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Periodic Functions

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Periodic Functions

Study Guide

Key Definition

A function $f$ is periodic if there exists a real number $P$ such that $f(x + P) = f(x)$ for all $x$.

Important Notes

A periodic function repeats its values over regular intervals.
The smallest positive value for which $f(x + P) = f(x)$ is called the period.
Common periodic functions include $\sin(x)$ and $\cos(x)$.
If you know a function over one period, you know it everywhere.
The graph of a periodic function is a repeating pattern.

Mathematical Notation

$f(x + P) = f(x)$ defines periodicity

$+$ represents addition

$-$ represents subtraction

$\sin$ is the sine function

$\cos$ is the cosine function

Remember to use proper notation when solving problems

Why It Works

Periodic functions allow us to model real-world phenomena that repeat in cycles, such as waves and oscillations, using $f(x + P) = f(x)$.

Remember

A periodic function repeats its values in a predictable cyclic pattern, described by its period $P$.

Quick Reference

Period of $\sin(x)$:$2\pi$

Period of $\cos(x)$:$2\pi$

Understanding Periodic Functions

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Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

A periodic function repeats its pattern after a fixed interval, known as the period $P$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the period of the function $f(x) = \sin(x)$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Imagine a Ferris wheel that completes one full rotation every 10 minutes. This motion can be modeled by a periodic function.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Suppose $f(x)$ is a periodic function with period $4$. What can you say about $f(x + 8)$?

Challenge Quiz

Single Choice Quiz

Advanced

If $f(x)$ has period $3$ and $f(2) = 5$, what is $f(11)$?

Recap

Watch & Learn

Review key concepts and takeaways