Question

Determining Differentiability In Exercises \(85-88\) , find the derivatives from the left and from the right at \(x=1\) (if they exist). Is the function differentiable at \(x=1 ?\) $$ f(x)=\left\\{\begin{array}{ll}{(x-1)^{3},} & {x \leq 1} \\ {(x-1)^{2},} & {x>1}\end{array}\right. $$

Short answer

The function \(f(x)\) is differentiable at \(x = 1\), because the derivative limits from the left and the right are both 0.

Step-by-step solution

Differentiate from the Left

The first part of the function, \((x-1)^{3}\), applies to \(x \leq 1\), so this will be used for the derivative limit from the left of \(x = 1\). First, find the derivative of this part of the function, which is \(3*(x - 1)^2\). Second, calculate the derivative as \(x\) approaches 1 from the left (written mathematically as \(x \to 1^-\)), which results in \((3*(1 - 1)^2 = 0\).

Differentiate from the Right

The second part of the function, \((x-1)^{2}\), applies to \(x > 1\), so this will be used for the derivative limit from the right of \(x = 1\). Find the derivative of this part of the function, which is \(2*(x - 1)\). Next, calculate the derivative as \(x\) approaches 1 from the right (written mathematically as \(x \to 1^+)\). The result is \(2*(1 - 1) = 0\).

Compare Derivative Limits

The function will be differentiable at \(x = 1\) if and only if the derivative limits from the left and the right are equal. Since both derivatives are 0, the function is differentiable at \(x = 1\).

Key concepts

Derivatives from the Left and Right

When determining the differentiability of a function at a particular point, it's crucial to consider the derivative limits from both sides of that point. The 'derivative from the left' refers to the slope of the tangent line to the function as you approach the point from the left, mathematically represented as the limit of the derivative as the independent variable, often denoted as 'x', approaches the point from the left. Similarly, the 'derivative from the right' is the slope as you approach the point from the right.
In the given exercise, the derivatives from the left and right at x=1 must be calculated separately for the corresponding intervals defined by the piecewise function. For the left-hand derivative, we examine the interval x ≤ 1, and for the right-hand derivative, the interval x > 1. An essential condition for differentiability at that point is that these two derivatives agree—that is, they result in the same value.
In our example, we found that both the left and right derivatives equal zero, indicating that the slopes of the tangent lines from both sides are the same at x = 1. This equal value is key; if the derivatives did not match, it would imply a 'kink' or 'corner' in the graph at the point, leading to non-differentiability there, despite the function being continuous.

Piecewise Functions

A piecewise function is defined by different expressions for different intervals of the independent variable. These types of functions can model complex situations but pose unique challenges when it comes to differentiability. Each 'piece' of the function may have its own derivative, and when we approach a point where the function 'pieces' meet, we must carefully consider the behavior from both sides of the point.
In the case of the exercise, f(x) is a piecewise function with two expressions: one applied when x<1, and the other when x≥1. Determining differentiability at the point where these pieces meet—in this case, x = 1—requires analyzing the limits and derivatives up to that point from both intervals.
Piecewise functions, with their distinct intervals and conditions, require this side-by-side derivative comparison to confirm continuity and a smooth transition at the juncture, translating to differentiability. This characteristic makes working with piecewise functions a staple in understanding how functions behave at crucial points.

Limit of a Function

The limit of a function is a fundamental concept in calculus, essential for understanding derivatives and differentiability. It describes the behavior of a function as the input approaches a particular value. In practical terms, it tells us the value that the function output is getting closer to, even if it never actually reaches that value. Limits are instrumental when analyzing functions at points where they are not explicitly defined or at boundary points of their intervals.
In our exercise's context, the limit process is pivotal for finding the derivatives from the left and right. Evaluating the limit of the derivative of the function as x approaches 1 from both directions allows us to determine if the slopes of the tangent lines from each side are heading towards the same value. When these limits are equal, the function is usually differentiable at that point. However, if they differ or do not exist, the function is not differentiable at that point.
In conclusion, the limit concept is not only about what happens right at a point but also about the immediate approach towards it. Understanding limits is, therefore, crucial in determining if a function has a well-defined tangent and is differentiable at a specific point.