Question

The given function \(f(t)\) is periodic with fundamental period \(T\); therefore, \(f(t+T)=f(t)\). Use the information given to sketch the graph of \(f(t)\) over the time interval \(0 \leq t \leq 4 T\). (a) \(f(t)=t(2-t), \quad 0 \leq t<2, \quad T=2\) (b) \(f(t)=\left\\{\begin{array}{ll}1, & 0 \leq t \leq \frac{3}{2} \\ 0, & \frac{3}{2}

Short answer

Question: For each given periodic function, identify the fundamental period and sketch the graph of the function over one complete period. (a) \(f(t) = t(2-t)\) in the interval \(0 \leq t < 2\), with \(T=2\). (b) A piecewise function: \(\displaystyle f(t)=\left\\{\begin{array}{ll}1, & 0 \leq t \leq \frac{3}{2} \\\ 0, & \frac{3}{2}<t<2\end{array}\right.\) in the interval \(0 \leq t < 2\), with \(T=2\). (c) A linear piecewise function: \(\displaystyle f(t)=\left\\{\begin{array}{cl}t, & 0 \leq t \leq 2 \\\ 4-t, & 2<t<4\end{array}\right.\) in the interval \(0 \leq t < 4\), with \(T=4\). (d) \(f(t) = -1+t\) in the interval \(0 \leq t < 2\), with \(T=2\). (e) \(f(t) = 2e^{-t}\) in the interval \(0 \leq t < 1\), with \(T=1\). (f) \(f(t) = \sin t\) in the interval \(0 \leq t < \pi\), with \(T=\pi\). (g) A piecewise function: \(\displaystyle f(t)=\left\\{\begin{array}{cl}2 \sin \pi t, & 0 \leq t \leq 1 \\\ 0, & 1<t<2\end{array}\right.\) in the interval \(0 \leq t < 2\), with \(T=2\).

Step-by-step solution

(a) Analyze the given function for part (a)

For this case, we are given \(f(t) = t(2-t)\) in the interval \(0 \leq t < 2\) and \(T=2\). Since the function is periodic, we know that \(f(t+T) = f(t)\). Our task is to sketch the function over the time interval \(0 \leq t \leq 4T\).

(a) Sketch the graph of the function for part (a)

Start by plotting the given function \(f(t) = t(2-t)\) in the interval \(0 \leq t < 2\). Then, sketch the function by copying the same pattern over the next three periods (\(2 \leq t < 4\), \(4 \leq t < 6\), and \(6 \leq t \leq 8\)) to cover the entire interval \(0 \leq t \leq 4T\).

(b) Analyze the given function for part (b)

For this case, we are given a piecewise function: \(\displaystyle f(t)=\left\\{\begin{array}{ll}1, & 0 \leq t \leq \frac{3}{2} \\\ 0, & \frac{3}{2}<t<2\end{array}\right.\) with the fundamental period \(T=2\). Our task is to sketch the function over the time interval \(0 \leq t \leq 4T\).

(b) Sketch the graph of the function for part (b)

Start by plotting the given piecewise function in the interval \(0 \leq t < 2\). Then, sketch the function by copying the same pattern over the next three periods (\(2 \leq t < 4\), \(4 \leq t < 6\), and \(6 \leq t \leq 8\)) to cover the entire interval \(0 \leq t \leq 4T\). Similarly, we can analyze and sketch the graphs for the remaining parts (c) to (g):

(c) Sketch the graph for part (c)

For part (c), we have a linear piecewise function: \(\displaystyle f(t)=\left\\{\begin{array}{cl}t, & 0 \leq t \leq 2 \\\ 4-t, & 2<t<4\end{array}\right.\) with the fundamental period \(T=4\). Sketch this function over the interval \(0 \leq t \leq 4T = 16\).

(d) Sketch the graph for part (d)

For part (d), we are given a linear function \(f(t) = -1+t\) in the interval \(0 \leq t < 2\) with the fundamental period \(T=2\). Sketch this function over the interval \(0 \leq t \leq 4T = 8\).

(e) Sketch the graph for part (e)

For part (e), we are given an exponential function \(f(t) = 2e^{-t}\) in the interval \(0 \leq t < 1\) with the fundamental period \(T=1\). Sketch this function over the interval \(0 \leq t \leq 4T = 4\).

(f) Sketch the graph for part (f)

For part (f), we are given a trigonometric function \(f(t) = \sin t\) in the interval \(0 \leq t < \pi\) with the fundamental period \(T=\pi\). Sketch this function over the interval \(0 \leq t \leq 4T = 4\pi\).

(g) Sketch the graph for part (g)

Finally, for part (g), we have another piecewise function: \(\displaystyle f(t)=\left\\{\begin{array}{cl}2 \sin \pi t, & 0 \leq t \leq 1 \\\ 0, & 1<t<2\end{array}\right.\) with the fundamental period \(T=2\). Sketch this function over the interval \(0 \leq t \leq 4T = 8\).

Key concepts

Fundamental Period

The fundamental period of a function is a key concept in understanding periodic functions. It refers to the smallest positive interval, denoted as \( T \), over which the function completes one full cycle and repeats itself. For any periodic function \( f(t) \), it holds that \( f(t + T) = f(t) \). This repetition property simplifies the analysis and sketching of these functions.

When dealing with real-world applications, the fundamental period allows us to predict the behavior of phenomena like sound waves, alternating current in electrical circuits, or even biological rhythms. It's important to identify this period as it provides insight into the oscillatory nature of the function. In the exercise, functions like \( f(t) = t(2-t) \) with a period \( T = 2 \), or \( f(t) = \sin t \) with a period \( T = \pi \), are examples of how the period dictates the pattern repeated across intervals.

Graph Sketching

Graph sketching of periodic functions requires understanding the behavior of the function across its fundamental period and then extending this pattern across multiple periods. The process starts with plotting the function within one period as precisely as possible, capturing its peaks, valleys, and any intercepts with the axes.

After sketching the initial period, copy this pattern to adjacent intervals to cover the entire specified domain. For instance, in the exercise, part (a) suggests sketching \( f(t) = t(2-t) \) initially on \( 0 \leq t < 2 \) and then repeating this sketch over \( 0 \leq t \leq 8 \).

• Begin with a clear plot of the function over one cycle, noting key points and symmetry.
• Repeat this plot for each period segment across the desired range.
• Ensure consistency by adhering to the fundamental period.

By following this methodical approach, sketching becomes a matter of recognizing cyclical patterns and ensuring precision in replication over the designated intervals.

Piecewise Functions

Piecewise functions are defined by different expressions over distinct intervals. Each interval has its own rule for how the function behaves, and this often results in pieces with different shapes or slopes.

For example, in part (b) of the exercise, we encounter a piecewise function defined as:\[ f(t) = \begin{cases} 1, & 0 \leq t \leq \frac{3}{2} \ 0, & \frac{3}{2} < t < 2 \end{cases} \]

The task in sketching piecewise functions involves plotting each segment separately, then combining them into a single graph over the entire period. The challenge often lies in neatly joining these segments, ensuring continuity where required, and accurately representing any discontinuities.

Care must be taken to evaluate the boundaries of each piece, paying attention to whether endpoints are included (denoted by closed or open intervals). This clarity is critical to accurately reflect the function’s behavior across differing segments.

Differential Equations

Though not directly solved in this exercise, differential equations represent the backbone in the study of periodic functions and their behavior as they can describe how these functions change. Solving a differential equation provides us with a function that can model systems exhibiting periodic behavior, such as oscillations.

In mathematical terms, a differential equation may look like:\[ \frac{d^2 y}{dt^2} + ky = 0 \] where solving involves finding a function \( y(t) \) that satisfies the equation. Cyclical and oscillatory solutions often arise, demonstrating either simple harmonic motion or more complex oscillations depending on the boundary conditions and initial values.

Differential equations are crucial in capturing not only the instantaneous rate of change but also the repetitive nature of phenomena, linking initial conditions to periodic solutions.

• They model scenarios like a vibrating string or a pendulum's swing.
• Understanding solutions involves recognizing connections between these equations and periodic functions.

Integrating knowledge of differential equations with periodic functions enhances comprehension of natural and engineered systems that exhibit periodicity.