Problem 36 - Applied Calculus

Question

Find the equation of the tangent line to the graph of \(f(x)=\frac{2 x-5}{1}\) at the point at which \(x=0\).

Step-by-Step Solution

Answer

The equation of the tangent line is \( y = 2x - 5 \).

Step 1: Understand the Function

The given function is \( f(x) = \frac{2x - 5}{1} \), which simplifies to \( f(x) = 2x - 5 \). This is a linear function, which means the graph is a straight line.

Step 2: Find the Slope of the Function

Since the function \( f(x) = 2x - 5 \) is already in the slope-intercept form \( y = mx + b \), the slope \( m \) of this line is 2.

Step 3: Determine the Point on the Function When x=0

To find the point where \( x = 0 \), substitute \( x = 0 \) into the function: \[ f(0) = 2(0) - 5 = -5 \]. Therefore, the point is \( (0, -5) \).

Step 4: Write the Equation of the Tangent Line

The equation of the tangent line at any given point for a linear function is the same as the function itself because the derivative (slope) is constant. Thus, the equation of the tangent line at \( (0, -5) \) is \( y = 2x - 5 \).

Key Concepts

Linear Functions

Linear functions are some of the simplest and most foundational elements in mathematics. A linear function is a type of mathematical function that creates a straight line when graphed. The general form of a linear function is expressed as \( y = mx + b \), where:

• \( y \) is the dependent variable (often representing the output of the function).
• \( m \) is the slope, which reflects the rate at which \( y \) changes with respect to \( x \).
• \( x \) is the independent variable (generally the input value).
• \( b \) is the y-intercept, which is the point where the line crosses the y-axis.

Understanding linear functions is crucial, as they are often used to model relationships with constant rates of change, making them applicable across various fields such as physics, economics, and statistics. In the provided exercise, the function \( f(x) = 2x - 5 \) is a linear function where the slope is 2, indicating a steady increase, and the y-intercept is -5.

Slope-Intercept Form

The slope-intercept form is a way of writing the equation of a line so that it is easy to graph. The form \( y = mx + b \) is particularly useful because it immediately gives information about:

• the slope \( m \), which shows how steep the line is, and
• the y-intercept \( b \), which indicates where the line crosses the y-axis.

Using this format allows you to quickly identify the characteristics of the line. For example, in the function \( f(x) = 2x - 5 \), the slope \( m \) is 2. This means for every unit increase in \( x \), \( y \) increases by 2 units. The y-intercept, \( b \), is -5, meaning the line crosses the y-axis at the point (0, -5).
These components provide a straightforward way to begin graphing the line or understanding its behavior.

Derivatives

Derivatives are a fundamental concept in calculus representing the rate at which a function changes at any given point. For linear functions, however, derivatives play a rather simple role. Since linear functions have constant rates of change, their derivatives are constant.
For instance, the derivative of \( f(x) = 2x - 5 \) is simply the slope of the line, which is 2. This constant derivative indicates that no matter where you are along the line, the rate of change remains fixed.
When discussing the tangent line in this exercise, the derivative tells us that at the point \( x = 0 \), the rate of change of the function (or slope of the tangent) is 2.
For linear functions, the derivative confirms that the tangent line is the same as the original line. This makes finding the tangent line straightforward since it merely requires identifying the linear equation itself.