Problem 84 - Brief Applied Calculus

Question

GENERAL: First-Class Mail In 2011, the U.S. Postal Service would deliver a first-class letter weighing \(3.5\) ounces or less for the following prices: not more than 1 ounce, \(44 \mathfrak{c} ;\) more than 1 ounce but not more than 2 ounces, \(64 \mathrm{c} ;\) more than 2 ounces but not more than 3 ounces, \(84 \epsilon ;\) and more than 3 ounces but not more than \(3.5\) ounces, $$\$ 1.04$$. This information determines the price (in cents) as a function of the weight (in ounces). At which values in open interval \((0,3.5)\) is this function discontinuous? Source: U.S. Postal Servi

Step-by-Step Solution

Answer

The function is discontinuous at \(\text{weight} = 1\), \(\text{weight} = 2\), and \(\text{weight} = 3\).

Step 1: Review Price Intervals

First, review the intervals for the pricing based on weight given in the problem: \(0<\text{weight} \leq 1\) for 44₡, \(1<\text{weight} \leq 2\) for 64₡, \(2<\text{weight} \leq 3\) for 84₡, and \(3<\text{weight} \leq 3.5\) for $1.04.

Step 2: Identify Potential Discontinuities

The price function is piecewise, changing at the endpoints of each interval. Check where the weight slightly passes the boundaries: \(\text{weight} = 1\), \(\text{weight} = 2\), and \(\text{weight} = 3\).

Step 3: Check for Discontinuity at Each Endpoint

Evaluate the price at \(\text{weight} = 1\) from both sides: As \(\text{weight}\) approaches 1 from the left, the price is 44₡; from the right, it changes to 64₡. Similar checks for \(\text{weight} = 2\) (84₡ from the right) and \(\text{weight} = 3\) (104₡ from the right) confirm changes in price.

Step 4: Conclusion on Discontinuities

Since the price changes abruptly at weights \(\text{weight} = 1\), \(\text{weight} = 2\), and \(\text{weight} = 3\) without a smooth transition, the function is discontinuous at these points.

Key Concepts

Piecewise Functions

Piecewise functions are a type of mathematical function defined by multiple sub-functions, each applying to a specific interval of the main function's domain. They are helpful when the behavior of a situation changes at certain points.

To construct a piecewise function, we define different cases based on certain conditions or intervals. Each case has its rule or formula that applies only within that case's interval. The key characteristic of a piecewise function is that it "pieces together" several functions, each corresponding to specific conditions or ranges.

The U.S. Postal Service's pricing based on weight is an excellent example of a piecewise function, with different prices set for different weight categories. This structure allows for more tailored pricing strategies rather than using a single function to cover all situations.

Price Intervals

Price intervals refer to the specific ranges or sections in which a particular pricing rule applies. In the context of the U.S. Postal Service, these are intervals of postal weight, each associated with a different price.

Understanding price intervals is crucial because it determines when a new pricing rule will take effect as we cross a weight threshold. For the given example:

• Weight from 0 to 1 ounce costs 44₡.
• Weight between 1 and 2 ounces costs 64₡.
• Weight between 2 and 3 ounces costs 84₡.
• Weight between 3 and 3.5 ounces costs $1.04.

This clear separation ensures that customers can anticipate exact charges depending on how much their mail weighs within the specified intervals.

U.S. Postal Service Pricing

The U.S. Postal Service pricing functions as an organized system based on weight categories, providing fair and predictable costs for mailing. It acts as a simple example of how differentiated pricing can be systematically applied.

The postal service strategically determines these weight categories to balance operational costs and customer accessibility. Customers pay more as their mailing weight increases and moves into a higher weight category.

Such structures are not just limited to the postal service but are widely applicable across various industries like shipping and telecommunications.

Function Discontinuity

Discontinuity in a function occurs when there is a "jump" or abrupt change at a point in the function's domain. For the U.S. Postal Service's pricing model, these discontinuities occur at the borders between the price intervals.

At weights exactly equal to 1, 2, and 3 ounces, the price "jumps" to a new value. For instance:

• Price jumps from 44₡ to 64₡ at 1 ounce.
• Price jumps from 64₡ to 84₡ at 2 ounces.
• Price jumps from 84₡ to 104₡ at 3 ounces.

These jumps demonstrate discontinuities, meaning the transition between different pricing rules is not gradual but rather sudden.