $ of a periodic function $ f(x) $ is the smallest value $ t $ such that $$ f(x+t) = f(x) $$

Graphically, its curve is repeated over the interval of each period. The function is equal to itself for every cycle of length $ t $ (it presents a pattern/graph that is repeated by translation).

The value of the period $ t $ is also called the periodicity of the function or fundamental period.

To find the period $ t $ of a signal or a function $ f(x) $, demonstrate that $$ f(x+t)=f(x) $$

__Example:__ The trigonometric function $ \sin(x + 2\pi) = \sin(x) $ so $ \sin(x) $ is periodic of period $ 2\pi $

Trigonometric/sinusoidal functions are usually periodic, with a period $ 2\pi $, to guess the period, try multiples of pi for value $ t $.

If the period is equal to 0, then the function is not periodic.

Any periodic function of period $ t $ repeats every $ t $ values. To predict the cycle value of a periodic function, for a value $ x $ calculate $ x_t = x \mod t $ (modulo t) and find the known value of $ f(x_t) = f(x) $

__Example:__ The function $ f(x) = \cos(x) $ has a period of $ 2\pi $, the value for $ x = 9 \pi $ is the same as for $ x \equiv 9 \pi \mod 2 \pi \equiv \pi \mod 2 \pi $ and therefore $ \cos(9 \pi) = \cos(\pi) = -1 $

The amplitude is the absolute value of the non-periodic part of the function.

__Example:__ $ a \sin(x) $ has for amplitude $ | a | $

The demonstration of the existence of a period $ t $ for a function $ f $ consists in calculating if the equation $ f(x+t)=f(x) $ is true.

If $ f $ is periodic, then it exists a real not null such as $$ f(x+t)=f(x) $$

Demonstration consists in proving that it is impossible. For example with a reductio ad absurdum or performing a calculation that leads to a contradiction.

The most common periodic functions are trigonometric functions based on sine and cosine functions (which have a period of 2 Pi).

Function | Period |
---|---|

Sine $ \sin(x) $ | $ 2\pi $ |

Cosine $ \cos(x) $ | $ 2\pi $ |

Tangent $ \tan(x) $ | $ \pi $ |

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*Period of a Function* on dCode.fr [online website], retrieved on 2024-06-24,

- Period of a Function Calculator
- What is the period of a function? (Definition)
- How to find the period of a function?
- How to find the value f(x) of a periodic function?
- How to find the amplitude of a periodic function?
- How to prove the periodicity of a function?
- How to prove that a function is not periodic?
- What are usual periodic functions?

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