Problem 53 - Calculus: Graphical, Numerical, Algebraic

Question

Airport Parking Valuepark charge \(\$ 1.10\) per hour or fraction of an hour for airport parking. The maximum charge per day is \(\$ 7.25\) (a) Write a formula that gives the charge for \(x\) hours with \(0 \leq x \leq 24 .\) (Hint: See Exercise \(52 . )\) (b) Graph the function in part (a). At what values of \(x\) is it continuous?

Step-by-Step Solution

Answer

The formula for parking charges is \(f(x) = 1.10 * \lceil x \rceil\) for \(0 \leq x < 6.59\) and \(f(x) = 7.25\) for \(6.59 \leq x \leq 24\). From the graph, the function is discontinuous at \(x = 6.59\) and continuous at all other points between 0 and 24.

Step 1: Understand the Charging Mechanism and Write a Formula

The charge for any parking hour or fraction of an hour is $1.10. This means that even a fraction of an hour, say 0.1 hour, is also charged as a full hour ($1.10). Therefore, the number of hours is always rounded up to the nearest whole number. Thus, it can be represented as \(\lceil x \rceil\), where \(\lceil x \rceil\) is the ceiling function that rounds a number up to the nearest integer. \n\nThe charge for a number of hours is $1.10 times the number of hours, until the maximum charge of $7.25 per day is reached. Hence, the function can be represented as two cases in mathematical form: \(f(x) = 1.10 * \lceil x \rceil\) for \(0 \leq x < 6.59\) and \(f(x) = 7.25\) for \(6.59 \leq x \leq 24\)

Step 2: Graph the Function

The function \(f(x)\) has two parts. From \(0 \leq x < 6.59\), it's an increasing function because the cost increases with the number of hours. This part of the function will look like steps rising from the origin to reach a height of $7.25 at \(x = 6.59\). \n\nFrom \(6.59 \leq x \leq 24\), the cost is flat at $7.25 because the cost doesn't increase beyond this point. This part of the function will be a flat line at a height of $7.25 from \(x = 6.59\) to \(x = 24\).

Step 3: Analyze Continuity

Looking at the graph, it can be observed that this function is discontinuous at \(x = 6.59\). This is because at \(x = 6.59\), the function jumps from $7.15 to $7.25. At all other points, \(f(x)\) is continuous because there are no jumps or breaks in the graph.

Key Concepts

Piecewise Function

A piecewise function is a type of mathematical function that is defined by multiple sub-functions, each applied to a certain interval of the main function's domain. In the case of airport parking charges, we are dealing with a piecewise function that reflects different pricing strategies based on the number of hours a car is parked.

For hours less than 6.59, the function uses the ceiling function, meaning any fraction of an hour is rounded up to the nearest whole number, and then this number is multiplied by $1.10 to calculate the parking fee. As a result, if you park for 3.2 hours, you are charged for 4 full hours.

Another rule is applied for any duration equal to or exceeding 6.59 hours, where the parking cost is capped at $7.25, which is the daily maximum. This means that no matter how long you park beyond this point, the cost will not increase.

Piecewise functions are incredibly useful in real-world situations where different rules apply at different stages or intervals. They provide a structured way to describe such situations mathematically.

Graphing Functions

Graphing functions is a crucial method for visualizing mathematical relationships and understanding how variables behave. When we graph the piecewise function for parking charges, we see two distinct parts on the graph.

The first part of the graph, representing the period from 0 to less than 6.59 hours, appears as a set of steps. This "staircase" pattern emerges because each fractional hour leads to a full incremental charge, characteristic of the ceiling function's impact. As time progresses, the cost increases linearly but in discrete steps.

In the second part, ranging from 6.59 to 24 hours, the graph flattens. This reflects the constant maximum charge of $7.25, regardless of how much time you remain parked after the threshold. This creates a horizontal line on the graph, showing no further increase in cost.

Graphing these functions helps us easily observe the changes in variables and how the piecewise rules affect the overall cost.

Function Continuity

Function continuity refers to how smooth or unbroken a graph is. A function is considered continuous if you can draw its graph without lifting your pencil off the paper. Discontinuity occurs where there are jumps, spikes, or holes in the graph.

For the airport parking function, continuity is maintained in different intervals except at a critical point. From 0 up to 6.59 hours, the function is continuous but not smooth, due to stepped increases caused by the ceiling function. However, it still flows without breaks within this domain.

The discontinuity happens exactly at 6.59 hours. At this point, there's an abrupt "jump" from the staggered growth of fees to a flat rate. This discontinuity is identified as a jump discontinuity because the value of the function jumps suddenly from $7.15 to $7.25, without passing through all intermediate values.

Despite this jump, other parts of the function remain continuous from 6.59 to 24 hours, where the graph remains horizontally steady at the daily max fee. Understanding these points assists in predicting and explaining behaviors over continuous and discontinuous domains.