Problem 55 - Calculus Early Transcendentals: Pearson New International Edition

Question

Suppose that a continuous function is periodic with period 1 and is linear between 0 and \(0.25\) and linear between \(-0.75\) and \(0 .\) In addition, it has the value 1 at 0 and 2 at \(0.25 .\) Sketch the function over the domain \([-1,1]\), and give a piecewise definition of the function.

Step-by-Step Solution

Answer

Piecewise periodic over [-1,1]: linear segments with slopes 4 and -4.

Step 1: Understanding the Problem

We need to define a continuous periodic function with period 1. It is linear between 0 to 0.25 and -0.75 to 0 with specific values at 0 and 0.25.

Step 2: Identify Domain Segments

The function is linear in two segments of one period: from 0 to 0.25 and from -0.75 to 0. We need to define it piecewise over the domain [-1, 1].

Step 3: Determine Linear Equations

The line segment from 0 to 0.25 starts at (0,1) and ends at (0.25,2). The slope is found using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2-1}{0.25-0} = 4 \). The equation of the line is \( f(x) = 4x + 1 \).

Step 4: Define Function from -0.75 to 0

The segment from -0.75 to 0 ends at (0,1) and must start at a calculated point such that it is periodic. This linear function is a reflection of the first segment: \( f(x) = -4x + 1 \).

Step 5: Checking Periodicity

Since the function repeats every 1 unit, where the segment from -0.75 to 0 is a reflection of the segment from 0 to 0.25, the function maintains periodicity. Ensure the function's values at these segments' ends align with the values of 1 and 2.

Step 6: Assemble Piecewise Function

The piecewise function over [-1, 1] is:\[ f(x) = \begin{cases} 4x + 1, & \text{if } 0 \leq x < 0.25 \-4x + 1, & \text{if } -0.75 \leq x < 0 \ f(x + 1), & \text{for other } x\end{cases}\]

Step 7: Sketch Function Over Domain

Draw the function over the domain from -1 to 1. For each segment, use the linear equations defined previously. Ensure to continue the periodic pattern beyond these segments to cover the whole domain. - It rises from 1 at x=0 to 2 at x=0.25 - It mirrors, rising from approximately 1 at x=-0.75 to 1 at x=0

Key Concepts

Piecewise Functions

Piecewise functions are defined by different expressions over different intervals. They allow you to define a single function with multiple behavior patterns within different ranges of the variable's domain. In our exercise, we have a piecewise function defined over the interval from -1 to 1, specifically breaking down into two main segments: from -0.75 to 0, and from 0 to 0.25.
There are unique linear expressions for each of these segments:

• From 0 to 0.25, the function follows the line equation \( f(x) = 4x + 1 \)
• From -0.75 to 0, it follows the line equation \( f(x) = -4x + 1 \)

By defining these separate expressions, piecewise functions give us flexibility to model real-life situations or complex functions where behavior changes can be neatly segmented.

Continuity in Functions

A function is said to be continuous when there are no jumps, breaks, or holes in its graph. This is a fundamental characteristic of the function given in the exercise. Since the function is periodic and continuous, it means that not only is it seamless at its transition points, but it also matches perfectly at the beginning and end of each period.
To ensure continuity over each piecewise segment:

• At \( x = 0 \), the value must transition smoothly between the segments defined by different equations.
• Similarly, at \( x = 0.25 \) and \( x = -0.75 \), the function must not have any abrupt changes.

Continuity ensures that each segment connects with the next without any breaks, which is critical especially around the boundaries where the linear segments meet the repetition from -1 to 1.

Linear Functions

Linear functions are essential as the building blocks for our piecewise function. They describe relationships where the change between two values is constant, forming a straight line when plotted on a graph. For example:

• The equation of the line between 0 and 0.25 as \( f(x) = 4x + 1 \) indicates a straight-line increase from the point (0, 1) to (0.25, 2).
• The line from -0.75 to 0, described by \( f(x) = -4x + 1 \), illustrates another linear relationship but in the opposite direction.

Understanding these linear patterns is crucial because we use them to model the smooth transitions required for piecewise and periodic functions. In our example, these lines provide a direct way to handle the transition between key points without introducing complexity.

Function Sketching

Function sketching is a powerful visual tool that helps in understanding and analyzing the behavior of functions over their domain. For the given problem, sketching the function from -1 to 1 involves drawing the straight-line segments based on their equations to represent the piecewise nature.

• Start by plotting the points and lines: from (0, 1) to (0.25, 2) and from (-0.75, approximately 1) to (0, 1).
• Ensure to capture the repeating pattern from -1 to 1.
• Check that the lines meet smoothly at \( x = 0 \) and \( x = 0.25 \) for continuity.

The sketch should then reflect the periodic nature, which repeats every unit along the x-axis. Sketching not only helps in visualizing the function but also in confirming the mathematical logic behind the piecewise and periodic structure.