Problem 47 - Calculus Single Variable

Question

Use this method to express the given multicase function with one formula. Plot the function. $$ R(x)=\left\\{\begin{array}{ll} 1 & \text { if } x>1 \\ x & \text { if } 0

Step-by-Step Solution

Answer

The multicast function is expressed as \( R(x) = x\cdot H(x) - x\cdot H(x-1) + H(x-1) \).

Step 1: Identify the Segments of the Piecewise Function

The piecewise function \( R(x) \) is divided into three segments based on the value of \( x \):- \( R(x) = 1 \) for \( x > 1 \).- \( R(x) = x \) for \( 0 < x \leq 1 \).- \( R(x) = 0 \) for \( x \leq 0 \).

Step 2: Use the Heaviside and Ramp Functions

The Heaviside step function \( H(x) \) is defined as:- 0 for \( x < 0 \)- 1 for \( x \geq 0 \)Express \( R(x) \) using Heaviside functions as:- \( x \cdot H(x) \) to capture \( 0 < x \leq 1 \).- \( H(x-1) \) to capture \( x > 1 \).- Overall, use \( x \cdot H(x) - x \cdot H(x-1) + H(x-1) \) to piece together all segments.

Step 3: Formulate the Single Expression

Combine the previous expressions:\[R(x) = x\cdot H(x) - x\cdot H(x-1) + H(x-1)\]This expression considers each segment and smoothly transitions between them using the properties of \( H(x) \).

Step 4: Plot the Function

Plot the single expression by:- For \( x \leq 0 \), \( R(x) \) remains at 0.- For \( 0 < x \leq 1 \), \( R(x) = x \) and forms a straight line through the origin.- For \( x > 1 \), \( R(x) \) becomes constant at 1.Use graph plotting software to draw these segments as described.

Key Concepts

Understanding the Heaviside Function

The Heaviside function, also known as the step function, is a common tool in mathematics to represent a sudden change in value. It is a piecewise function defined by its straightforward rules. The function is typically denoted as \( H(x) \), and it assigns the value 0 for all values where \( x < 0 \) and assigns the value 1 where \( x \geq 0 \). This function is particularly useful in defining systems or phenomena that switch on or off at a specified point.

In the context of piecewise and multicase functions, the Heaviside function helps create compact expressions to represent different segments of the function. By utilizing the Heaviside function, you can neatly transition between different conditions of a function based on ranges of \( x \). For instance, for the function \( R(x) \), we use \( H(x) \) to help manage the conditions where different expressions apply at different regions.

Introducing the Ramp Function

The ramp function is a specific type of linear function that starts at a certain value and increases linearly beyond that point. In simple terms, it resembles a ramp, as its name suggests, steadily climbing from a baseline. It is frequently expressed as \( R(x) = x \) for the positive range of \( x \).

In the defined piecewise function \( R(x) \), the segment where \( 0 < x \leq 1 \) showcases the ramp function as it directly outputs the value of \( x \). This behavior results in a line that passes diagonally through the origin until it reaches \( x = 1 \). Beyond this interval, the behavior of \( R(x) \) shifts due to the next segment dictated by the Heaviside function.

Decoding Multicase Functions

Multicase or piecewise functions consist of multiple sub-functions, each valid over a certain interval of the input variable's domain. They allow us to define functions that exhibit different behaviors across various domains without needing separate equations entirely. This makes them perfect for complex scenarios like step changes or segmented linear growths.

In our exercise, \( R(x) \) is a piecewise or multicase function. It consolidates three cases into one concise expression: \( R(x) = 0 \) when \( x \leq 0 \), the ramp function \( R(x) = x \) when \( 0 < x \leq 1 \), and finally \( R(x) = 1 \) when \( x > 1 \). Utilizing the Heaviside function helps join these cases seamlessly into a single formula.

Techniques for Function Plotting

Plotting functions involves creating a visual representation of the function's behavior across its domain. This visualization helps understand how the function behaves under different conditions. For piecewise functions, this process includes plotting each segment distinctly while ensuring continuity where applicable.

For \( R(x) \) in this problem, when creating a graph:-

• For \( x \leq 0 \), you'll plot a constant line at \( y = 0 \).
• For the interval \( 0 < x \leq 1 \), plot a line beginning at \( y=0 \) and moving linearly to \( y=1 \) at \( x=1 \).
• For \( x > 1 \), plot a horizontal line at \( y=1 \).

By using graphing software or plotting manually, these steps capture the piecewise nature of \( R(x) \) and illustrate how each section corresponds to its domain.