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Chapter 7: Problem 1

\(f(t)=t, \quad 0

Short Answer

Expert verified
The periodic function \(f(t) = t\) for the interval \(0 < t < 2\) with a period of 2 can be extended to \(f(t) = t- 2n\) for \(t > 2\), where \(n\) is an integer, and \(2n\) is the closest value under \(t\).

Step by step solution

01

Function Definition

Define the given function \(f(t) = t\) in the interval \(0 < t < 2\).
02

Extension of the function

Since \(f(t)\) has period 2, it will repeat its pattern every 2 units. Hence, the value of \(f(t)\) for \(2 < t < 4\) would be the same as \(f(t)\) for \(0 < t < 2\). Therefore, for \(t > 2\), the function can be defined as \(f(t) = t - 2n\) where \(n\) is an integer, and \(2n\) is the closest value under \(t\).
03

Representation of the function

The function can be graphically represented by a line in the range \(0 < t < 2\) and then repeating this line every 2 units in both the positive and negative directions

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Periodic Functions
A periodic function repeats its values in regular intervals. In the exercise, the function is given as \( f(t) = t \) with a period of 2. This means every 2 units on the \( t \)-axis, the function will undergo the same cycle. Periodic functions are fundamental in signal processing and many areas of physics, because they model repetitive or cyclical phenomena.
  • Definition: A function \( f(t) \) is periodic with period \( T \) if \( f(t) = f(t + T) \) for all \( t \).
  • Example: The sine and cosine functions are classic examples of periodic functions with a period of \( 2\pi \).
Understanding the periodicity means recognizing this pattern's presence in visual or graphical representations.
Function Extension
Function extension refers to expanding the definition of a function beyond its original interval or domain while maintaining its properties. In the case of the function \( f(t) = t \) with a period of 2, the extension means repeating the defined line pattern over intervals that are multiples of the period. This ensures the function behaves consistently over an infinite range of \( t \).
  • To extend \( f(t) = t \) from the interval \( 0 < t < 2 \) to the interval \( 2 < t < 4 \), use \( f(t) = t - 2n \) where \( n \) is an integer.
  • This method ensures the extended function mimics the original behavior at every shift by the period of 2.
The concept of extending functions is crucial in solving problems involving repeating signals or patterns.
Function Representation
Function representation involves expressing the essence of the function visually or in an alternative form. For \( f(t) = t \), the function can be graphically depicted as a line segment within its defined interval, and it will continue this line pattern periodically.
When visualized:
  • In \( 0 < t < 2 \), \( f(t) \) is a line increasing at a constant rate.
  • In successive intervals like \( 2 < t < 4 \), the same pattern repeats, starting again from the lower bound of the interval.
Graphical representation helps in understanding the function's behavior over multiple cycles. It allows one to "see" the periodicity in a clear format, aiding in comprehension and practical application.
Step-by-Step Solutions
Step-by-step solutions break down the problem-solving process into manageable segments, allowing a student to track the logical progression of a solution. For the function given, the solution proceeds as follows:
  • Step 1: Clearly define the function in its original interval, \( 0 < t < 2 \).
  • Step 2: Extend the function to cover additional periods while maintaining the function's fundamental rules and characteristics.
  • Step 3: Graphically represent the function to visualize how it repeats over its range.
This methodic approach ensures that each aspect of the problem is addressed sequentially, leading to a thorough understanding of the exercise鈥檚 requirements. Step-by-step solutions are invaluable for retaining knowledge and applying similar logic to new problems.

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