Question

A cab company charges \(\$ 2.50\) for the first \(\frac{1}{4}\) mile and \(\$ 0.20\) for each additional \(\frac{1}{8}\) mile. Sketch a graph of the cost of a cab ride as a function of the number of miles driven. Discuss the continuity of this function.

Short answer

The function is continuous and starts at $2.50, linearly increasing by $1.60 per mile after the first 1/4 mile.

Step-by-step solution

Calculate the Initial Cost

The cab company charges $2.50 for the first \(\frac{1}{4}\) mile. This means that for any distance up to \(\frac{1}{4}\) mile, the charge is a constant \(2.50\).

Determine the Cost per Additional Mile

After the first \(\frac{1}{4}\) mile, the cab charges \(0.20\) for every additional \(\frac{1}{8}\) mile. To understand this, let's relate additional miles to cost: for an additional \(\frac{1}{8}\) mile, the cost increases by \(0.20\). As \(1\) mile equals \(8/8\) mile, you would need to drive \(\frac{3}{4}\) more miles beyond the initial \(\frac{1}{4}\) mile to make a full mile, incurring additional charges at the \(0.20\) per \(\frac{1}{8}\) mile rate.

Define the Function

The cost function \(C(x)\) in relation to the distance \(x\) (in miles) is given by: \[ C(x) = \begin{cases} 2.50, & 0 < x \leq \frac{1}{4} \ 2.50 + 0.20 \left(8 (x - \frac{1}{4})\right), & x > \frac{1}{4} \end{cases} \] Here, \(8 (x - \frac{1}{4})\) calculates how many additional \(\frac{1}{8}\) miles are covered beyond the initial \(\frac{1}{4}\).

Discuss the Continuity of the Function

The function is continuous at all points as there are no jumps, skips, or infinite values; each \(\frac{1}{8}\) mile increment smoothly increases the cost. At \(x = \frac{1}{4}\), the cost function transitions from a constant to a linearly increasing function without interruption.

Sketch the Graph

The graph of the function starts at \((\frac{1}{4}, 2.50)\) and transitions smoothly into a sloping line with a slope of \(1.60\) per mile for \(x > \frac{1}{4}\). It looks like a straight line starting from \((0, 2.50)\) to reach \(\frac{1}{4}\) on the x-axis and then rises linearly as x increases.

Key concepts

Continuity of Functions

In mathematics, a function is continuous when you can draw its graph without lifting your pencil off the paper. This means there are no breaks, jumps, or holes in the graph. A function is said to be continuous at a point if the left-hand limit, the right-hand limit, and the value of the function at that point are all equal.
For the cab fare cost function problem, continuity is an essential concept. The function here is continuous because it smoothly transitions from the initial flat cost of $2.50 to the increasing costs as more miles are added.
The function, defined as piecewise, has two parts:

• A constant cost for the first quarter-mile (no change in cost, hence it's a horizontal line on the graph).
• A linearly increasing cost with each additional eighth of a mile beyond the first quarter-mile.

At the point where the function transitions from constant to increasing, which is at one-quarter mile, the cost function remains continuous because the limit from both the left side and the right side at this point are equal, ensuring a smooth graph.

Graphing Functions

Graphing serves as a visual representation of a function, allowing us to actually see the behavior and pattern of changes. For the cab fare problem, drawing the graph will help you understand how costs accumulate with distance traveled.
To graph the cab fare cost function:

• Start at the point \( (0, 2.50) \) because, at zero miles, the cost is $2.50.
• Maintain this cost horizontally until reaching \( x = \frac{1}{4} \) mile.
• From this point, the line begins to slope upwards, showing a steady increase in cost after each additional eighth mile.

The slope of the line for \( x > \frac{1}{4} \) is calculated by the rate of the additional \( 0.20 \) per \( \frac{1}{8} \) mile, equating to a slope of 1.60 per mile.

Calculus Applications

In calculus, understanding how functions behave helps in analyzing real-world situations, such as costs and charges by distance. Calculus provides tools to find important features like slopes, areas under curves, and continuity of functions.
With the cab cost function:

• The initial flat rate represents a simple horizontal line segment, requiring no calculus techniques.
• The slope of the line for distances greater than the initial \( \frac{1}{4} \) mile is derived using basic algebra techniques but involves the concept of rate change, which ties closely to differentiation in calculus.

Differentiation could further be used to evaluate how the cost changes with respect to mileage, representing a practical application of calculus for understanding incremental costs.

Step Functions

A step function is a piecewise function that has constant values over specific intervals and jumps or steps at certain points. In real-world situations, step functions can be highly useful because they allow for defining scenarios where changes happen at set intervals.
For the cab service example:

• Initially, it looks like a step function since for \( 0 < x \leq \frac{1}{4} \) mile, the cost is fixed at \(2.50.
• However, beyond \( \frac{1}{4} \) mile, the cost increases smoothly with each \( \frac{1}{8} \) mile, which differs from a traditional step function that's defined by distinct, constant steps.

Even though the cab cost function transitions to a linear graph when extra distance is considered, the initial segment showcases the basic idea of a step function, where a specific condition (distance less than or equal to \( \frac{1}{4} \) mile) results in a fixed output (cost of \)2.50).