Problem 66
Question
Explain, in two different ways, without using the rules of differentiation, why the derivative of the linear function \(f(x)=3 x-5\) must be \(f^{\prime}(x)=3\). [Hint: Think of the slope of the line \(y=m x+b\) that represents this function, and also of the instantaneous rate of change of a function that increases linearly.]
Step-by-Step Solution
Verified Answer
The derivative is 3 because it is the constant slope of the linear function.
1Step 1: Identify the Line's Slope
The function \(f(x) = 3x - 5\) is a linear function, which follows the general format of \(y = mx + b\), where \(m\) represents the slope of the line. For our given function, \(m = 3\). The slope of a linear function is constant and represents the change in \(y\) for each unit change in \(x\). Thus, the derivative, which represents the rate of change, is the slope of the line: \(f'(x) = 3\).
2Step 2: Instantaneous Rate of Change Explanation
Consider the concept of instantaneous rate of change, which for any linear function is constant across all points. Since the function increases linearly, the rate at which \(f(x)\) changes as \(x\) changes is uniform. This constant rate of change over any small interval \(\Delta x\) is the derivative of the function. For \(f(x) = 3x - 5\), this rate is constant and equals 3, which matches the slope of the function.
Key Concepts
linear functionsloperate of change
linear function
A linear function is one of the simplest types of functions you will encounter in mathematics. It is expressed generally as \( y = mx + b \).
The linear function forms a straight line when plotted on a graph. This function has two main components: the slope \( m \) and the y-intercept \( b \).
The slope \( m \) determines the steepness and direction of the line. A positive slope indicates the line is rising as it moves from left to right. A negative slope means it is falling.
This proportionality is what gives the line its constant slope, a key feature that simplifies understanding the derivative of linear functions.
The linear function forms a straight line when plotted on a graph. This function has two main components: the slope \( m \) and the y-intercept \( b \).
The slope \( m \) determines the steepness and direction of the line. A positive slope indicates the line is rising as it moves from left to right. A negative slope means it is falling.
- The y-intercept \( b \) is the point where the line crosses the y-axis, indicating the value of \( y \) when \( x = 0 \).
This proportionality is what gives the line its constant slope, a key feature that simplifies understanding the derivative of linear functions.
slope
The slope of a line is a measure of its steepness and direction, a critical concept in understanding linear functions. For a linear equation of the form \( y = mx + b \), the slope is represented by \( m \).
But what exactly does this slope mean? Consider it as a ratio of rise over run, or the change in \( y \) divided by the change in \( x \).
In the context of derivatives, understanding that the slope is constant for linear functions highlights why the derivative \( f'(x) \) of the function \( f(x) = 3x - 5 \) is \( 3 \). It reflects the constant change in \( y \) with respect to \( x \).
But what exactly does this slope mean? Consider it as a ratio of rise over run, or the change in \( y \) divided by the change in \( x \).
- Calculating the slope involves taking two points on the line, \( (x_1, y_1) \) and \( (x_2, y_2) \), and finding \( \frac{y_2 - y_1}{x_2 - x_1} \).
- For example, if the line passes through points (1, 2) and (3, 8), the slope will be \( \frac{8-2}{3-1} = 3 \).
In the context of derivatives, understanding that the slope is constant for linear functions highlights why the derivative \( f'(x) \) of the function \( f(x) = 3x - 5 \) is \( 3 \). It reflects the constant change in \( y \) with respect to \( x \).
rate of change
The rate of change is a fundamental concept that connects closely with the idea of slope in linear functions. It basically tells you how much one quantity changes in response to a change in another quantity.
For linear functions, like our example \( f(x) = 3x - 5 \), the rate of change is constant.
This steady rate of change equates to the slope of the line, \( m \), and in our specific function, this rate is \( 3 \).
In differentiation, the instantaneous rate of change of a function at any point is its derivative. For our linear function, the derivative \( f'(x) \) equals the slope, further cementing the understanding of why derivatives of linear functions are simply their slopes.
For linear functions, like our example \( f(x) = 3x - 5 \), the rate of change is constant.
- This constant rate means that for any change in \( x \), the change in \( f(x) \) remains the same over the entire domain of the function.
This steady rate of change equates to the slope of the line, \( m \), and in our specific function, this rate is \( 3 \).
In differentiation, the instantaneous rate of change of a function at any point is its derivative. For our linear function, the derivative \( f'(x) \) equals the slope, further cementing the understanding of why derivatives of linear functions are simply their slopes.
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