Problem 7 - Applied Calculus

Question

Find the slope (if it is defined) of the line determined by each pair of points. $$ (2,3) \text { and }(4,-1) $$

Step-by-Step Solution

Answer

The slope of the line is -2.

Step 1: Understand the Slope Formula

The slope formula for a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]where \(m\) represents the slope.

Step 2: Identify Given Points

Identify the coordinates from the given problem.The points in consideration are \((x_1, y_1) = (2, 3)\) and \((x_2, y_2) = (4, -1)\).

Step 3: Substitute Values into the Slope Formula

Plug the values from the points into the slope formula: \[ m = \frac{-1 - 3}{4 - 2} \]This simplifies to:\[ m = \frac{-4}{2} \]

Step 4: Simplify the Expression

Simplify the expression obtained:\[ m = \frac{-4}{2} = -2 \]

Step 5: Interpret the Result

The result \( m = -2 \) means the slope of the line is \(-2\), which indicates the line falls 2 units down for every 1 unit it moves to the right.

Key Concepts

Slope Formula

Understanding the slope formula is crucial in calculus problem-solving, especially when dealing with lines. The slope formula helps us find the steepness or incline of a line determined by two points. If you're given two points, \((x_1, y_1)\) and \((x_2, y_2)\), the formula is written as: \[ m = \frac{y_2 - y_1}{x_2 - x_1}\]where:

• \(m\) is the slope of the line.
• \(y_2 - y_1\) is the change in the \(y\)-coordinates, often called 'rise'.
• \(x_2 - x_1\) is the change in the \(x\)-coordinates, known as 'run'.

Basically, the slope \(m\) tells you how much the line goes up or down for each step it takes to the right. If \(m\) is a positive number, the line goes upward. If \(m\) is negative, the line goes downward. Zero means the line is flat, and an undefined slope means the line is vertical, having no movement in the \(x\) direction.

Point-Slope Calculation

Let's delve into point-slope calculation, a method used to compute the line equation. Once you've determined the slope \(m\) using the slope formula, the next step is to define your line using one of the given points.The point-slope form of a line's equation is:\[y - y_1 = m(x - x_1)\]where:

• \((x_1, y_1)\) are coordinates of a point through which the line passes.
• \(m\) is the slope you've calculated.

This form is particularly useful when you have the slope and one point, making it possible to write the equation of the line fairly straightforwardly. By substituting the known slope and a chosen point into the formula, you can derive the line’s equation that represents the line graphically.

Line Equations

Understanding line equations is an essential part of calculus problem solving. Whether you're working with simple, linear graphs or complex systems, grasping how to create line equations can greatly enhance your mathematical toolbox.The standard form of a line equation (with some algebra) relates directly to the concepts we've covered:\[Ax + By = C\]where:

• \(A\), \(B\), and \(C\) are integers.
• \(x\) and \(y\) are variables representing any point on the line.

However, when starting with slope-intercept form, \(y = mx + b\), it becomes immediately useful because \(m\) gives the slope and \(b\) the y-intercept (the point where the line crosses the \(y\)-axis).Working with line equations allows you to predict any particular point on a line or solve for one variable when the other is known. This ability to handle line equations plays a big role in broader mathematical concepts and real-world applications, such as calculating trends or understanding relationships between variables.