Problem 69 - Calculus

Question

Give an example of:A linear function with derivative 2 at \(x=0\).

Step-by-Step Solution

Answer

An example is \( f(x) = 2x + 1 \).

Step 1: Understand the Definition of a Linear Function

A linear function is of the form \( f(x) = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept of the line.

Step 2: Find the Derivative of a Linear Function

The derivative of a function \( f(x) = mx + c \) is a constant, \( f'(x) = m \). This derivative represents the rate of change of the function.

Step 3: Set the Derivative Equal to the Given Value

We want the derivative of the function at \( x = 0 \) to be 2. Since the derivative of a linear function is constant, it will be 2 for all \( x \). Therefore, set \( m = 2 \).

Step 4: Construct the Function

Using the slope \( m = 2 \) found in the previous step, the function becomes \( f(x) = 2x + c \). Use any value for \( c \) as it does not affect the derivative.

Step 5: Verify the Function Satisfies the Condition

Check that the derivative is indeed \( 2 \) at \( x = 0 \). We know \( f'(x) = 2 \), which satisfies the condition given in the exercise.

Key Concepts

Linear Function

A linear function is a fundamental concept in calculus, represented by the formula \( f(x) = mx + c \). This equation defines a straight line when plotted on a graph.
Here are some key points about linear functions:

• **Slope (\(m\))**: It dictates the steepness and direction of the line.
• **Y-intercept (\(c\))**: The point where the line crosses the y-axis.

Linear functions are widely used in mathematics to model relationships with a constant rate of change. They form the foundation for more complex mathematical concepts.

Derivative

In calculus, the derivative is a measure of how a function changes as its input changes. For a function \( f(x) \), the derivative is denoted \( f'(x) \).
In the context of linear functions, such as \( f(x) = mx + c \), the derivative is simply the coefficient \(m\). This is because a linear function changes at the same rate throughout its entire domain.

• **Constant Derivative**: For linear functions, the derivative \( f'(x) = m \) is constant, indicating that the slope does not change.
• **Meaning**: The derivative reflects the function's sensitivity to changes in \( x \).

The simplicity of derivatives in linear functions makes them an appealing starting point for understanding calculus.

Slope

The slope is a critical concept of linear functions, revealing how much the function's value increases or decreases as \( x \) changes.
The formula for slope in a linear function is the coefficient \(m\) in \( f(x) = mx + c \).

• A positive slope (\(m>0\)) indicates the function is increasing.
• A negative slope (\(m<0\)) shows the function is decreasing.
• A zero slope means the function is constant and has a horizontal line.

Understanding slope helps to anticipate the behavior of the function without graphing. It's a straightforward yet crucial component of analyzing functions.

Rate of Change

Rate of change in calculus refers to how quickly one quantity changes in relation to another.
For a linear function, this is given by the slope \(m\).
The rate of change can be:

• **Constant**: In linear functions there is a uniform rate of change.
• **Identical to the derivative**: As it expresses the rate at which the dependent variable responds to changes in an independent variable.

Rate of change is important for interpreting real-world scenarios, such as speed in physics or economic trends, through a mathematical lens. It allows us to quantify changes and make predictions based on linear models.