Problem 43 - Calculus Single Variable

Question

A multicase function \(f\) is defined. Is \(f\) differentiable at \(x=0 ?\) Give a reason for your answer. $$ f(x)=\left\\{\begin{array}{ccc} x^{2} & \text { if } & x \leq 0 \\ x & \text { if } & x>0 \end{array}\right. $$

Step-by-Step Solution

Answer

The function \(f\) is not differentiable at \(x=0\) because the left-hand and right-hand derivatives are not equal.

Step 1: Understand the Definition of Differentiability

To check if a function is differentiable at a point, the function must not only be continuous at that point but also the derivative must exist as the limit from both sides must be equal. We need to compute the left-hand and right-hand derivatives at \(x=0\) and see if they are equal.

Step 2: Verify Continuity at x=0

For the function to be differentiable, it must first be continuous at \(x=0\). To check continuity, evaluate the limit from the left as \(x\) approaches 0 for \(f(x)=x^2\), and from the right for \(f(x)=x\). Both limits should equal \(f(0)\) which is \(0^2=0\). \[\lim_{{x \to 0^-}} f(x) = 0^2 = 0 \\lim_{{x \to 0^+}} f(x) = 0 = 0 \\text{Since both limits are } 0, \text{ f is continuous at } x=0.\]

Step 3: Find the Left-Hand Derivative at x=0

The left-hand derivative is calculated as \(\lim_{{h \to 0^-}} \frac{f(h)-f(0)}{h} = \lim_{{h \to 0^-}} \frac{h^2-0}{h} = \lim_{{h \to 0^-}} h = 0 \). Thus, the left-hand derivative is 0.

Step 4: Find the Right-Hand Derivative at x=0

The right-hand derivative is computed using the function's definition for \(x>0\), as \(\lim_{{h \to 0^+}} \frac{f(h)-f(0)}{h} = \lim_{{h \to 0^+}} \frac{h-0}{h} = \lim_{{h \to 0^+}} 1 = 1 \).Thus, the right-hand derivative is 1.

Step 5: Compare Left-Hand and Right-Hand Derivatives

For \(f(x)\) to be differentiable at \(x=0\), both left-hand and right-hand derivatives must be equal. Here, the left-hand derivative is 0 and the right-hand derivative is 1, which are not equal. Therefore, \(f\) is not differentiable at \(x=0\).

Key Concepts

Continuity

In the realm of calculus, continuity is a key concept that helps us understand how functions behave at specific points. A function is said to be continuous at a point if there is no abrupt change in its value; in simpler terms, you could draw it without lifting your pencil from the paper.
To determine continuity at a point, we need to verify that the limit of the function approaching from the left (negative side) and right (positive side) are the same and equal to the function’s value at that point.
For our multicase function at the point \(x = 0\), we evaluate

• The left limit as \(x\) approaches 0: \[\lim_{{x \to 0^-}} f(x) = 0^2 = 0\]
• The right limit as \(x\) approaches 0:\[\lim_{{x \to 0^+}} f(x) = 0 = 0\]
• This matches the function's value at \(x = 0\), which is also 0.

Since these conditions are satisfied, the function is continuous at \(x = 0\). But remember, continuity is only one part of determining differentiability.

Left-Hand Derivative

The left-hand derivative focuses on understanding how the function's behavior changes as it approaches a point from the left. Mathematically, it is defined by the limit:

• \[\lim_{{h \to 0^-}} \frac{f(h) - f(0)}{h}\]

For our function, as \(h\) approaches 0 from the left, within the domain \(x \leq 0\), we compute:

• \(f(h) = h^2\), and \(f(0) = 0^2 = 0\)
• Substitute into the limit expression:\[\lim_{{h \to 0^-}} \frac{h^2 - 0}{h} = \lim_{{h \to 0^-}} h = 0\]

Thus, the left-hand derivative at \(x = 0\) is 0. This indicates that from the left, the function's approach towards \(x = 0\) is smooth, with the slope of the tangent being 0.

Right-Hand Derivative

The right-hand derivative is similar to the left-hand one, but focuses on approaching a specific point from the right side. It's calculated through the expression:

• \[\lim_{{h \to 0^+}} \frac{f(h) - f(0)}{h}\]

In the case of our function, for \(x > 0\):

• \(f(h) = h\), and \(f(0) = 0\)
• The limit becomes:\[\lim_{{h \to 0^+}} \frac{h - 0}{h} = \lim_{{h \to 0^+}} 1 = 1\]

Therefore, the right-hand derivative at \(x = 0\) is 1. This highlights a different approach behavior than from the left side, creating a contrast.

Multicase Function

A multicase function is one that is defined by different expressions for different intervals of the input variable \(x\). This type of segmentation often allows functions to model complex systems or behaviors.
Our exercise involves such a function:

• For \(x \leq 0\), the function is defined as \(f(x) = x^2\).
• For \(x > 0\), the function follows \(f(x) = x\).

This segmentation is crucial when assessing continuity and differentiability because it directly impacts how limits and, subsequently, derivatives are calculated.
In our example, while the function is continuous at \(x = 0\), the differing behaviors (or slopes) indicated by the left-hand and right-hand derivatives lead to a lack of differentiability. Such behaviors are common in multicase functions at transition points, affecting how seamlessly a function can bridge two distinct functional rules.