Problem 74 - 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. $$ \lim _{x \rightarrow 2.6^{-}} p(x), \lim _{x \rightarrow 2.6^{+}} p(x), \lim _{x \rightarrow 2.6} p(x) $$

Step-by-Step Solution

Answer

\( \lim _{x \rightarrow 2.6^{-}} p(x) = \$1.40 \), \( \lim _{x \rightarrow 2.6^{+}} p(x) = \$1.40 \), \( \lim _{x \rightarrow 2.6} p(x) = \$1.40 \).

Step 1: Understanding the Function

The cost function, \( p(x) \), is a step function defined piecewise depending on the weight \( x \) of the envelope. For each ounce over 1 up to 13 ounces, an additional \(0.21 is added. Thus, for the weight interval \( 2 < x \leq 3 \), the cost is given as \( p(x) = \\)1.40 \). The graph of \( p(x) \) consists of horizontal steps, each step representing a specific cost for a specific weight interval.

Step 2: Finding \( \lim _{x \rightarrow 2.6^{-}} p(x) \)

The limit \( \lim _{x \rightarrow 2.6^{-}} p(x) \) asks for the value of \( p(x) \) as \( x \) approaches 2.6 from the left. Since 2.6 is within the interval \( 2 < x \leq 3 \), if \( x \) approaches 2.6 from values less than 2.6, the cost remains \( \\(1.40 \). Therefore, \( \lim _{x \rightarrow 2.6^{-}} p(x) = \\)1.40 \).

Step 3: Finding \( \lim _{x \rightarrow 2.6^{+}} p(x) \)

The limit \( \lim _{x \rightarrow 2.6^{+}} p(x) \) asks for the value of \( p(x) \) as \( x \) approaches 2.6 from the right. Since we are still within the interval \( 2 < x \leq 3 \), approaching from slightly more than 2.6 does not change the defined cost. Thus, \( p(x) \) does not increase to the next cost tier until \( x > 3 \). Therefore, \( \lim _{x \rightarrow 2.6^{+}} p(x) = \$1.40 \).

Step 4: Finding \( \lim _{x \rightarrow 2.6} p(x) \)

This limit exists if both the left-hand and right-hand limits are equal, which they are: \( \lim _{x \rightarrow 2.6^{-}} p(x) = \lim _{x \rightarrow 2.6^{+}} p(x) = \\(1.40 \). Therefore, \( \lim _{x \rightarrow 2.6} p(x) = \\)1.40 \).

Key Concepts

Step Functions

In calculus, a **step function** is a type of function that increases or decreases in steps, rather than smoothly. Imagine a staircase, where each step represents a specific range of values and maintains a constant height. This is how step functions behave. They jump from one value to the next, creating a series of horizontal lines or "steps" when graphed.
Step functions are commonly encountered in real-world scenarios, such as pricing systems where costs are tiered based on usage. In the given exercise, the cost of mailing changes at specific weight intervals, creating a step function. Each interval corresponds to a different postal charge, thus forming distinct "steps" on the price graph.

Piecewise Functions

**Piecewise functions** are functions that have different expressions or rules for different parts of their domain. Think of them as instructions on how to calculate the function's value depending on the input you provide. In essence, they are multiple step functions combined into one.
This is particularly useful when a function cannot be described using a single expression over its entire domain. For example, in mailing costs like the exercise discussed, the price changes with the weight of the envelope. Each weight range has its own formula for cost, making it a classic piecewise function. Here, each piece of the function corresponds to a specific weight range and computes the cost accordingly.

Limit Calculation

**Limit calculation** is a fundamental concept in calculus used to analyze how a function behaves as it approaches a certain point. It's like trying to figure out the speed you're going just before you come to a complete stop at a traffic light.
In a practical sense, calculating limits helps understand the behavior of a function at points of interest, including discontinuities or changes, like the steps in our postage cost function. The limits help determine how closely a function approaches a particular value as the input "x" gets closer to a point from either direction.

• They help understand the continuity and stability of functions.
• Limits are essential in defining derivatives and integrals.

In our exercise, understanding the limit as the weight approaches 2.6 ounces from different sides allows us to determine that the cost remains consistent at $1.40.

Left-Hand Limit

The **Left-Hand Limit** refers to the value that a function approaches as the input comes from the left side of a certain point. If you picture approaching a crossroad, the left-hand limit is like coming up to that point strictly from the left lane.
Mathematically, we express it as \( \lim_{x \to a^-} f(x) \), which reads "the limit of \( f(x) \) as \( x \) approaches \( a \) from the left."
In the context of our mailing cost function, when calculating \( \lim _{x \to 2.6^{-}} p(x) \), we are examining what value the function "p(x)" is getting close to when "x" gets closer to 2.6 ounces from below. Since 2.6 is within the 2 to 3 ounces step, the cost remains at $1.40 even when coming from just a smidgen below.

Right-Hand Limit

Conversely, the **Right-Hand Limit** is focused on the value a function reaches as the input comes from the right direction. Imagine making your way to a destination, sticking to the right lane.
In notation, it's written as \( \lim_{x \to a^+} f(x) \), meaning "the limit of \( f(x) \) as \( x \) approaches \( a \) from the right."
Within our exercise, \( \lim _{x \to 2.6^{+}} p(x) \) looks at the function's value as "x" advances to 2.6 ounces from the right. Since we're still in the same weight bracket of 2 to 3 ounces, the postal cost doesn't increase just yet. Thus, the function also approaches $1.40 from the right, underscoring the consistency of the step function over this interval.