Question

True or False? In Exercises \(93-96\) , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If a function has derivatives from both the right and the left at a point, then it is differentiable at that point.

Short answer

The statement 'If a function has derivatives from both the right and the left at a point, then it is differentiable at that point.' is True

Step-by-step solution

Understand the Concept of Derivatives

The derivative of a function at a specific point is a measure of the rate at which the function is changing at that point. It is found by taking the limit as the change in x approaches 0 of the function's rate of change at that point.

Understand the Concept of Differentiability

A function is differentiable at a point when it has a derivative at that point. More specifically, if the function has a derivative when approached from the right (or from values larger than the point) and from the left (or from values smaller than the point) and these two derivatives are equal, then the function is differentiable at that point.

Test the Statement

The statement 'If a function has derivatives from both the right and the left at a point, then it is differentiable at that point.' is actually true according to our understanding of differentiability. If the function's right-hand derivative and left-hand derivative at a point are equal, then the function is differentiable at that point because it meets the definition of differentiability.

Key concepts

Derivatives of Functions

Understanding the derivatives of functions is essential for comprehending the behavior of functions and their graphs. The derivative represents the slope of the tangent line to the function at any given point and indicates how a function is changing at that moment.

When we talk about the derivative of a function, we're discussing the instantaneous rate of change, akin to finding the speed of a car at an exact moment. Mathematically, we express this idea using the limit process, defining the derivative of a function at a point x as the limit of the average rate of change of the function over an interval as that interval approaches 0. The formula for this limit is \[ \lim_{{h \to 0}} \frac{{f(x+h) - f(x)}}{h} \] where h represents the interval's width approaching zero. If this limit exists, the function is said to have a derivative at x.

Right-Hand Derivative

The right-hand derivative of a function at a point deals with the behavior of the function as it approaches from values greater than the point. It is a specific use of the derivative concept that focuses on the limit from the right side of the point.

To illustrate, imagine stepping forward on the number line, the right-hand derivative tells you the slope of the function if you were to only look ahead. The technical description would be \[ \lim_{{h \to 0^+}} \frac{{f(x+h) - f(x)}}{h} \] where h approaches zero from the positive side. If this limit exists, the function has a right-hand derivative at that point.

Left-Hand Derivative

In contrast to the right-hand derivative, the left-hand derivative examines what occurs as the function approaches a point from values less than the point, or essentially, as you would move backwards on the number line. The left-hand derivative provides the slope from this perspective.

To mathematically define the left-hand derivative at a point x, we use the limit as follows: \[ \lim_{{h \to 0^-}} \frac{{f(x+h) - f(x)}}{h} \] Here, h approaches zero from the negative side. Just like the right-hand derivative, if this limit exists, the function has a left-hand derivative at that point.

Rate of Change

The rate of change is a universal concept that measures how a quantity changes with respect to another quantity. In the context of functions, it's typically how y, or the function's output, changes as x, the input, changes.

For linear functions, the rate of change is constant and represents the slope of the line. However, for nonlinear functions, the rate of change varies along the curve, which is where derivatives come into play. Derivatives provide a way to speak about the rate of change at any specific instant, giving us a tool to analyze instantaneous changes in complex systems, such as physics, economics, and biology.

Limit

The idea of a limit is foundational in understanding derivatives and the concept of continuity in calculus. A limit describes the value that a function approaches as the input (or x-value) approaches some value.

Limits are essential for defining both derivatives and integrals and for the study of continuity. They are the glue that holds together all the foundational bricks of calculus, making sense of what happens between gaps, at points of discontinuity, or at infinity. For instance, the formal definition of a derivative is actually a limit: \[ \frac{df}{dx} = \lim_{{h \to 0}} \frac{{f(x+h) - f(x)}}{h} \] Here, the limit captures the 'approaching behavior' of the average rate of change of a function as the interval h shrinks to zero.