Problem 76 - Calculus and its applications

Question

The cost of sending a large envelope via U.S. first-class mail in 2014 was \(\$ 0.98\) for the first ounce and \(\$ 0.21\) for each additional ounce (or fraction thereof). (Source: www.usps.com.) If \(x\) represents the weight of a large envelope, in ounces, then \(p(x)\) is the cost of mailing it, where $$ \begin{array}{l} p(x)=\$ 0.98, \text { if } \quad 0 < x \leq 1, \\ p(x)=\$ 1.19, \text { if } \quad 1 < x \leq 2, \\ p(x)=\$ 1.40, \text { if } 2 < x \leq 3, \end{array} $$ and so on, up through 13 ounces. The graph of \(p\) is shown below. Using the graph of the postage function, find each of the following limits, if it exists. $$ \lim _{x \rightarrow 3.4} p(x) $$

Step-by-Step Solution

Answer

The limit is \(1.61\).

Step 1: Understand the Function

The cost function \( p(x) \) is a piecewise function, meaning it changes its expression based on the weight of the envelope \( x \). For each additional ounce beyond the first, the cost increases by \( \\(0.21 \). Specifically, \( p(x) = \\)1.40 \) when \( 2 < x \leq 3 \) and \( p(x) = \$1.61 \) when \( 3 < x \leq 4 \).

Step 2: Determine Which Piece to Use

The limit specified is \( \lim_{x \rightarrow 3.4} p(x) \). Since \( 3.4 \) is greater than \( 3 \), identify that it falls within the range where \( 3 < x \leq 4 \). Therefore, this segment uses \( p(x) = \$1.61 \).

Step 3: Evaluate the Limit

Since the function \( p(x) \) is constant within the interval \( 3 < x \leq 4 \), the limit as \( x \) approaches \( 3.4 \) from either side within this interval is simply the value of the function at that interval. Hence, \( \lim_{x \rightarrow 3.4} p(x) = 1.61 \).

Key Concepts

Limit Evaluation

Limit evaluation is a fundamental concept in calculus, helping us understand the behavior of functions as we approach certain values. In our given piecewise function, the goal is to find the limit of the cost function \( p(x) \) as \( x \) approaches 3.4. A limit examines the value that the function approaches as the input gets indefinitely close to a particular point. Notably, for piecewise functions like \( p(x) \), it's crucial to identify which piece to use when evaluating limits. For the interval \( 3 < x \leq 4 \), the cost remains constant at \( \$1.61 \). Consequently, as \( x \) approaches 3.4, we're operating within this range, meaning \( \lim_{x \to 3.4} p(x) = 1.61 \). Understanding this stability within a segment of a piecewise function underscores why limits are often simple to compute for constant intervals.
Limits seamlessly link the values of a function to its "neighborhood," providing insights into the function's continuity and any jumps or breaks.

Piecewise Continuity

Piecewise continuity describes the smoothness of a function made up of distinct segments. In the function \( p(x) \) that determines mailing costs, each segment reflects a constant price over specific ounce ranges: \(0.98\) for \(0 < x \leq 1\), with each additional range adding \(0.21\).
This structure means the cost graph will consist of horizontal line segments with abrupt jumps between each, translating to discontinuities at each boundary. For instance, moving from \( x = 3 \) (\\(1.40) to \( x = 3.1 \) shifts the cost handily to \( \\)1.61 \) due to this abrupt jump.

• Continuity within a segment: The function remains constant.
• Discontinuity at transitions: Occurs where the cost increases suddenly to a new price tier.

Exploring these breaks is essential for understanding piecewise continuity, where limits take a key role in recognizing these transition points.

Function Graph Analysis

Analyzing the graph of a piecewise function like \( p(x) \) allows us to visually interpret data structures and behaviors manifesting in formulaic representations. Each segment in the postage cost graph represents a flat stretch over a defined range, interspersed with sudden jumps at each transition beyond an ounce.

• Horizontal Segments: These segments showcase constant value spans, directly translating into cost levels for specific weights.
• Vertical Jumps: The graph's distinct upward movements occur at weight cutoff points, highlighting increased mailing costs.

By dissecting these graphical components, students recognize how piecewise functions output different values easily, depending on the range considered. Observing the graph also allows one to anticipate limits better, particularly as they visually confirm function behaviors through local segments and step-changes indicative of price adjustments.