Problem 4 - Calculus: Graphical, Numerical, Algebraic

Question

In Exercises \(1-4,\) find the coordinate increments from \(A\) to \(B\) $$A(0,4), \quad B(0,-2)$$

Step-by-Step Solution

Answer

The coordinate increments from A to B are \(Δx=0\) and \(Δy=-6\).

Step 1: Identifying the X and Y coordinates

First, identify the 'x' and 'y' coordinates of both point A and point B. For A we have \(x=0\) and \(y=4\). For B we have \(x=0\) and \(y=-2\).

Step 2: Calculating the increments in x-coordinate

The increment in 'x' coordinates (Δx) is the difference in the 'x' coordinates of B and A. Calculate this by subtracting the x-coordinate of A from that of B: \(0 - 0 = 0\).

Step 3: Calculating the increments in y-coordinate

The increment in 'y' coordinates (Δy) is the difference between 'y' coordinates of B and A. Calculate this by subtracting the y-coordinate of A from that of B: \(-2 - 4 = -6\).

Key Concepts

Coordinate Increments

In coordinate geometry, understanding increments is a crucial part of moving between two points. The term "coordinate increments" refers to the change or difference in either the x or y values when moving from one point to another on the Cartesian plane.

• The change in the x-coordinate is usually denoted as \( \Delta x \), and it’s calculated by subtracting the x-coordinate of the initial point from the x-coordinate of the final point.
• The change in the y-coordinate is similarly represented by \( \Delta y \), found by subtracting the y-coordinate of the initial point from the y-coordinate of the final point.

This calculation helps in determining the direction and magnitude of travel from one point to another and is foundational for lines and vectors.

X and Y Coordinates

A point on a Cartesian plane is defined by two numbers, its x-coordinate and y-coordinate. These numbers form an ordered pair \( (x, y) \), where:

• The x-coordinate indicates movement along the horizontal axis and shows how far right or left the point is from the origin.
• The y-coordinate represents movement along the vertical axis to show how far up or down the point is from the origin.

In the exercise given, point A is \( (0, 4) \) and point B is \( (0, -2) \). This means for both points, the x-coordinate is 0, indicating they lie vertically aligned along the y-axis at different y-coordinate values, 4 and -2 respectively.

Point Calculation

Calculating the increments between two points involves understanding the differences in their corresponding coordinates. In this scenario, the calculated increments \( \Delta x \) and \( \Delta y \) provide insights into how the points are positioned relative to each other.
For point A \( (0, 4) \) and point B \( (0, -2) \):

• The x-coordinate increment \( \Delta x = 0 - 0 = 0 \), showing no horizontal change between the points. They align vertically.
• The y-coordinate increment \( \Delta y = -2 - 4 = -6 \) indicates a vertical movement downward from A to B of 6 units.

These calculations not only provide a numerical value for the distance moved but also indicate the direction of the move, which is essential for understanding geometric relationships.