Question

Given the function \(y=5 x-2\) (a) find the difference quotient \(\Delta y / \Delta x\). What type of function is it? (b) since the expression \(\Delta x\) does not appear in the function \(\Delta y / \Delta x\) in part \((a),\) does it make any difference to the value of \(\Delta y / \Delta x\) whether \(\Delta x\) is large or small? Consequently, what is the limit of the difference quotient as \(\Delta x\) approaches zero?

Short answer

(a) The difference quotient is 5, and the function is linear. (b) The value of the quotient is constant, so \( \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x} = 5 \).

Step-by-step solution

Identify the Components for Difference Quotient

The given function is \( y = 5x - 2 \). To find the difference quotient, we need the expression for \( \Delta y = y(x + \Delta x) - y(x) \).

Calculate \( y(x + \Delta x) \)

Substitute \( x + \Delta x \) into the function: \( y(x + \Delta x) = 5(x + \Delta x) - 2 \). Expanding it, we get: \( y(x + \Delta x) = 5x + 5\Delta x - 2 \).

Find \( \Delta y \)

Using the expressions from Steps 1 and 2, \( \Delta y = (5x + 5\Delta x - 2) - (5x - 2) \). Simplify to get \( \Delta y = 5\Delta x \).

Calculate the Difference Quotient \( \Delta y / \Delta x \)

Divide \( \Delta y \) by \( \Delta x \): \( \frac{\Delta y}{\Delta x} = \frac{5\Delta x}{\Delta x} \). Simplifying, we get \( \frac{\Delta y}{\Delta x} = 5 \).

Determine Type of Function

Since the difference quotient \( \frac{\Delta y}{\Delta x} = 5 \) is constant and does not depend on \( x \), the function \( y = 5x - 2 \) is a linear function.

Analyze the Value for Different \( \Delta x \)

The expression for \( \frac{\Delta y}{\Delta x} = 5 \) does not contain \( \Delta x \), indicating its value is constant regardless if \( \Delta x \) is large or small.

Limit as \( \Delta x \to 0 \)

As \( \Delta x \to 0 \), \( \frac{\Delta y}{\Delta x} = 5 \) remains the same since it is independent of \( \Delta x \). Thus, the limit of the difference quotient is 5.

Key concepts

Linear Functions Explained

Linear functions are one of the simplest types of mathematical functions. They are called 'linear' because their graph is a straight line. A linear function can be written in the form of \( y = mx + b \), where \( m \) and \( b \) are constants.

• \( m \) is the slope of the line. It tells us how steep the line is and the direction in which it increases or decreases.
• \( b \) is the y-intercept. It represents the point where the line crosses the y-axis.

In our example, the linear function is \( y = 5x - 2 \). Here, the slope \( m \) is 5, indicating the line rises steeply, and the y-intercept \( b \) is -2, showing it crosses the y-axis 2 units below the origin.
Linear functions have a constant rate of change, meaning as \( x \) increases by 1 unit, \( y \) increases by \( m \) units. In this case, for every 1 unit increase in \( x \), \( y \) increases by 5 units. This consistency is key to understanding the behavior of linear functions and why the difference quotient (change in y over change in x) is always constant, as seen in this exercise.

Understanding the Limit of a Function

The limit of a function is a fundamental concept in calculus. It describes the value that a function approaches as the input (or x-value) approaches a certain point. Limits allow us to understand the behavior of functions at points that might not be easily accessible, like where they are not defined.
In this exercise, we're interested in the limit of the difference quotient as \( \Delta x \to 0 \). The difference quotient is \( \frac{y(x + \Delta x) - y(x)}{\Delta x} \), and it simplifies to \( 5 \) for the given linear function \( y = 5x - 2 \).
Because the difference quotient is a constant value of 5, regardless of the value of \( \Delta x \), its limit as \( \Delta x \to 0 \) is also 5. This behavior is typical of linear functions, where the rate of change (slope) does not depend on the x-values and remains constant. Understanding limits is crucial for analyzing how functions behave under small changes, providing the groundwork for derivatives in calculus.

Essential Calculus Concepts

Calculus is a branch of mathematics that deals with continuous change and is used for calculating quantities related to change. Key ideas in calculus include differentiation and integration.

• Differentiation involves finding the rate at which a quantity changes. It's closely related to limits and the difference quotient. The difference quotient like the one in this exercise is the foundation of the derivative, which represents an instantaneous rate of change.
• Integration is about accumulation, such as finding the area under a curve.

These calculus concepts enable us to model and solve problems involving continuous change effectively. For our exercise, understanding the difference quotient and its limit helps us grasp the core idea of derivatives in calculus. It shows how the function behaves as changes become infinitesimal, which is crucial for applications in physics, engineering, economics, and beyond. Engaging with these concepts fosters deeper insights into how calculus helps in making predictions and understanding dynamics in various fields.