Problem 31 - Calculus for Biology and Medicine

Question

Draw an open disk with radius 2 centered at \((1,-1)\) in the \(x-y\) plane, and give a mathematical description of this set.

Step-by-Step Solution

Answer

The open disk is described by \((x-1)^2 + (y+1)^2 < 4\).

Step 1: Understand the Concept of Open Disk

An open disk in the plane is the set of all points that are strictly within a specified distance (radius) from a given center point, excluding the boundary. The disk is 'open' because it does not include the edge or circumference.

Step 2: Identify the Given Parameters

In this problem, the open disk is centered at the point (1, -1) and has a radius of 2. These are the key parameters for defining the set mathematically.

Step 3: Use Distance Formula for Mathematical Description

The mathematical description of the open disk involves all points (x, y) such that their distance to the center (1, -1) is less than the radius, 2. The distance from a point (x, y) to the center is given by the formula: \[ \sqrt{(x-1)^2 + (y+1)^2} < 2 \] This inequality describes the set of all points inside the open disk.

Step 4: Simplify the Mathematical Description

To simplify the inequality, we can remove the square root by squaring both sides: \[ (x-1)^2 + (y+1)^2 < 4 \] This describes the set of all points (x, y) that are inside the open disk with the specified center and radius.

Key Concepts

Distance Formula

The distance formula is a tool used in mathematics to determine the distance between two points in a coordinate plane. It's derived from the Pythagorean theorem. For two given points \( (x_1, y_1) \) and \( (x_2, y_2) \), the distance \( d \) between them can be calculated using the formula:

• \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)

This formula is crucial when describing geometric shapes, such as circles and open disks. In our exercise regarding the open disk, it helps determine which points are within the disk by ensuring their distance from the center is less than a specified radius.
Understanding how to apply the distance formula ensures that you can find and describe these points accurately. It's fundamental in defining more complex regions defined by distance, which is an essential aspect of geometry.

Inequalities

Inequalities are expressions that describe the relative size or order of two values and are a fundamental component of algebra. They are not equations because they don't show equality between expressions; instead, they explore relationships where one side is larger or smaller than the other.
When dealing with inequalities, it's important to understand the symbols used:

• \( > \): Greater than
• \( < \): Less than
• \( \geq \): Greater than or equal to
• \( \leq \): Less than or equal to

In the context of our exercise with an open disk, the inequality \( (x-1)^2 + (y+1)^2 < 4 \) is used to express that points must be strictly inside the circle defined by this inequality.
This approach demonstrates how mathematical inequalities can describe regions or sets in the plane, defining boundaries and spaces effectively.

Mathematical Set Description

A mathematical set description is a way of specifying a collection of objects or numbers. In geometry, sets can describe specific areas in a coordinate plane, like the open disk in our exercise.
In mathematical terms, a set can be written to include all elements that satisfy certain conditions. For instance, the set of points forming our open disk is defined by the inequality \( (x-1)^2 + (y+1)^2 < 4 \). This means:

• Every point \( (x, y) \) satisfies the given condition, being less than the radius squared from the center.
• The description excludes points on the circle's boundary, making it 'open'.

Such set descriptions provide a concise way to capture and communicate the nature of various geometric figures and regions.
By understanding how to apply these concepts, you'll be better equipped to describe and analyze mathematical areas and their properties.