Problem 65 - Calculus

Question

Give an example of: A family of linear functions all with the same derivative.

Step-by-Step Solution

Answer

Functions of the form \( f(x) = 3x + c \) share the same derivative, where \( c \) is any real number.

Step 1: Understand the derivative of linear functions

A linear function can be represented as \( f(x) = mx + c \) where \( m \) is the slope and \( c \) is the y-intercept. The derivative of a linear function is the constant \( m \), which represents the rate of change or the slope.

Step 2: Set a constant slope value

To create a family of linear functions with the same derivative, we choose a constant slope value for all functions. Let's choose \( m = 3 \). So, the derivative for each function in this family is \( f'(x) = 3 \).

Step 3: Define a general form for the family

The general form of the linear functions will be \( f(x) = 3x + c \), where \( c \) can be any real number. Each different \( c \) will give a different function within the family, but all will have the same derivative of 3.

Step 4: Provide examples

Using the form \( f(x) = 3x + c \), we can give specific examples: 1. If \( c = 1 \), then \( f(x) = 3x + 1 \).2. If \( c = -5 \), then \( f(x) = 3x - 5 \).3. If \( c = 0 \), then \( f(x) = 3x \).Each of these functions has the same derivative, \( f'(x) = 3 \).

Key Concepts

Derivative

The concept of a derivative is essentially the core of calculus and provides us with a measure of how a function changes as its input changes. For linear functions, the derivative is straightforward. A linear function, expressed as \( f(x) = mx + c \), has a derivative of \( f'(x) = m \). This means for any linear function, the slope \( m \) is its derivative.

In simpler terms, the derivative tells you how steep a line is at any given point. For example, if \( m = 3 \), the line rises 3 units vertically for every 1 unit it moves horizontally. Thus, the derivative is a constant value for linear functions, indicating a uniform rate of change.

Slope

Slope is a fundamental concept that helps describe how a line behaves in a graph. In the equation of a linear function \( f(x) = mx + c \), the slope \( m \) determines the angle of the line. A higher absolute value of \( m \) represents a steeper line.

There are a few key characteristics of slope to remember:

• A positive slope means the line goes upwards as you move from left to right.
• A negative slope means the line goes downwards as you move from left to right.
• A zero slope is a horizontal line with no rise or fall.

Understanding slope is crucial because it defines the direction and steepness of a line.

Rate of Change

Rate of change is essentially another way to describe the slope of a function. In the context of a linear function \( f(x) = mx + c \), the rate of change is the same as the slope, \( m \).

It measures how much the dependent variable \( y \) changes for a unit change in the independent variable \( x \). For instance, if a line has a constant rate of change of 3, it indicates that for every unit increase in \( x \), \( y \) increases by 3 units. The constant rate of change makes linear functions straightforward to analyze, maintaining the same behavior across their domain.

Y-Intercept

The y-intercept of a line is the point where the line crosses the y-axis. It provides a starting point for the line within the graph. In the equation for a linear function \( f(x) = mx + c \), \( c \) is the y-intercept.

For example, if we have the function \( f(x) = 3x + 2 \), the y-intercept is 2. That means the line will cross the y-axis at the point (0, 2).

Understanding the y-intercept is important because it gives us a point of reference from which we can determine the rest of the line, especially when combined with the slope.

Family of Functions

A family of functions is a group of functions that share certain characteristics. In the context of our example, a family of linear functions can all have the same slope, but differ in their y-intercepts.

For instance, the functions \( f(x) = 3x + c \) represent a family of functions where \( m = 3 \) for all members, but \( c \) can be any real number.

Each unique value of \( c \) results in a different line within the same family.

• \( f(x) = 3x + 1 \)
• \( f(x) = 3x - 5 \)
• \( f(x) = 3x \)

All these examples are lines that have the same derivative of 3, indicating they belong to the same family. Families of functions are useful for understanding how slight changes in parameters can affect the behavior of the function.