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 \(x=1 ?\) $$ f(x)=\left\\{\begin{array}{ll}{x,} & {x \leq 1} \\ {x^{2},} & {x>1}\end{array}\right. $$

Short answer

The function \(f(x)\) isn't differentiable at \(x = 1\) as the left and right derivatives (1 and 2) do not match.

Step-by-step solution

Define the Functions and Limits

First, define the two different functions in the given piecewise function. Function one, \(f_1(x)\) is \(x\) for \(x \leq 1\), and function two, \(f_2(x)\) is \(x^2\) for \(x > 1\). Define \(x = 1\) as the limit of differentiation.

Calculate the Left and Right Derivatives

Next, calculate the derivatives from the left and right at \(x = 1\). The derivative of \(f_1(x)\) is 1. The derivative of \(f_2(x)\) is \(2x\), which in terms of the question should be \(2 \cdot 1 = 2\), since \(x = 1\). These are the left and right derivatives, respectively.

Compare the Left and Right Derivatives

The question states that the function is differentiable at \(x = 1\) if the left and right derivatives are equal. Since they are 1 (from the left) and 2 (from the right), they are not equal, and so the function is not differentiable at \(x = 1\).

Key concepts

Piecewise Functions

Piecewise functions are mathematical expressions defined by multiple sub-functions, each of which applies to a certain interval of the main function's domain. Imagine a piecewise function as a road map that shows different paths (sub-functions) depending on where you're coming from (input values). In the case of the exercise, we're dealing with a function that changes its rule of operation at a specific point, namely at \(x = 1\).

On one side of the point, \(x \leq 1\), the function follows the formula \(f_1(x) = x\), which gives a straight line with a slope of 1. On the other side, for \(x > 1\), it takes on the formula \(f_2(x) = x^2\), forming a parabola. The behavior of piecewise functions near the points where their formula changes is of particular interest in calculus since it can affect limits, continuity, and differentiability.

Left-Hand Derivative

When examining the differentiability of functions at a specific point, we need to consider how the function behaves as it approaches that point from the left. This concept is known as the left-hand derivative. It is essentially the slope of the function as you approach the point of interest from the left side on the graph.

In our example, to find the left-hand derivative at \(x = 1\), we look at the sub-function \(f_1(x) = x\) and take the derivative. Derivatives represent the rate of change, so in this case, since \(f_1(x)\) is a straight line, its rate of change or slope is constant at 1. This means that as \(x\) approaches 1 from the left, the slope remains 1—a crucial piece of information for determining differentiability.

Right-Hand Derivative

Just as with the left-hand derivative, to assess differentiability, one must also understand the right-hand derivative. This is the slope of the function as you approach the point from the right. Visually on the graph, it's looking at how steep the line or curve is as you move towards the point of interest from higher values of \(x\).

For the right-hand derivative at \(x = 1\) for our function \(f_2(x) = x^2\), we determine the slope by taking the derivative of \(f_2(x)\), which gives us \(2x\). At \(x = 1\), this evaluates to 2, indicating that as we approach \(x = 1\) from the right, the slope or rate of change is twice as steep as when approaching from the left.

Differentiation of Functions

Differentiation is a cornerstone of calculus and, broadly speaking, it is the process of finding the derivative of a function. A derivative measures how a function changes as its input changes, representing the slope of the function at any given point for a smooth curve. The act of differentiation tells us a lot about the behavior of functions, such as the rate of change, finding maxima and minima, and understanding motion in physics.

In culmination, when we discuss the differentiability of a function at a point, we're asking whether the function smoothly transitions through that point. This requires that the function is continuous at that point and that the left-hand and right-hand derivatives at that point are equal. If they're not—as seen in our example with different left-hand (1) and right-hand (2) derivatives—the function is said to be non-differentiable at that point due to the 'corner' or cusp that exists there.