Problem 1 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems \(1-4\), plot the given points in the coordinate plane and then find the distance between them. \((3,1),(1,1)\)

Step-by-Step Solution

Answer

The distance between the points (3,1) and (1,1) is 2.

Step 1: Plot the Given Points

First, identify and plot the points \(3,1\) and \(1,1\) on the coordinate plane. These points are located at the coordinates where \(x=3, y=1\) and \(x=1, y=1\). Mark these two points clearly on the graph.

Step 2: Use the Distance Formula

Recall the distance formula for the distance \(d\) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) in a coordinate plane:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]Substitute the points \( (3, 1) \) and \( (1, 1) \) into the formula.

Step 3: Substitute Point Values into the Formula

For points \( (3,1) \) and \( (1,1) \), calculate the difference in the x-coordinates and the y-coordinates:- \(x_2 - x_1 = 1 - 3 = -2\)- \(y_2 - y_1 = 1 - 1 = 0\)Place these values into the distance formula: \[d = \sqrt{(-2)^2 + (0)^2}\]

Step 4: Calculate the Distance

Calculate the square values and the square root:- \((-2)^2 = 4\)- Add the values: \((4 + 0 = 4)\)- Finally, take the square root: \(\sqrt{4} = 2\)The distance between the points is therefore 2.

Key Concepts

Coordinate Plane

The coordinate plane is a two-dimensional surface where we can visualize and interpret mathematical ideas and relationships. It consists of two intersecting lines, typically called the x-axis (horizontal) and the y-axis (vertical). These axes divide the plane into four quadrants. Each point in the plane is defined by an ordered pair of numbers, usually written as \(x, y\).

• The x-coordinate shows how far to move horizontally from the origin (0,0).
• The y-coordinate shows the vertical movement from the origin.

Understanding the coordinate plane is essential as it provides a visual way to represent relationships between numbers. You can think of it like a map, where each point tells you how to get from the starting point (origin) to the specific location by taking steps along the x-axis and y-axis.

Plotting Points

Plotting points on the coordinate plane is a fundamental skill in graphing and analyzing data. It begins with identifying the values of the x and y coordinates from a given ordered pair. For example, the point \(3, 1\) means moving 3 units to the right along the x-axis (since x = 3) and then 1 unit up along the y-axis (since y = 1).

• Locate the x-coordinate on the horizontal axis.
• From there, move vertically to the corresponding y-coordinate.
• Mark the point where these movements intersect.

Plotting points accurately is crucial for interpreting data correctly, as each point represents valuable information. By mastering this skill, students gain a clearer understanding of how data can be used to make informed decisions.

Distance Calculation

Calculating the distance between two points on a coordinate plane may seem challenging at first, but with the distance formula, it becomes straightforward. The formula is derived from the Pythagorean theorem and is written as: \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]. This helps you find the straight-line distance between two points, \((x_1, y_1)\) and \((x_2, y_2)\).

• Subtract the x-coordinates of the two points to find the difference along the x-axis.
• Subtract the y-coordinates of the two points to find the difference along the y-axis.
• Square both differences, then add those squares together.
• Finally, take the square root of that sum to find the distance.

Using this formula allows for precise calculation of distances, a vital skill in both mathematics and real-world applications where measuring distances accurately is important.