Problem 41 - Calculus Early Transcendentals: Pearson New International Edition

Question

The French Railroad Suppose that Paris is located at the origin of the \(x y\) -plane. Rail lines emanate from Paris along all rays, and these are the only rail lines. Determine the set of discontinuities of the following functions. (a) \(f(x, y)\) is the distance from \((x, y)\) to \((1,0)\) on the French railroad. (b) \(g(u, v, x, y)\) is the distance from \((u, v)\) to \((x, y)\) on the French railroad.

Step-by-Step Solution

Answer

Discontinuities are off the positive x-axis for \(f(x,y)\) and non-scalar multiples for \(g(u,v,x,y)\).

Step 1: Understanding Railway Paths and Distances

In this problem, consider that railroad paths are rays originating from the origin (0,0) and extending outward through any point (x,y). Travel along the railroad means moving along these straight lines. Distance on the railroad between two points is measured as the sum of distances to and from the origin.

Step 2: Analyzing function f(x,y)

For a point (x,y), traveling to (1,0) involves moving from (x,y) to the origin and then to (1,0). The distance function is \(f(x, y) = \sqrt{x^2 + y^2} + |1| = \sqrt{x^2 + y^2} + 1\). The discontinuity occurs when the ray through (x,y) differs from the ray through (1,0), which is any point not on the x-axis, excluding the positive x-ray.

Step 3: Identifying Discontinuities in f(x,y)

The function \(f(x, y)\) has discontinuities for all points not lying on the positive x-axis. For a point (x,y) on other rays, the distance on the railroad is not direct, leading to discontinuity.

Step 4: Exploring function g(u,v,x,y)

For a point (u,v) to travel to (x,y), the distance is \(g(u,v,x,y) = \sqrt{u^2 + v^2} + \sqrt{x^2 + y^2}\). This function has discontinuities whenever (u,v) and (x,y) do not lie on the same ray from the origin, meaning when the vectors (u,v) and (x,y) are not scalar multiples of each other.

Step 5: Identifying Discontinuities in g(u,v,x,y)

The function \(g(u,v,x,y)\) will have discontinuities for all pairs (u,v) and (x,y) not aligned on the same ray from the origin. Therefore, discontinuities exist whenever there is no linear path connecting these via the origin.

Key Concepts

Distance Functions

Distance functions help us measure how far apart two points are, which is a fundamental concept in geometry and calculus.
In the context of the French railroad problem, the distance function helps determine the length of the path traveled via specific routes.
In classical Euclidean geometry, the distance between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is calculated using the Pythagorean theorem:

• The distance is \( \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \).

In our problem, however, the distance is measured uniquely because travel occurs along predefined rays emanating from the origin.
This modifies the distance calculation by summing the distances from the starting point to the origin, and from the origin to the end point.For example, the function \( f(x, y) = \sqrt{x^2 + y^2} + 1 \) calculates the distance from \( (x, y) \) to \( (1, 0) \) by first measuring to the origin and then onward.

Discontinuities in Functions

Discontinuities in functions occur where a function is not continuous — that is, where there are abrupt changes in the output values.
For railroad distance functions, understanding the discontinuities helps us identify points where the travel path changes.For the function \( f(x, y) \), discontinuities happen at points not on the positive x-axis.
This happens because, for points not along this axis, the travel from \( (x, y) \) to the origin and then to \( (1, 0) \) is not straightforwardly linear, creating a disruption in expected distance measurement.Similarly, the function \( g(u, v, x, y) \) exhibits discontinuities for pairs of points not aligned on the same ray from the origin:

• Alignment here means the vectors \( (u, v) \) and \( (x, y) \) must be scalar multiples of each other.
• Otherwise, the path is disrupted, leading to a discontinuity.

Understanding these discontinuities is crucial for accurately solving problems involving such unique distance measurements on a coordinate plane.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, uses algebra to explore geometric problems.
It allows for a detailed analysis of relationships between points, lines, and other geometric elements via a coordinate plane.In the context of our problem, the \( xy \)-plane is essential for determining the paths starting at the origin and radiating outward.
The coordinates of any point represent its position and thus, guide us in computing the distance within this peculiar system of rays.

• Points are manipulated using their coordinates, which helps mathematicians determine relationships and solve problems involving distances or alignment.
• In our case, the alignment or lack thereof between two rays can be denoted by equivalent angle positions or scalar multiples of their vector forms.

This approach allows insight into how points connect via the origin and aids in plotting positions accurately on this unique rail system.

Mathematical Problem Solving

Mathematical problem-solving involves the application of various mathematical techniques to tackle complex problems and find solutions.
It combines logical reasoning, pattern recognition, and the application of formulas and theorems. Tackling the French railroad problem requires understanding how non-standard distance functions work.
The integration of real-world constructs, like rail paths, challenges typical Euclidean distance norms, compelling us to rethink intuitions about geometry.

• Problem-solving here starts by clearly defining the rules and constraints—know the rays originate from the origin.
• Next, identify points of discontinuity to understand where conventional wisdom about straight paths fails.
• Finally, apply calculations of distance in this unique setup to fully comprehend how geometry shifts when routes have fixed, radial paths.

Through this dynamic problem-solving process, students not only tackle mathematical challenges but also sharpen their analytical thinking, crucial for mastering advanced calculus and related fields.