Question

Using one-sided derivatives, show that the function \(f(x)=\left\\{\begin{array}{ll}{x^{3},} & {x \leq 1} \\ {3 x,} & {x>1}\end{array}\right.\) does not have a derivative at \(x=1\)

Short answer

The one-sided derivatives at \(x=1\) are not equal, hence the function \(f(x)\) does not have a derivative at \(x=1\).

Step-by-step solution

Calculate the left-hand derivative

The left-hand derivative at \(x\) equals 1 is calculated from the function \(x^{3}\). So, it can be computed as: \(\lim_{h \to 0} [(1-h)^{3} - 1^{3}] / (1-h-1) = \lim_{h \to 0}[-h^{3}]/[-h]= -h^{2}\). As \(h\) approaches 0, \(-h^{2}\) also approaches 0.

Calculate the right-hand derivative

The right-hand derivative at \(x\) equals 1 is calculated from the function \(3x\). So, it can be computed as: \(\lim_{h \to 0} [3*(1+h) - 3*1] / (1+h-1) = \lim_{h \to 0}3h/h= 3\).

Compare the left-hand and right-hand derivatives

The left-hand derivative and the right-hand derivative at \(x=1\) are not equal. The left-hand derivative is 0 and the right-hand derivative is 3. Therefore, the function does not have a derivative at \(x=1\).

Key concepts

Derivative at a Point

Understanding the concept of a derivative at a point is crucial for deepening your grasp of calculus. A derivative at a particular point measures the rate at which a function's output value is changing at that specific location along the function. Mathematically, it is the limit of the function's average rate of change as the interval considered approaches zero.

For a function to have a derivative at a point, both the left-sided and the right-sided limits of the difference quotient must exist and be equal. If they differ or if one does not exist, as in the given exercise, the function does not have a derivative at that point. In the example of the piecewise function, we find that the left-hand limit, as we approach from values less than 1, is 0, and the right-hand limit is 3. The discrepancy between the limits indicates a discontinuity in the slope, which signifies the absence of a derivative at the point.

Calculus

The branch of mathematics known as calculus focuses on rates of change (differential calculus) and accumulation of quantities (integral calculus). Calculus provides the tools for modeling and solving problems in physics, engineering, economics, and various other fields where change is a fundamental concept.

The fundamental principle used here involves differential calculus, which deals with the derivative— measuring how a function changes as its input changes. The beauty of calculus lies in its ability to describe the dynamic nature of the physical world, such as velocity, growth rates, and slopes of curves.

Limits

Limits are a foundational concept in calculus, describing the behavior of a function as the input approaches a certain value. Limits are used to define both derivatives and integrals, making them an indispensable part of understanding calculus.

In the context of the exercise, limits were taken as the variable 'h' approaches zero from the left and right, giving us the left-hand and right-hand derivatives. Limit calculations provide the precise value that the function's output approaches, providing us with the slope of the tangent line at any given point—if it exists. When the function behaves differently from the left and from the right, the limits diverge, and hence the derivative at the point does not exist.

Piecewise Functions

Piecewise functions are defined by different expressions or 'pieces,' depending on the interval of the input variable. They are powerful tools allowing the expression of complex functions that behave differently under various conditions.

For derivatives of piecewise functions, continuity is a key aspect. If a function switches the rule it follows at a certain point, like the one in the exercise, it may not have a continuous derivative at that point. The evaluation of the derivatives of the separate pieces around these critical points is essential to understand the overall behavior of the function. In calculus, the ability to analyze such functions is critical for dealing with real-world phenomena where certain conditions lead to different outcomes or behaviors for a function.