Problem 25 - A Graphical Approach to Precalculus with Limits

Question

By considering the graph of the function but not calculating any limits, give the value of \(f^{\prime}(2)\) for each function. $$f(x)=-x$$

Step-by-Step Solution

Answer

\(f'(2) = -1\)

Step 1: Identify Function Type

The given function is a linear function: \(f(x) = -x\). Linear functions have constant slopes except at discontinuities, which do not exist for this function.

Step 2: Determine the Slope from the Linear Function

For a linear function \(f(x) = mx + b\), the derivative \(f'(x)\) is simply the constant \(m\), representing the slope. Here, \(f(x) = -x\) implies that \(m = -1\).

Step 3: Evaluate the Derivative at the Given Point

Since the derivative \(f'(x)\) is constant at \(-1\) for any \(x\), it follows that \(f'(2) = -1\).

Key Concepts

Linear Functions

A linear function is one of the simplest and most fundamental types of functions represented by the formula \( f(x) = mx + b \). In this equation:

• \( m \) is the slope of the line
• \( b \) is the y-intercept, which is the point where the line crosses the y-axis

Linear functions create straight lines when graphed, making them easy to analyze. Since they form a straight line, they have a constant rate of change. This simplicity is why they serve as a perfect introduction to other mathematical concepts like differentiation.

Slope

The slope of a line is a measure of its steepness. It tells us how much the y-value of a function changes for a change in the x-value. For a linear function represented by \( f(x) = mx + b \), the slope is the constant \( m \).
Here's how slope is helpful:

• A positive slope means the line rises as it moves from left to right.
• A negative slope means the line falls as it moves from left to right.
• If the slope is zero, the line is horizontal, indicating no vertical change.

In the function \( f(x) = -x \), the slope is \( -1 \). This means that for every 1 unit increase in \( x \), the y-value decreases by 1.

Graphical Analysis

Graphical analysis involves examining the graph of a function to understand its behavior. With linear functions, this means observing the line, its slope, and its y-intercept.
Some key points include:

• Linear functions yield straight lines.
• The steepness of the line is determined by the slope.
• The position where the line crosses the y-axis is the y-intercept.

In our example, the function \( f(x) = -x \) results in a line with a negative slope. If we plotted this, the line would descend from left to right, perfectly matching the mathematical description that its slope is \(-1\). Analyzing graphs is an intuitive way to grasp the underlying properties of functions.

Differentiation

Differentiation is a process in calculus that helps us find the derivative of a function. The derivative is a tool that measures how a function's output changes as its input changes. For a linear function, the derivative is straightforward—it is simply the slope \( m \).
For any function \( f(x) = mx + b \):

• The derivative \( f'(x) = m \).
• This derivative is constant because linear functions have a uniform rate of change.

In our problem, since \( f(x) = -x \), the derivative is \( f'(x) = -1 \). This tells us that the function changes at a constant rate of \(-1\) no matter the value of \( x \). Differentiation provides a powerful way to quantify change, forming the foundation for more complex calculus concepts.