Problem 36 - Calculus for Biology and Medicine

Question

In Problems , graph each function and, on the basis of the graph, guess where the function is not differentiable. (Assume the largest possible domain.)$$ f(x)=\left\\{\begin{array}{cl} x & \text { for } x \leq 0 \\ x+1 & \text { for } x>0 \end{array}\right. $$

Step-by-Step Solution

Answer

The function is not differentiable at \(x=0\) due to the jump discontinuity.

Step 1: Understand the Piecewise Function

The function is defined as a piecewise function with two different rules: for \(x \leq 0\), the function is \(f(x) = x\), and for \(x > 0\), the function is \(f(x) = x + 1\). This indicates a potential point of interest at \(x = 0\), where the rule of the function changes.

Step 2: Graph the Piecewise Functions Separately

Graph each piece of the function separately. For \(x \leq 0\), graph \(f(x) = x\), which is a straight line through the origin with slope 1. For \(x > 0\), graph \(f(x) = x + 1\), which is a straight line starting at \( (0, 1) \) with the same slope. These lines are parallel and do not connect smoothly at \(x=0\).

Step 3: Examine the Continuity at x=0

At \(x=0\), check the left-hand limit and the right-hand limit. For \(x\leq 0\), \(f(x)\) approaches \(0\). For \(x > 0\), \(f(x)\) at \(x=0\) is \(1\). Therefore, there is a jump discontinuity at \(x=0\). This typically indicates non-differentiability.

Step 4: Check Differentiability Criteria

A function is non-differentiable at a point if there is a discontinuity or a sharp corner (cusp). Since there's a jump discontinuity at \(x=0\), the function is not differentiable at \(x=0\). There are no other points where non-differentiability might occur since the function is otherwise composed of straight lines.

Key Concepts

Non-Differentiability

A function is non-differentiable at a point when its graph has certain features that prevent the derivative, or slope of the tangent line, from being defined. These features include:

• A corner point or cusp, where the direction of the graph abruptly changes
• Vertical tangent lines, where the slope becomes infinite
• Discontinuities, where the function makes a jump or has a gap

In the provided piecewise function, non-differentiability occurs at the junction point, which is at \(x = 0\). The reason is due to the jump discontinuity present there. As you trace the graph from left to right, the line \(f(x) = x\) ends at the origin (0,0). Just after this point, the line \(f(x) = x + 1\) begins at point (0,1).
The gap of 1 unit between these two points means the graph lacks a smooth transition, which results in non-differentiability. So if you trace along the graph, there is a visible jump at \(x = 0\) which confirms the lack of differentiability at that point.

Graphing Functions

Graphing a piecewise function involves plotting each segment of the function separately on the same coordinate plane. This technique can reveal important characteristics such as discontinuities or inflection points.

Steps to Graph the Function:

• Start with \(f(x) = x\) for \(x \leq 0\). It is a simple linear function with a slope of 1, passing through the origin. So, for these values, the graph will be a straight line extending left from \(x = 0\).
• Next, plot \(f(x) = x + 1\) for \(x > 0\). This line has the same slope of 1 but is shifted upward on the y-axis by 1 unit. It starts from the point (0,1) and moves to the right.

Both segments are straight lines. They are parallel and do not connect at \(x=0\), highlighting the discontinuity here. This gap is crucial for identifying where the function is non-differentiable.

Continuity and Discontinuity

Understanding continuity and discontinuity helps to predict the behavior of a graph around given points. A function is continuous at a point if the left-hand limit, right-hand limit, and the function's value at that point are all equal.

In this piecewise function:

• The left-hand limit as \(x \to 0^-\) (approaching from the left) results in \(f(x) = 0\).
• The right-hand limit as \(x \to 0^+\) (approaching from the right) gives \(f(x) = 1\).
• The value of the function at \(x=0\) is not equal to the right-hand limit since the function dictates no specific value at this exact point for \(x \leq 0\).

There is a discrepancy between these values, specifically a jump of 1 unit. This indicates a jump discontinuity at \(x=0\). Discontinuities are important because they are often points of non-differentiability. In simple terms, when a graph jumps or has a gap, it cannot have a tangent, united slope, resulting in non-differentiability.