Problem 7 - Calculus Early Transcendentals

Question

A parking lot charges \(\$ 3\) for the first hour (or part of an hour) and \(\$ 2\) for each succeeding hour (or part), up to a daily maximum of \(\$ 10\) . (a) Sketch a graph of the cost of parking at this lot as a function of the time parked there. (b) Discuss the discontinuities of this function and their significance to someone who parks in the lot.

Step-by-Step Solution

Answer

Cost increases by \(\$2\) each hour until \(\$10\), with jumps at every hour.

Step 1: Understand the Problem

We need to sketch a graph showing how the cost to park is calculated based on the hours spent in the parking lot. The first hour costs \(\\(3\), each additional hour costs \(\\)2\), and the maximum daily charge is \(\$10\). We also need to identify the discontinuities in this graph and explain their significance.

Step 2: Determine Cost Function Segments

For the first hour (0 to 1 hour), the cost is \(\\(3\). For each additional hour, it will cost an extra \(\\)2\). The cost increases linearly with each hour after the first until it reaches a daily maximum of \(\$10\).

Step 3: Sketch the Graph

Draw the x-axis representing time in hours and the y-axis representing cost in dollars. Begin with a point at \((0, 3)\) for the first hour, then plot additional points at each hour, adding \(\\(2\) for each until it reaches \(\\)10\). The graph will have steps at integer hours until \(\$10\). Connect the points with horizontal lines to show the constant cost until the next hour starts.

Step 4: Analyze Graph Discontinuities

The discontinuities occur at every integer hour where the cost jumps by \(\$2\). These 'jump discontinuities' mark where the cost increases suddenly from the previous rate to the next hourly rate.

Step 5: Discuss Discontinuities

The significance of these discontinuities is that the cost increases immediately at the start of each new hour, affecting decisions by parkers to minimize cost. For someone parked just under a full hour, it may suddenly become more expensive if they stay even a minute longer.

Key Concepts

Cost Function

In the context of the parking lot scenario, the cost function describes how much you will pay for parking over a certain period. It's a mathematical representation of the parking fees and is defined as a function of the number of hours parked:

• For the first hour or part thereof, the cost is fixed at $3.
• For each additional hour or part thereof, add $2 until reaching the maximum charge of $10.

This structure shows how parking fees increase, clearly mapping out what you can expect to pay based on how long you stay. Modeling it as a cost function allows us to calculate and predict expenses more reliably.

Piecewise Function

A piecewise function is a type of function that has different expressions based on different intervals of the input. In this parking lot context, the cost is calculated using a piecewise function:

• The first segment (0 to 1 hour) has a cost formula of $3.
• For subsequent hours, the function adds $2 per hour until the cap is reached.
• The function stops increasing after reaching $10, the daily maximum.

This kind of function is useful for situations where the cost doesn't change smoothly or continuously, such as in step functions like this parking fee example.

Discontinuity

Discontinuities in a function are points at which the function isn't continuous; it makes a sudden jump or change. In the parking lot cost function, discontinuities happen at each hour mark. These are known as jump discontinuities because the cost jumps by $2 at the start of each new hour.

• They occur at 1, 2, 3 hours, etc., until the cost hits the daily cap.
• These points highlight where the fee increases without a gradual transition.

Understanding these discontinuities is crucial, as they can affect a parker's decision; for example, parking just over an hour results in a sudden price hike.

Graphical Analysis

Graphical analysis involves studying a graph to understand the behavior and characteristics of a function. In a graphical representation of the parking cost function, we see a step-like graph:

• The x-axis represents time in hours.
• The y-axis represents the cost in dollars.
• Each horizontal line segment indicates a constant cost over the hour.
• The steps occur at integer hours, reflecting the jump in cost.

By analyzing this graph, you can easily visualize the increasing cost pattern, thanks to its clear steps. This method of analysis helps to quickly demonstrate where and when costs increase, aiding in a better understanding of the fee structure.