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Chapter 7: Problem 546

\(f(t)= \begin{cases}\sin t, & 0 \leq t<2 \pi \\ 0 . & t \geq 2 \pi\end{cases}\)

Short Answer

Expert verified
The function \(f(t)\) is \(\sin t\) for \(0 \leq t < 2\pi\) and \(0\) for \(t \geq 2\pi\).

Step by step solution

01

Understanding the Problem

We are asked to analyze a piecewise function, \(f(t)\). The function is defined differently for the intervals \(0 \leq t < 2 \pi\) and \(t \geq 2\pi\). Our goal is to understand how \(f(t)\) behaves on these intervals.
02

Analyzing the First Interval

For the interval \(0 \leq t < 2 \pi\), the function is given by \(f(t) = \sin t\). This means that in this interval, \(f(t)\) behaves like the sine function, which oscillates between -1 and 1.
03

Analyzing the Second Interval

For the interval \(t \geq 2 \pi\), the function is defined as \(f(t) = 0\). Therefore, for any value of \(t\) greater than or equal to \(2\pi\), the function becomes a constant zero.
04

Sketching the Function

To visualize \(f(t)\), plot \(\sin t\) from \(t = 0\) to \(t = 2\pi\), where it completes one full oscillation. Then, plot a horizontal line along the \(t\)-axis for \(t \geq 2\pi\). This visualization helps to understand the abrupt change at \(t = 2\pi\).

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Key Concepts

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

Function Behavior Analysis
Analyzing a function's behavior is essential to understand how it performs under different conditions. A piecewise function like \(f(t)\) divides the domain into separate intervals, each with its own behavior and rules. In our exercise, we have two distinct intervals:
  • From \(0 \leq t < 2\pi\) where the function follows the behavior of the sine wave.
  • From \(t \geq 2\pi\) where the function flattens to zero.
Understanding the behavior in each segment allows you to predict and graph the function accurately. In mathematical terms, behavior analysis involves investigating if the function increases, decreases, oscillates, or remains steady within its intervals. In this exercise, \(f(t)\) oscillates in the first interval and stays constant in the second.
Sine Function
The sine function, denoted as \(\sin t\), is one of the most fundamental trigonometric functions. It generates a wave-like pattern, continuously oscillating between -1 and 1.

Important characteristics of the sine function include:
  • A period of \(2\pi\), which means it completes one full cycle over this interval.
  • An amplitude of 1, representing the peak values of the wave at +1 and -1.
  • Intercepts at multiple points with the horizontal axis (such as at 0, \(\pi\), and \(2\pi\)).
In our function \(f(t) = \sin t\) for \(0 \leq t < 2\pi\), it showcases a single period of the sine curve. Recognizing these elements helps understand how the function will behave and appear when graphed.
Function Intervals
Function intervals specify the domains for which different rules or expressions apply within a piecewise function. They are critical in breaking down the behavior of the function into manageable segments.

In our piecewise function \(f(t)\), there are two intervals clearly defined:
  • \(0 \leq t < 2\pi\) where \(f(t) = \sin t\)
  • \(t \geq 2\pi\) where \(f(t) = 0\)
By understanding each interval, we know where to apply different rules and predict the output. The first interval behaves dynamically, showing oscillations typical of sine, while the second becomes static, simplifying to zero and continuing indefinitely.
Graphing Functions
Graphing a piecewise function requires careful consideration of each segment's unique behavior. Let鈥檚 visualize \(f(t)\) by plotting these intervals:
  • First, sketch the sine curve starting from \(t = 0\) up to \(t = 2\pi\), noting its smooth wave pattern reaching from -1 to 1.
  • Next, at \(t = 2\pi\), draw a horizontal line along the x-axis, indicating that the function becomes zero from this point forward.
This graphical representation provides insight into both the continuity and discontinuity of the function. It highlights how \(f(t)\) smoothly transitions from an oscillatory pattern to a constant, emphasizing the importance of understanding both components in a piecewise function.

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Most popular questions from this chapter

Use the Laplace transform to solve the given differential equation subject to the indicated initial conditions. $$ y^{\prime}+4 y=e^{-4}, \quad y(0)=2 $$

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