Problem 55 - Calculus Early Transcendentals: Pearson New International Edition

Question

A cell phone company charges \(\$ 0.12\) for connecting a call plus \(\$ 0.08\) per minute or any part thereof (e.g., a phone call lasting 2 minutes and 5 seconds costs \(\$ 0.12+3 \times \$ 0.08)\). Sketch a graph of the cost of making a call as a function of the length of time \(t\) that the call lasts. Discuss the continuity of this function.

Step-by-Step Solution

Answer

The cost function is a step function with jumps at whole minutes due to rounding up partial minutes.

Step 1: Understanding the Problem

The problem involves determining the cost of making a phone call based on its duration. The cost consists of a fixed connection charge and a variable charge based on each minute, or part thereof, of the call. We need to define the cost function with respect to time and then sketch its graph.

Step 2: Defining the Cost Function

The cost function for a call can be expressed as:\[ C(t) = 0.12 + 0.08 \times \lceil t \rceil \]where \(C(t)\) is the cost and \(\lceil t \rceil\) denotes the ceiling function of time \(t\), which rounds up to the nearest whole number. This accounts for rounding up partial minutes.

Step 3: Graphing the Cost Function

To sketch the graph, plot the cost \(C(t)\) against time \(t\). The graph will be a step function, with jumps occurring at each integer value of \(t\). Each step has a height of \(0.08\) units (each additional minute charge) and crosses the y-axis at \(0.12\), the initial connection charge.

Step 4: Analyzing the Continuity

The step function is not continuous because it jumps at each integer value of \(t\). As such, the function has discontinuities at every point \(t\) where the minute changes. These jumps represent the additional cost when moving from one full minute to the next.

Key Concepts

Step Functions

Step functions are a type of piecewise function known for their unique, staircase-like graphs. These graphs are characterized by horizontal segments that jump or "step" up or down at specific points.

• The defining feature of a step function is its discontinuities. It remains constant over intervals but changes value at specific points.
• In the context of the given problem, the cost of a phone call is a step function. The cost increases in discrete amounts at every full minute, which means that the function takes on a new constant value at each of these points.
• When plotting a step function, such as the phone call cost function given, you'll observe steps that remain level between changes. In our case, there are increments of $0.08, occurring at every minute mark.

This characteristic makes step functions useful for modeling scenarios where variables don't change smoothly but rather in sudden jumps.

Cost Function

A cost function in mathematics represents the total cost associated with the production or purchase of goods and services, correlating this cost with one or more variables. In the scenario of the cell phone company:

• The cost function is defined by the expression \( C(t) = 0.12 + 0.08 \times \lceil t \rceil \), where \(C(t)\) is the cost and \(\lceil t \rceil\) is the ceiling of time \(t\).
• This formula indicates a base cost of \(0.12 for connecting the call, plus \)0.08 for each full or partial minute.
• With this setup, it's easy to calculate the costs for different durations using simple arithmetic. For example, a call that lasts 2 minutes and 5 seconds is rounded up to 3 minutes, resulting in a total cost of \(0.12 + 3 \times 0.08 = 0.36\).

Such functions are commonly used to model real-world scenarios where costs are dependent on variable factors, like time in this case.

Discontinuity in Functions

Discontinuity in functions refers to points where a function is not continuous, meaning there is an abrupt change in its value.

• For the cost function of the phone company, these discontinuities appear at each integer value of time, \(t\), where the cost suddenly jumps by $0.08.
• This happens because the cost is calculated using a ceiling function, \(\lceil t \rceil\), which rounds partial minutes up to the next whole minute.
• Such discontinuities are hallmarks of step functions: the graph consists of distinct, separate intervals with jumps between them.

Understanding and identifying discontinuities is crucial when interpreting graphs in mathematics and real-world scenarios. It allows one to pinpoint exactly where changes occur and how they impact the overall behavior of the function.