Problem 13 - Differential Equations and Linear Algebra

Question

Make a sketch of the given function on \([0, \infty)\) and express it in terms of the unit step function. $$f(t)=\left\\{\begin{array}{cc}0, & 0 \leq t<2, \\\3-t, & 2 \leq t<4 ,\\\\-1, & t \geq 4, \end{array}\right.$$

Step-by-Step Solution

Answer

In summary, the given function can be sketched on \([0, \infty)\) as a horizontal line at 0 for \(0 \leq t<2\), a straight line with a slope of -1 for \(2 \leq t<4\), and a horizontal line at -1 for \(t \geq 4\). The function can be expressed in terms of the unit step function as: \[f(t)=0\cdot u(t) + (3-t)\cdot [u(t-2) - u(t-4)] -1\cdot u(t-4).\]

Step 1: Understanding the unit step function

The unit step function, also known as the Heaviside step function, is defined as: \[ u(t) = \left\\{ \begin{array}{cc} 0, & t<0 , \\\\ 1, & t \geq 0 . \end{array} \right. \] The unit step function is useful for describing piecewise functions, and combinations of unit step functions can be used to define the given function.

Step 2: Analyze the given function

The given function is defined as: \[ f(t) = \left\\{ \begin{array}{cc} 0, & 0 \leq t<2 , \\\\ 3-t, & 2 \leq t<4 ,\\\\ -1, & t \geq 4 . \end{array} \right. \] We can note that the function has three separate pieces: \(0\) for \(0 \leq t<2\), \(3-t\) for \(2 \leq t<4\), and \(-1\) for \(t \geq 4\).

Step 3: Sketch the function

To sketch the given function on \([0, \infty)\), follow these steps: 1. For \(0 \leq t<2\), the function is equal to \(0\). The graph will be a horizontal line segment along the x-axis between 0 and 2. 2. For \(2 \leq t<4\), the function is equal to \(3-t\). The graph will be a straight line segment with a slope of -1, starting at t=2 with a value of 1 and ending at t=4 with a value of -1. 3. For \(t\geq 4\), the function is equal to \(-1\). The graph will be a horizontal line segment parallel to the x-axis at a value of \(-1\) going from t=4 to \(\infty\).

Step 4: Express the given function in terms of the unit step function

To express the function in terms of the unit step function, we can represent each piece as a combination of unit step functions: 1. \(0\) for \(0 \leq t<2\): this part can be represented as \(0 \cdot u(t)\). 2. \(3-t\) for \(2 \leq t<4\): this part can be represented as \((3-t) \cdot [u(t-2) - u(t-4)]\). 3. \(-1\) for \(t \geq 4\): this part can be represented as \((-1) \cdot u(t-4)\). Now, we combine all these parts into a single expression: \[f(t)=0\cdot u(t) + (3-t)\cdot [u(t-2) - u(t-4)] -1\cdot u(t-4).\]

Key Concepts

Heaviside step function

The Heaviside step function, commonly denoted as the unit step function, is a fundamental mathematical tool that greatly simplifies the representation of piecewise functions.

Imagine flipping a switch: before the flip, the switch is 'off' (0), and after the flip, it's 'on' (1). Mathematically, this is expressed as:
\[ u(t) = \begin{cases} 0, & t<0, \1, & t \geq 0. \end{cases} \]
The Heaviside function 'turns on' at a certain point, marking a transition from zero to one. This can be shifted along the time axis, so if we want the function to turn on at time 2, we'd write \(u(t-2)\).
In the context of our exercise, we place unit step functions at the transitions of our piecewise function to properly define it for different intervals. The unit step function eases the task of handling complex piecewise behaviors by allowing us to 'activate' or 'deactivate' certain function parts depending on the value of \(t\).

Piecewise functions

Piecewise functions are like a mosaic, consisting of different function pieces equipped to handle different intervals of input values. They allow for the display of multiple behaviors or rules within a single function, based on the input.

For the given function \(f(t)\), we have three rules:

• \(0\) for \(0 \leq t<2\)
• \(3-t\) for \(2 \leq t<4\)
• \(-1\) for \(t \geq 4\)

The function switches its behavior at \(t=2\) and \(t=4\). By stating the function in terms of the Heaviside step function, each 'piece' is matched with an interval where it's active, and then they are smoothly assembled into a concise formula that represents the entire piecewise function.

This approach emphasizes clarity and precision, ensuring that we only 'activate' the part of the function that's relevant to the interval of \(t\) we’re examining. It's a powerful technique for analyzing systems that exhibit distinct states or phases.

Graphing functions

Graphing is not just about plotting points; it's about visualizing the story a function tells. A good graph conveys the behavior of a function across different domains clearly and effectively.

In this exercise, the graph takes on three different states:

• A flat line at \(y=0\) from \(t=0\) to \(t=2\)
• A descending line from \(t=2\) to \(t=4\) , reflecting the \(3-t\) relationship
• Another flat line, this time at \(y=-1\), starting at \(t=4\) stretching off to infinity

To graph piecewise functions like \(f(t)\), you identify the domain for each 'piece' and graph accordingly. For the descending line, you'd observe the slope (which is \(-1\)); this tells us the function decreases by 1 unit in the y-direction for each unit increase in \(t\).

Graphing helps illustrate the effectiveness of representing functions with step functions, as transitions between states are seen at the 'steps' of the graph. These visual cues align beautifully with the piecewise definition, making it easier to understand and analyze the behavior of the function over its entire domain.