Question

a. Graph the function \(f(x)=\left\\{\begin{array}{cc}x & \text { for } x \leq 0 \\ x+1 & \text { for } x > 0\end{array}\right.\) b. For \(x < 0,\) what is \(f^{\prime}(x) ?\) c. For \(x > 0,\) what is \(f^{\prime}(x) ?\) d. Graph \(f^{\prime}\) on its domain. e. Is \(f\) differentiable at \(0 ?\) Explain.

Short answer

Provide a brief explanation for your response.

Step-by-step solution

a. Graph the function f(x)

To graph the function, we need to plot both parts separately: 1. When \(x \leq 0\), \(f(x) = x\) - This is a linear function with a slope of 1 and passes through the origin (0,0). 2. When \(x > 0\), \(f(x) = x + 1\) - This is also a linear function with a slope of 1 and passes through the point (0, 1). Keep in mind that the graph must be continuous, so draw a solid line for the first piece and a dashed line for the second piece.

b. For \(x < 0,\) find \(f^{\prime}(x)\)

When \(x < 0\), \(f(x) = x\). To find the derivative, we differentiate \(f(x)\) with respect to \(x\): \(f^{\prime}(x) = \frac{d}{dx}(x) = 1\). So, for \(x < 0\), \(f^{\prime}(x) = 1\).

c. For \(x > 0,\) find \(f^{\prime}(x)\)

When \(x > 0\), \(f(x) = x + 1\). To find the derivative, we differentiate \(f(x)\) with respect to \(x\): \(f^{\prime}(x) = \frac{d}{dx}(x + 1) = 1\). So, for \(x > 0\), \(f^{\prime}(x) = 1\).

d. Graph \(f^{\prime}\) on its domain

Now let's graph \(f^{\prime}(x)\): 1. When \(x < 0\), \(f^{\prime}(x) = 1\). This is a constant function with a constant value of 1. Draw a horizontal line at \(y = 1\) for all \(x \leq 0\). 2. When \(x > 0\), \(f^{\prime}(x) = 1\). This is also a constant function with a constant value of 1. Draw another horizontal line at \(y = 1\) for all \(x > 0\). The graph should show a continuous horizontal line at \(y = 1\) with an open circle when \(x = 0\).

e. Is \(f\) differentiable at \(0 ?\) Explain.

To check if \(f\) is differentiable at \(x = 0\), we need to do the following: 1. Check if \(f(x)\) is continuous at \(x = 0\): \(f(0) = 0\) and the limit of \(f(x)\) as \(x\) approaches 0 from the left is 0, and the limit as \(x\) approaches 0 from the right is 1. So, \(f(x)\) is not continuous at \(x = 0\). 2. Check if the derivative exists at \(x = 0\): \(f^{\prime}(0)\) doesn't exist as we have an open circle at \(x = 0\) on our graph of \(f^{\prime}(x)\). Since \(f(x)\) is not continuous and \(f^{\prime}(0)\) doesn't exist, we can conclude that \(f\) is not differentiable at \(x = 0\).

Key concepts

Graphing Piecewise Functions

Graphing piecewise functions involves plotting multiple sub-functions, each defined over a certain interval. These functions are pieced together to form the entire function.

Consider the function provided in the exercise, which consists of two linear pieces. The first piece is defined for all values of x less than or equal to zero, and the second piece for x greater than zero. A common mistake is failure to properly represent the 'break' in the function at the boundary point, which is x=0 in this case. Making sure the line is solid where the function is defined (including the boundary for the left piece) and dashed where it is not (excluding the boundary for the right piece) is crucial.

Graphing piecewise functions accurately is essential for understanding their behavior, particularly as it pertains to evaluating limits, continuity, and differentiability at the points where the function pieces meet.

Derivatives of Piecewise Functions

The derivative of piecewise functions must be considered separately for each piece. Since differentiation is a local operation, you calculate the derivative for each interval without crossing boundaries in the domain.

In our exercise, both pieces of the function are linear with a constant slope of 1. The derivative of each piece individually yields a constant value, which results in a horizontal line. The key point here is that while the derivative within each interval is the same, verifying the behavior of the derivative at the boundary is the next crucial step—this pertains to checking differentiability, as explored further in the subsequent sections. Understanding that the derivatives for each piece of the function can be calculated independently helps avoid confusion when working with more complex piecewise functions.

Continuity of Piecewise Functions

Continuity of piecewise functions is another aspect integral to understanding how these functions behave. A piecewise function is continuous at a point if there are no jumps, gaps, or vertical asymptotes at that point.

For the function in our exercise, while both pieces are continuous by themselves, there is a step discontinuity at x=0 where the two pieces meet—specifically, there's a jump of 1 unit. It's this type of discontinuity that students often overlook. When graphing, observing the values of the function from the left and the right sides of the boundary point can help you identify potential issues with continuity, a vital step before considering differentiability.

Concept of Differentiability

Differentiability refers to the existence of a unique tangent line to the curve at a given point; in analytical terms, this corresponds to the existence of a derivative at that point. A function can be continuous but not differentiable; a classic example is the absolute value function, which is continuous everywhere but not differentiable at x=0.

In the exercise, while the derivatives on either side of x=0 are equal, the derivative at x=0 itself doesn't exist due to the step discontinuity. This is a key takeaway: for a function to be differentiable at a point, it must first be continuous there. Examining both continuity and the behavior of the derivative at the points where different pieces of the function meet is crucial for understanding whether a piecewise function is differentiable at those points.