Problem 12 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

For each function in Problems 12 through 14 : (a) Sketch the graph of \(f\). (b) Find \(f^{\prime}(x)\). Are there any values of \(x\) for which \(f^{\prime}\) is undefined? $$ f(x)=x+|x| $$

Step-by-Step Solution

Answer

The graph of the function is a straight line with a constant slope of 2 for \(x \geq 0\) and zero for \(x < 0\). The derivative of the function \(f(x)=x+|x|\) is \(f^{\prime}(x) = 0\) for \(x < 0\) and \(f^{\prime}(x) = 2\) for \(x \geq 0\). There are no points at which the derivative is not defined.

Step 1: Sketching the function

To sketch the function \(f(x)=x+|x|\), it is crucial to know how an absolute value function behaves. When \(x \geq 0\), absolute function will be \(x\). And, when \(x < 0\), absolute function will be \(-x\). Therefore, f(x) will be \(2x\) when \(x \geq 0\) and \(0\) when \(x < 0\) . The graph will start from origin and will have a constant slope of 2 for all values of \(x \geq 0\) showing a straight line.

Step 2: Find the derivative of the function

The derivative of the function \(f^{\prime}(x)\) can be found by differentiating the individual terms. Since the function is defined differently for positive and negative values of \(x\), we take the derivative separately. For \(x < 0\), \(f(x) = 0\), hence \(f^{\prime}(x) = 0\). For \(x \geq 0\), \(f(x) = 2x\), so \(f^{\prime}(x) = 2\).

Step 3: Check for Undefined Derivative

The derivative should be checked for undefined points. In this case, however, there are no fractions, radicals, or logarithms in the derivative, so the derivative is defined for all values of \(x\).

Key Concepts

Absolute Value Function

The absolute value function, denoted by \(|x|\), is a fundamental concept in calculus and mathematics. It's defined as the distance of a number from zero on the number line, regardless of its direction. This intrinsic property makes absolute value functions behave uniquely compared to linear or polynomial functions.

• For positive numbers and zero: \(|x| = x\).
• For negative numbers: \(|x| = -x\).

In the function \(f(x) = x + |x|\), the absolute value function dictates the expression's form depending on the value of \(x\).

Behavior Analysis

When \(x \geq 0\), \(|x|\) equals \(x\), so \(f(x) = 2x\), illustrating a linear form with slope 2. Meanwhile, when \(x < 0\), \(|x|\) turns into \(-x\), simplifying the function to \(f(x) = 0\). This creates a piecewise function that changes its behavior at the transition point of \(x = 0\).

Derivative

Understanding derivatives is a central part of calculus, as they represent the rate of change or slope of a function at any given point. The process of finding the derivative is called differentiation.For the function \(f(x) = x + |x|\), we determine its derivative considering the two distinct cases formed by the absolute value:

Finding Derivative by Piece

• For \(x < 0\): Since \(f(x) = 0\), the derivative \(f'(x)\) is zero.
• For \(x \geq 0\): The function simplifies to \(2x\), leading to a constant derivative \(f'(x) = 2\).

Points of Discontinuity

In this scenario, \(f(x)\) is a combination of linear functions with no discontinuities in the derivative as expressed. It's crucial to examine junction points like \(x = 0\) for potential issues, but here, the simple transition implies there are no undefined points.

Graphing Functions

Graphing functions is a visual method to understand how functions behave across different domains. It helps identify characteristics like intercepts, slopes, and continuity. In our exercise, the function \(f(x) = x + |x|\) showcases unique graphing aspects because of its dependence on the absolute value.

Piecewise Graphing

When graphing \(f(x)\), consider the separate functional expressions for different values of \(x\):

• For \(x < 0\): The graph is a flat line at \(y = 0\), indicating no change as \(f(x)\) remains constant at zero.
• For \(x \geq 0\): The graph is a straight line with a slope of 2, starting from the origin and extending into the first quadrant.

Interpreting the Transition

The transition at \(x = 0\) is notable because it's where the function's behavior shifts. Graphically, it's where the flat section switches to an incline. This kind of exercise is excellent for understanding how piecewise functions behave and transition smoothly, emphasizing the importance of breaking down tasks into smaller parts for clearer insights.