Problem 9 - 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. $$f(t)=\left\\{\begin{array}{lr}t^{2}, & 0 \leq t<1 \\\1, & t \geq 1\end{array}\right.$$

Step-by-Step Solution

Answer

The given function can be expressed in terms of the unit step function as: $$f(t) = t^2(u(t)-u(t-1)) + 1 \cdot u(t-1)$$ The sketch of the given function \(f(t)\) on the interval \([0, \infty)\) consists of a parabola \(t^2\) for \(0 \leq t < 1\) and a horizontal line at \(y=1\) for \(t \geq 1\).

Step 1: Sketch the given function

The function \(f(t)\) is defined as: $$f(t)=\left\\{\begin{array}{lr}t^{2}, & 0 \leq t<1 \\\1, & t \geq 1\end{array}\right.$$ We can start by sketching the two parts of the function: 1. For \(0 \leq t < 1\), the function is defined as \(f(t) = t^2\). This represents a parabola that starts at the origin \((0,0)\) and opens upwards. 2. For \(t \geq 1\), the function is defined as \(f(t) = 1\). This is a horizontal line at the level \(y=1\) which extends to the right from where \(t=1\). Make sure to draw these two parts on the same graph and connect them to form the complete sketch of the given function.

Step 2: Express the function in terms of the unit step function

The unit step function, also known as the Heaviside function, is defined as: $$u(t) = \left\{\begin{array}{lr}0, & t<0 \\\1, & t \geq 0\end{array}\right.$$ Now, we need to express the given function \(f(t)\) in terms of the unit step function. 1. To represent the first part of the function \(t^2\) when \(0 \leq t < 1\), we can multiply this function with the expression \((u(t) - u(t-1))\), which results in the function equaling zero once \(t \geq 1\). This means that: $$t^2(u(t) - u(t-1)) = \left\{\begin{array}{lr}t^{2}, & 0 \leq t<1 \\\0, & t \geq 1\end{array}\right.$$ 2. To represent the second part of the function \(1\) when \(t \geq 1\), we simply multiply it by \(u(t-1)\). This leads to: $$1 \cdot u(t-1) = \left\{\begin{array}{lr}0, & t<1 \\\1, & t \geq 1\end{array}\right.$$ Combining these two expressions: $$f(t) = t^2(u(t)-u(t-1)) + 1 \cdot u(t-1)$$

Key Concepts

Piecewise Functions

Piecewise functions are a type of mathematical function defined through multiple sub-functions, each corresponding to a specific domain part. In the given exercise, the function is divided into two distinct rules, which apply depending on the value of the variable:

• For values of the variable between 0 and 1, the function is defined by the equation \(f(t) = t^2\), forming a parabola curve.
• For values equal to or greater than 1, the function is described simply by \(f(t) = 1\), which results in a horizontal line.

By using these two sub-functions, you can accurately describe the entire behavior of the function across its domain. Understanding how to break down a function into pieces can simplify the analysis and graphing of complex equations. The key is to carefully determine the boundaries of each piece and the rule that applies within those boundaries.

Heaviside Function

The Heaviside function, also called the unit step function, is a special mathematical function that represents a sudden jump in value. This step function is defined as \( u(t) \), and it is pivotal in expressing piecewise functions like the one in the exercise. The function is mathematically described as:\[ u(t) = \begin{cases} 0, & t < 0 \ 1, & t \geq 0 \end{cases} \]Using the Heaviside function, you can express changes over specific intervals by multiplying it with expressions. In the current exercise:

• The expression \( (u(t) - u(t-1)) \) helps constrain the quadratic part \( t^2 \) only between \( 0 \) and \( 1 \), since it transitions from 1 to 0 exactly at \( t = 1 \).
• The expression \( u(t-1) \) helps activate the constant part \( 1 \) from \( t = 1 \) onwards.

This clever use of the Heaviside function enables us to neatly combine the different "pieces" of the piecewise function into a single expression.

Function Sketching

Function sketching involves visually representing a function based on its mathematical description. It is an accessible way of understanding the function's behavior and characteristics.When sketching the given piecewise function from the exercise, you should:

• First, draw the parabola \( f(t) = t^2 \) for the interval \( 0 \leq t < 1 \). It starts at the origin and rises to \( (1,1) \).
• Next, draw a straight horizontal line at \( y = 1 \). This line should start at \( t = 1 \), extending indefinitely towards the right for all \( t \geq 1 \).
• Ensure the two pieces connect precisely at the point \( (1,1) \), giving a smooth transition between the different behaviors of the function.

This graphical representation helps you see where and how the function transitions between its different sub-function components, making it easier to comprehend the overall structure of the piecewise function.