Problem 22 - Calculus Single Variable

Question

Calculate \(f^{\prime}(x),\) and sketch the graph of the equation \(y=f^{\prime}(x)\). $$ f(x)=3 x-|x| $$

Step-by-Step Solution

Answer

The derivative is piecewise: \( f'(x) = 2 \) if \( x \geq 0 \) and \( f'(x) = 4 \) if \( x < 0 \). The graph has two horizontal lines at \( y=2 \) and \( y=4 \).

Step 1: Identify the Function Components

The function given is \( f(x) = 3x - |x| \). This function is composed of two components: \( 3x \), a linear function, and \( |x| \), an absolute value function. We need to consider how these components behave for different values of \( x \).

Step 2: Understand the Absolute Value Function

The absolute value function \( |x| \) behaves differently depending on whether \( x \) is positive or negative. \( |x| = x \) when \( x \geq 0 \) and \( |x| = -x \) when \( x < 0 \). This distinction will help us express the function \( f(x) \) without the absolute value.

Step 3: Express the Function in Piecewise Form

Based on the behavior of \( |x| \), we can express \( f(x) \) as a piecewise function:- If \( x \geq 0 \): \( f(x) = 3x - x = 2x \).- If \( x < 0 \): \( f(x) = 3x - (-x) = 4x \).

Step 4: Determine the Derivative for Each Case

Now, find the derivative \( f'(x) \) for each piece of the piecewise function:- If \( x \geq 0 \): \( f(x) = 2x \) implies \( f'(x) = 2 \).- If \( x < 0 \): \( f(x) = 4x \) implies \( f'(x) = 4 \).

Step 5: Construct the Derivative Function

Combine the derivatives into a piecewise function:- \( f'(x) = \begin{cases} 2 & \text{if } x \geq 0 \ 4 & \text{if } x < 0 \end{cases} \).

Step 6: Sketch the Graph of the Derivative Function

To graph \( y = f'(x) \), plot two constant lines:- A horizontal line at \( y = 2 \) for all \( x \geq 0 \).- A horizontal line at \( y = 4 \) for all \( x < 0 \). These two lines represent the piecewise nature of \( f'(x) \) and indicate a step change at \( x = 0 \).

Key Concepts

Piecewise Functions

Piecewise functions are mathematical expressions that have different forms for different parts of their domain. They allow us to construct functions that can change behavior at specific points. In our exercise, we have a function \( f(x) = 3x - |x| \), which can be expressed as different linear expressions based on the value of \( x \).
- When \( x \geq 0 \), \( f(x) \) simplifies to \( 2x \).- When \( x < 0 \), \( f(x) \) becomes \( 4x \).
This kind of breakdown showcases a core characteristic of piecewise functions: they manage different conditions or intervals separately but within the same overall function. This piecewise nature makes it easier to calculate derivatives, as each piece is a standard function, either a line or a polynomial, allowing for straightforward differentiation.

Derivatives

In calculus, derivatives represent the rate at which a function changes. They are crucial for understanding the behavior of functions. For piecewise functions like our example \( f(x) = 3x - |x| \), derivatives can be determined separately for each piece.

• For \( f(x) = 2x \) when \( x \geq 0 \), the derivative \( f'(x) \) is simply \( 2 \), representing a constant rate of change.


• For \( f(x) = 4x \) when \( x < 0 \), the derivative \( f'(x) \) is \( 4 \), again a constant but at a steeper slope.

Thus, the derivative function \( f'(x) \) itself is a piecewise function:\[ f'(x) = \begin{cases} 2 & \text{if}\ x \geq 0 \ 4 & \text{if}\ x < 0 \end{cases} \]
Knowing the derivative allows us not only to graph the function's rate of change but also to anticipate how the function values develop over time and different intervals.

Absolute Value Functions

The absolute value function, denoted as \(|x|\), is fundamental in calculus as it measures the "distance" from zero, ignoring direction. It transforms values to non-negative equivalents, like converting \(-5\) to \(5\).
- When \( x \geq 0 \), \(|x| = x\).- When \( x < 0 \), \(|x| = -x\).
This duality captures the essence of absolute value: treating all numbers, whether positive or negative, in terms of their magnitude only. This characteristic allows \(|x|\) to effectively alter expressions and equations, such as in our function \( f(x) = 3x - |x| \), transforming a linear equation into a versatile piecewise function. Absolute values contribute significantly to piecewise functions because they introduce necessary conditions based on the behavior of the input values, distinguishing how the function reacts to different parts of its domain.