3.1: Periodic Phenomena | AP Precalculus

How it works

A periodic function repeats after a fixed positive input shift. The smallest positive repeat time, when it exists, is the fundamental period. A repeated output value alone is insufficient to establish a full repeat of the function.

f(t+T)=f(t)\quad\text{for every }t\text{ in the repeating domain}

Worked examples

Example 1

A wheel completes a rotation in $\displaystyle 30$ seconds. What period does a seat-height model have?

The seat-height cycle repeats every $\displaystyle 30$ seconds.

Example 2

For $\displaystyle f(t)=\cos(\pi t/4)$ , find the fundamental period.

$\displaystyle T=2\pi/(\pi/4)=8$ .

Problems

Problem 3.1.1

For $\displaystyle D(t)=5+2\cos(\pi t/6)$ , find a repeat time and evaluate $\displaystyle D(0),D(6),D(12)$ .

Depth oscillates between 3 and 7 meters over a 12-hour repeat time. — Time is in hours and depth in meters.

Show worked solution

The period is $\displaystyle 12$ . The depths are $\displaystyle 7,3,7$ meters respectively.

Depth oscillates between 3 and 7 meters over a 12-hour repeat time. The solution adds labeled relationships. — Maximum depth = 7 m, minimum = 3 m, period = 12 h.

Problem 3.1.2

Does $\displaystyle f(0)=f(2)$ alone prove f has period $\displaystyle 2$ ?

Show worked solution

No. A period requires $\displaystyle f(t+2)=f(t)$ for every relevant t. For example, $\displaystyle (t-1)^2$ has equal values at $\displaystyle 0$ and $\displaystyle 2$ but is not periodic.

Problem 3.1.3

A signal repeats every $\displaystyle 0.25$ seconds. Find its frequency in cycles per second.

Show worked solution

Frequency is the reciprocal of period: $\displaystyle 1/0.25=4$ cycles per second.

Lesson Problems PDF Solutions PDF Unit MCQ

How it works

Worked examples

Example 1

Example 2

Problems

Problem 3.1.1

Problem 3.1.2

Problem 3.1.3

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