Problem 6 - Calculus

Question

decide if the function is differentiable at \(x=0 .\) Try zooming in on a graphing calculator, or calculating the derivative \(f^{\prime}(0)\) from the definition. $$f(x)=\left\\{\begin{array}{ll} -2 x & \text { for } x<0 \\ x^{2} & \text { for } x \geq 0 \end{array}\right.$$

Step-by-Step Solution

Answer

The function is not differentiable at \(x=0\).

Step 1: Understand the Definition of Differentiability

A function is differentiable at a point if it has a derivative at that point. This means the function has a well-defined tangent at that point, and the left-hand and right-hand derivatives are equal there.

Step 2: Find the Left-Hand Derivative

For the left-hand derivative, we consider the part of the function where \(x<0\), which is \(-2x\). The derivative of \(-2x\) is \(-2\). Thus, the left-hand derivative as \(x\) approaches 0 from the left is \(-2\).

Step 3: Find the Right-Hand Derivative

For the right-hand derivative, we look at the function where \(x \geq 0\), which is \(x^{2}\). The derivative of \(x^{2}\) is \(2x\). At \(x=0\), the right-hand derivative is \(2(0) = 0\).

Step 4: Compare the Left and Right Derivatives

For the function to be differentiable at \(x=0\), the left-hand derivative and right-hand derivative must be equal. Here, the left-hand derivative is \(-2\) and the right-hand derivative is \(0\). Since these are not equal, \(f\) is not differentiable at \(x=0\).

Step 5: Verify with Graphing Calculator

Zooming in on a graphing calculator at \(x=0\) will show a sharp point or cusp rather than a smooth transition, confirming that the function is not differentiable at this point.

Key Concepts

Derivative

The concept of a derivative is fundamental in calculus. It represents the rate at which a function changes at any given point. Imagine you are driving a car, the derivative would be the speedometer, showing the speed at each instant.
In a graph, the derivative at a point represents the slope of the tangent line at that point. A tangent line is a straight line that "just touches" the curve at a single point without crossing it.

• A positive derivative means the function is increasing at that point.
• A negative derivative means the function is decreasing.
• If the derivative is zero, the function might be at a local maximum or minimum, or a saddle point.

Differentiability, therefore, requires that the derivative exists and is a real number. It asks whether you can reliably determine this "instantaneous speed" at a specific point.

Left-Hand Derivative

The left-hand derivative focuses on what happens to a function as you approach a point from the left-hand side. Suppose you are walking towards a door from the left; the left-hand derivative is akin to noticing changes in speed as you reach that entrance.
For a function, this means calculating the derivative using values to the left of a particular point, namely the limit of the function's slopes as you near that point from the left.
Mathematically, for a function \( f(x) \), the left-hand derivative at \( x = a \) is defined as:\[\lim_{{h \to 0^-}} \frac{{f(a + h) - f(a)}}{h}\]In our example with \( f(x) = -2x \) for \( x < 0 \), the derivative is constant, meaning the behavior is perfectly predictable, and so the left-hand derivative at \( x=0 \) is \(-2\).

Right-Hand Derivative

The right-hand derivative is about approaching a value from the right side. Continuing from our previous analogy, imagine walking towards that same door but this time from the right. The right-hand derivative captures how the speed changes as you near the entrance from the right.
This refers to the derivative calculated using values to the right of the point in question, specifically the limit of function's slopes as you approach the point from the right.

Mathematically, we define the right-hand derivative at \( x = a \) as:

\[\lim_{{h \to 0^+}} \frac{{f(a + h) - f(a)}}{h}\]For \( f(x) = x^2 \) for \( x \geq 0 \), when evaluating at \( x = 0 \), the derivative results in \( 0 \).
This contrasting behavior with the left-hand slope indicates a lack of smooth transition, contributing to the function's non-differentiability at a given point.

Graphing Calculator

Using a graphing calculator is a reliable method for visualizing function behaviors. It helps us "see" the mathematical concepts, such as differentiability, more tangibly.
When you zoom in to a graph at a particular point of interest, the transition should appear more like a straight line if it’s differentiable. However, a clear indication of non-differentiability is when a sharp turn, cusp, or crease appears.

• Visual tool: It provides a handy visual approximation.
• Verification: Confirms analytic calculations.
• Exploration: Offers a way to explore functions behavior variably by zooming and manipulating views.

In our exercise, observing the graph at \( x=0 \) using a graphing calculator reveals a sharp turn, illustrating why the function \( f(x) \) with its piecewise components, lacks a smooth transition and thus,is not differentiable at \( x=0 \).