Problem 11 - Calculus with Concepts in Calculus

Question

Determine the distance between the given points. \((a, a)\) and \((b, b)\)

Step-by-Step Solution

Answer

Distance is \(\sqrt{2} |b-a|\).

Step 1: Identify the Coordinates

We are given two points: - Point 1: - Coordinates: \((a, a)\)- Point 2: - Coordinates: \((b, b)\)

Step 2: Apply the Distance Formula

To find the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\), we use the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Substituting our specific points \((a, a)\) and \((b, b)\) into the formula, we get: \[ d = \sqrt{(b - a)^2 + (b - a)^2} \]

Step 3: Simplify the Expression

Simplify the expression under the square root: \[ d = \sqrt{2(b - a)^2} \]

Step 4: Simplify Further

Taking the square root of the expression: - We know \(\sqrt{x^2} = |x|\). - So, \(\sqrt{2(b-a)^2} = \sqrt{2} \cdot |b-a|\). Thus, the distance is: \[ d = \sqrt{2} |b-a| \]

Key Concepts

Distance Formula

The Distance Formula is a tool used in geometry to calculate the distance between two points in a plane. It's derived from the Pythagorean theorem. Imagine you are looking at two points on a graph, \((x_1, y_1)\) and \((x_2, y_2)\). The distance \(d\) between these points can be found by: \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]This may look complex, but it just means you find the difference in the x-coordinates and the difference in the y-coordinates of the two points. Next, square those differences, add them together, and take the square root of the total. Remembering this helps when you are given coordinates, and need to find how far apart they are on a grid. Getting comfortable with the formula can make it easier to visualize and solve real-world problems too.

Coordinate Geometry

Coordinate Geometry, also known as analytic geometry, is a branch of mathematics that uses algebra to investigate geometric problems. With coordinate geometry, we describe the positions of points on a plane using ordered pairs \((x, y)\). Each point on the plane is associated with exactly one pair of numbers, which helps create a connection between algebra and geometry.

• The x-coordinate tells you how far to move horizontally from the origin.
• The y-coordinate tells you how far to move vertically.

These coordinates can then be used to find other geometrical notions like distance, slopes, and midpoints. It's like having a map and using grid locations to navigate to different places. Understanding these concepts is crucial as it forms the backbone of more advanced geometry and even calculus.

Simplifying Expressions

Simplifying expressions is an essential technique in mathematics that makes complex expressions easier to work with or understand. It often involves combining like terms or reducing advanced expressions into simpler forms. In the context of the distance formula, reducing calculated results to their simplest form can make it easier to see patterns and find solutions.In our exercise, the expression \(\sqrt{2(b-a)^2}\) was simplified further by using properties of radicals and absolute values:

• The square of a number \((b-a)^2\) is always non-negative, and its square root is \(|b-a|\).
• Multiplying by the square root of 2, we get \(\sqrt{2} |b-a|\), which is the simplest form we present the distance.

Simplifying expressions helps in reducing errors and finding cleaner, more elegant solutions in mathematics. Developing this skill enhances problem-solving effectiveness and precision.