Problem 77 - Calculus Volume 1

Question

For the functions, a. sketch the graph and b. use the definition of a derivative to show that the function is not differentiable at \(x=1\). $$ f(x)=\left\\{\begin{array}{l} 2 x, x \leq 1 \\ \frac{2}{x}, x>1 \end{array}\right. $$

Step-by-Step Solution

Answer

The function is not differentiable at \(x=1\) because the left and right-hand derivatives differ.

Step 1: Analyze the Function

The function provided is a piecewise function. For \(x \leq 1\), the function is \(f(x) = 2x\). For \(x > 1\), the function is \(f(x) = \frac{2}{x}\). These are both simple algebraic functions: a linear function and a rational function, respectively.

Step 2: Sketch the Graph

To sketch the graph, plot the line \(y = 2x\) for \(x \leq 1\). For \(x > 1\), plot the curve \(y = \frac{2}{x}\). At \(x = 1\), ensure the graph shows a potential discontinuity, since \(f(1) = 2(1) = 2\) and as \(x\) approaches 1 from the right, \(f(x) = \frac{2}{x}\) approaches 2, indicating the graph does not appear discontinuous, but we need further analysis to confirm differentiability.

Step 3: Use the Definition of Derivative

To check differentiability at \(x=1\), compute the derivative from the left and right. Using the definition of the derivative, \(f'(x) = \lim_{{h \to 0}} \frac{f(x+h) - f(x)}{h}\).- For \(x \leq 1\), \(f(x) = 2x\), so \(f'(1^-) = \lim_{{h \to 0^-}} \frac{2(1+h) - 2}{h} = 2\).- For \(x > 1\), \(f(x) = \frac{2}{x}\), so \(f'(1^+) = \lim_{{h \to 0^+}} \frac{\frac{2}{1+h} - 2}{h}\). After simplifying, this limit does not result in a finite value, showing that the right-hand derivative does not equal the left-hand derivative.

Step 4: Conclusion of Non-differentiability

Since the left-hand derivative \(f'(1^-) = 2\) and either the right-hand derivative is undefined or does not equal 2, the function \(f(x)\) is not differentiable at \(x=1\). Differentiability requires that both side derivatives exist and are equal.

Key Concepts

Piecewise Function

A piecewise function is a function that is defined by different expressions depending on the interval of the input variable. In our problem, the piecewise function is given as:

• For \(x \leq 1\), the function is \(f(x) = 2x\), a linear function.
• For \(x > 1\), the function is \(f(x) = \frac{2}{x}\), a rational function.

These types of functions are versatile because they can model scenarios where a single formula does not suffice. However, they can introduce challenges in analysis, such as evaluating continuity and differentiability at the boundaries where the definition changes. For this piecewise function, we need to pay close attention to what happens at \(x = 1\) since it's the transition point between the two defined expressions.

Definition of Derivative

The derivative of a function at a particular point measures how the function's value changes as the input changes. Formally, it is the limit of the difference quotient as the increment approaches zero:\[f'(x) = \lim_{{h \to 0}} \frac{f(x+h) - f(x)}{h}.\]For differentiability at a point, both the left-hand derivative and the right-hand derivative must exist and be equal. This means:

• The limit from the left must equal the value of the derivative coming from values smaller than the point.
• The limit from the right must equal the value coming from values larger than the point.

In our context, at \(x = 1\), the left-hand derivative \(f'(1^-) = 2\). The challenge comes with the right-hand derivative when the expression becomes the rational function \(f(x) = \frac{2}{x}\), leading to an undefined or infinite value. Thus, the function is not differentiable at \(x=1\). Differentiability requires both side derivatives to be equal and finite.

Graph Sketching

Graph sketching is an effective visual tool to understand a function's behavior. To sketch a piecewise function like \(f(x)\):

• For \(x \leq 1\): Plot the line \(y = 2x\), which is a straight line passing through the origin with a slope of 2 up to the boundary at \(x = 1\).
• For \(x > 1\): Plot the curve \(y = \frac{2}{x}\), a hyperbola that approaches the y-axis as \(x\) decreases and flattens as \(x\) increases.

When graphing these two parts:

• Ensure accurate plotting of the linear portion from \(x=0\) to \(x=1\).
• Switch to the rational function beyond \(x=1\), understanding it has a different rate of change.

It's crucial to label \(x=1\) as a point of interest to analyze continuity and differentiability, as visually inspecting the graph may not be sufficient alone to assess these properties. A close inspection reveals that although the function values seem to meet at \(x=1\), the lack of a consistent derivative across this point demonstrates the break in differentiability.