Question

Sketch the graph of \(f(t)\), state the period of \(f(t)\), and find \(\mathcal{L}\\{f(t)\\}\). $$ f(t)=|\sin 2 t| $$

Short answer

Answer: The period of the function \(f(t)=|\sin(2t)|\) is \(\pi\), and its Laplace Transform is \(\mathcal{L}\{f(t)\} = \frac{s^2}{s^4+8s^2+16}\).

Step-by-step solution

Sketch the graph of \(f(t)\)

To sketch the graph of \(f(t)=|\sin(2t)|\), we need to first analyze the behavior of the sine function and then apply the transformation given by the absolute value. The sine function has a period of \(2\pi\). When we look at \(\sin(2t)\), we observe that the function is compressed horizontally by a factor of 2, leading to a new period of \(\pi\). However, since we are considering the absolute value of the sine function, the graph will be non-negative, having a mirror effect from the horizontal line \(y=0\). Now, let's sketch the graph of the function \(f(t)=|\sin(2t)|\) considering these properties.

State the period of \(f(t)\)

For the sine function, the period is \(2\pi\). When we have the function \(\sin(2t)\), the period becomes half of the original sine function, which is \(\pi\). Since the absolute value doesn't affect the period, the period of the function \(f(t)=|\sin(2t)|\) remains \(\pi\).

Find Laplace Transform of \(f(t)\)

To find the Laplace Transform of \(f(t) = |\sin(2t)|\), we'll first need to express the absolute value function as a piecewise function: $$ f(t) = \left\{ \begin{array}{ll} \sin(2t) & \quad 0 \le 2t \le \pi \\ -\sin(2t) & \quad \pi \le 2t \le 2\pi \end{array} \right. $$ With the piecewise function, we can define the Laplace Transform as follows: $$\mathcal{L}\{f(t)\} = \mathcal{L}\{\sin(2t)\} \Big|_0^{\pi} - \mathcal{L}\{\sin(2t)\} \Big|_{\pi}^{2\pi}$$ Using the Laplace Transform property for the sine function: \(\mathcal{L}\{\sin(at)\}=\frac{a}{s^2 + a^2}\), we can write: $$ \mathcal{L}\{f(t)\} = \frac{2}{s^2+4} \Big|_0^{\pi} - \frac{2}{s^2+4} \Big|_{\pi}^{2\pi} $$ Now, let's find the Laplace Transform of the function for each interval: $$\mathcal{L}\{f(t)\} = \int_0^{\pi} e^{-st}(\sin(2t))dt - \int_{\pi}^{2\pi} e^{-st}(-\sin(2t))dt$$ After solving the integrals, we get: $$\mathcal{L}\{f(t)\} = \frac{s^2}{s^4+8s^2+16}$$ So, the Laplace Transform of the function \(f(t)=|\sin(2t)|\) is \(\mathcal{L}\{f(t)\} = \frac{s^2}{s^4+8s^2+16}\).

Key concepts

Absolute Value Function

The absolute value function is a fascinating concept in mathematics, often symbolized by vertical bars, like \(|x|\). Its key characteristic is that it turns negative values into positive values, while positive numbers and zero remain unchanged. This is achieved by considering the distance a number is from zero on the number line, ignoring the direction.
For instance, the absolute value of -3, written as \(|-3|\), is 3, because -3 is three units away from zero.
The function is essential in various mathematical fields and helps deal with expressions that involve different sign configurations without the negative influence on subsequent operations.
When applied to a function, like \(f(t) = |\sin(2t)|\), the absolute value affects output values. This absolute transformation reflects any part of the sine wave usually below the horizontal axis (negative values) upwards, maintaining all other aspects of the sine wave intact. This change creates a graph that is always non-negative, following the sine wave's basic shape but only in its positive depiction.

Periodicity

Periodicity refers to the idea of a function repeating its values at regular intervals, known as periods. Think of it as the rhythm or cycle that a function follows over its graph.
For the sine function \(\sin(2t)\), its periodicity is influenced by its argument \(2t\). The regular base sine function \(\sin(t)\) repeats every \({2\pi}\). However, \(\sin(2t)\) compresses this cycle, effectively doubling its frequency, and thus halves the period to \(\pi\).
This means that every \(\pi\) units along the horizontal axis, the sine function completes a full wave cycle.
Even when transformed by an absolute value to \(|\sin(2t)|\), the periodicity remains unchanged. The function still completes its cycle every \(\pi\). This is crucial when analyzing functions over intervals as it ensures predictability and regularity in behavior.

Piecewise Function

A piecewise function comprises multiple sub-functions, each defined over a specific interval of the domain. This is particularly effective when detailing functions that behave differently based on input values.
In the case of \(f(t) = |\sin(2t)|\), the absolute value can be expressed as a piecewise function: If \(|\sin(2t)|\) is broken down, it takes the value of \(\sin(2t)\) when \(0 \leq 2t \leq \pi\), and \(-\sin(2t)\) when \(\pi \leq 2t \leq 2\pi\).
This breakdown helps simplify the process of applying the Laplace Transform or solving other complex integrations. By addressing each section independently within preset intervals, one can ensure accurate calculations and transformations, crucial for solving problems in calculus and engineering.
Thus, the piecewise function format provides a powerful structure to effectively handle diverse function behaviors over distinct domain intervals.