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Magazine Chapter 2: Problem 65
Give an example of: A family of linear functions all with the same derivative.
Short Answer
Expert verified
Functions of the form \( f(x) = 3x + c \) share the same derivative, where \( c \) is any real number.
Step by step solution
01
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.
02
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 \( f'(x) = 3 \).
03
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.
04
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, \( f'(x) = 3 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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.
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:
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.
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.
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.
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.
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.
- \( f(x) = 3x + 1 \)
- \( f(x) = 3x - 5 \)
- \( f(x) = 3x \)
