Problem 96 - Essential Calculus

Question

A line with slope \(m\) passes through the point (0,-2) . (a) Write the distance \(d\) between the line and the point (4,2) as a function of \(m\). (b) Use a graphing utility to graph the equation in part (a). (c) Find \(\lim _{m \rightarrow \infty} d(m)\) and \(\lim _{m \rightarrow-\infty} d(m)\). Interpret the results geometrically.

Step-by-Step Solution

Answer

The distance function is \(d(m)=\frac{|4 - 4m|}{\sqrt{m^2 + 1}}\), which must be graphed using a graphing utility. The limits of \(d(m)\) as \(m\) approaches infinity and negative infinity are -1 and 1 respectively.

Step 1: Calculate the distance d

The slope intercept form of a straight line is \(y = mx + c\). Given the slope \(m\) and knowing that the line passes through the point (0,-2), the equation of the line is \(y=mx-2\). The distance \(d\) between the point (4,2) and the line can be calculated using the distance formula \(d(m)=\frac{|y_1 - mx_1 + c|}{\sqrt{m^2 + 1}}\), which results in \(d(m)=\frac{|2 - m*4 + 2|}{\sqrt{m^2 + 1}} = \frac{|4 - 4m|}{\sqrt{m^2 + 1}}\)

Step 2: Graph d(m)

To graph the function \(d(m)=\frac{|4 - 4m|}{\sqrt{m^2 + 1}}\), you can use a graphing utility and plot the function for a range of \(m\) values. Note that because of the absolute value | |, this is a mirrored graph around \(m=1\).

Step 3: Calculate limits

To calculate the limit of \(d(m)\) as \(m\) tends towards infinity and negative infinity, use the limit properties and rules. The function \(d(m)=\frac{|4 - 4m|}{\sqrt{m^2 + 1}}\) is a quotient of functions, so the limit can be found using the quotient rule of limits. Calculating \(\lim _{m \rightarrow \infty} d(m) = \lim _{m \rightarrow \infty} \frac{|4 - 4m|}{\sqrt{m^2 + 1}} = \frac{-\infty}{\infty}= -1 \). Calculating \(\lim _{m \rightarrow -\infty} d(m) = \lim _{m \rightarrow -\infty} \frac{|4 - 4m|}{\sqrt{m^2 + 1}} = \frac{\infty}{\infty}=1\). The geometric interpretation of these limits is that as the slope of the line gets steeper, in both the positive and negative direction, the distance to the point (4,2) gets closer to a constant value.

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form is crucial for working with linear equations. The formula for a line in slope-intercept form is given by: \[ y = mx + c \] where:

• \(m\) is the slope of the line, representing how steep the line is.
• \(c\) is the y-intercept, which is where the line crosses the y-axis.

For the given exercise, the line passes through the point (0, -2). This tells us that the y-intercept \(c\) is -2. So, the equation becomes \( y = mx - 2 \). This equation allows us to explore various slopes \(m\) while maintaining the same intercept. This is helpful, especially when calculating the distance from another fixed point to this line.

Limits in Calculus

Limits are an essential concept in calculus, used to analyze the behavior of functions as inputs approach a particular point. In this exercise, we are interested in what happens to the distance \(d(m)\) as the slope \(m\) of the line increases or decreases indefinitely.To calculate limits, we look at:

• \(\lim_{m \to \infty} d(m)\): This examines the behavior of the distance as the slope becomes very large. In our function, this results in a distance approaching -1.
• \(\lim_{m \to -\infty} d(m)\): This investigates the distance as the slope becomes very negative. Here, the distance approaches 1.

These limits suggest that as the line becomes steeper, regardless of the direction, the distance approaches a consistent value. This geometric interpretation implies that the steeper the line, the closer it gets to a specific distance from the point.

Graphing Functions

Graphing functions can provide a visual understanding of the mathematical relationships. In this exercise, plotting the distance function \(d(m) = \frac{|4 - 4m|}{\sqrt{m^2 + 1}}\) helps comprehend how the distance changes with varying slopes.To graph this, a graphing utility is used:

• Enter the given function into the utility.
• Explore it across various values of \(m\).

The absolute value in the function \(|4 - 4m|\) indicates symmetry. Thus, the graph will mirror about a specific line parallel to the y-axis, creating a symmetric shape. Observing this graphically can help confirm the results from limits, illustrating how the distance stabilizes as \(m\) approaches large magnitudes either positively or negatively.