Problem 10 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

For Problems 10 through 16, give the set of functions all having the speci ed characteristics. Example: the set of nonvertical lines passing through \((3,2) .\) Solution: \(y-2=m(x-3)\), so \(f(x)=m(x-3)+2\), where \(m\) is any constant. The set of nonvertical lines passing through \((0, \pi)\)

Step-by-Step Solution

Answer

The set of all nonvertical lines passing through the point \((0, \pi)\) can be represented by the equation: \(y = mx + \pi\), where \(m\) represents any constant.

Step 1: Identify the given information

In the exercise, it can be understood that the given point through which all nonvertical lines pass is \((0, \pi)\).

Step 2: Apply the point-slope form of the linear equation

Now, replace \(x_1\) and \(y_1\) in the point-slope form equation with the given coordinates, which yields \(y - \pi = m(x - 0)\)

Step 3: Simplify the equation

The equation can be further simplified to \(y = mx + \pi\)

Key Concepts

Point-Slope Form

The point-slope form is a very handy equation when you're dealing with linear functions. It's particularly useful when you know a specific point on a line and the slope of the line itself. The formula looks like this: \[ y - y_1 = m(x - x_1) \] Where:

• \( (x_1, y_1) \) is a point that lies on the line.
• \( m \) represents the slope of the line.

Using this form, you can easily plug in the values for one point on the line and the slope to get the equation for the entire line.
For instance, if you have a line passing through the point \((0, \pi)\) with any slope \( m \), you'd replace \( x_1 \) with 0 and \( y_1 \) with \( \pi \) in the point-slope formula.

Nonvertical Lines

Nonvertical lines are all the lines on a plane except those that go straight up and down. These lines have a defined slope, meaning they slant in some direction—either upward from left to right or downward. A vertical line, on the other hand, would have an undefined slope because it doesn't move horizontally at all.
When dealing with a problem requiring you to find the equation of all nonvertical lines passing through a point, like \((0, \pi)\), the first thing you can do is rule out the vertical lines. Every line through this point will have a slope \( m \), which can be any real number except infinity.
This allows for the creation of infinite linear equations, each with its own unique slope, making nonvertical lines both flexible and varied while still maintaining measurable properties like slope.

Equation of a Line

The equation of a line is what people generally use to describe or predict any point that falls on that line. It helps define how the line behaves on a two-dimensional surface.
From the point-slope form, \( y - y_1 = m(x - x_1) \), one can also refine the equation into the slope-intercept form, which is perhaps the most recognized form:\[ y = mx + b \] Here, \( m \) is the slope, and \( b \) is the y-intercept, indicating where the line crosses the y-axis.
In practical terms, if you're given the point \((0, \pi)\) through which the line passes, fitting into the equation becomes straightforward: you get \( y = mx + \pi \), showing a range of nonvertical lines passing through this point, each governed by its slope \( m \). This form allows anyone to quickly graph or understand the behavior of the line based on how steep it is and where it starts on the y-axis.