Is A Function Differentiable At A Vertical Tangent

Is a Function Differentiable at a Vertical Tangent?

The question of differentiability at a point where a function possesses a vertical tangent is a crucial concept in calculus. Understanding this requires a firm grasp of the definition of differentiability and the geometric interpretation of derivatives. This article delves into the intricacies of this topic, exploring the reasons why a function is not differentiable at points with vertical tangents, exploring related concepts, and providing illustrative examples.

Understanding Differentiability

A function f(x) is differentiable at a point x = a if its derivative, f'(a), exists at that point. The derivative represents the instantaneous rate of change of the function at a. Geometrically, f'(a) is the slope of the tangent line to the graph of f(x) at x = a.

The formal definition of the derivative uses the limit of the difference quotient:

f'(a) = lim_h→0 [(f(a + h) – f(a)) / h]

For the derivative to exist, this limit must exist and be finite. This means that the limit must approach the same value from both the left and the right (i.e., the left-hand and right-hand limits must be equal).

Vertical Tangents and the Breakdown of Differentiability

A vertical tangent line has an undefined slope. The slope is given by the derivative, and an undefined slope implies that the derivative does not exist. Therefore, if a function has a vertical tangent at a point, it is not differentiable at that point.

Why the Limit Fails

Let's examine why the limit in the definition of the derivative fails to exist when there's a vertical tangent. At a point with a vertical tangent, the difference quotient [(f(a + h) – f(a)) / h] will become arbitrarily large as h approaches 0. This is because the change in f(x), (f(a + h) – f(a)), will not approach 0 at the same rate as h. Instead, the numerator grows much faster than the denominator, resulting in the limit approaching either positive or negative infinity, which is not a finite value. Therefore, the limit does not exist, meaning the derivative is undefined.

Examples Illustrating Non-Differentiability at Vertical Tangents

Let's look at specific functions exhibiting vertical tangents to solidify the concept.

Example 1: The Cube Root Function

Consider the function f(x) = x^1/3. This function has a vertical tangent at x = 0. Let's analyze the derivative using the limit definition:

f'(0) = lim_h→0 [(h^1/3 – 0) / h] = lim_h→0 (h^-2/3)

As h approaches 0, h^-2/3 approaches infinity. The limit does not exist; therefore, f(x) = x^1/3 is not differentiable at x = 0.

Example 2: A Piecewise Function

Consider the piecewise function:

f(x) = { x^1/3, x ≥ 0

•   { -x<sup>1/3</sup>, x < 0*
  

This function also has a vertical tangent at x = 0. While the left and right hand derivatives exist separately, they are not equal resulting in a non-differentiable point. The limit from the right is positive infinity and the limit from the left is negative infinity.

Example 3: The Absolute Value Function

The function f(x) = |x| has a sharp point, or a cusp, at x = 0. While the function is continuous at x = 0, it is not differentiable there. The left-hand derivative is -1, and the right-hand derivative is 1. The limit from both sides does not converge to a single value, indicating non-differentiability. Although this is not a vertical tangent, it illustrates a case where non-differentiability occurs due to the lack of a unique tangent.

Differentiability and Continuity: A Crucial Distinction

It is important to note that while a function must be continuous at a point to be differentiable there, continuity does not guarantee differentiability. The functions shown above are all continuous at the points where they have vertical tangents, but they are not differentiable at those points. Continuity simply means that the function has no breaks or jumps at a point, but differentiability requires a smooth, continuous curve without sharp corners or vertical tangents.

Higher-Order Derivatives and Vertical Tangents

The concept extends to higher-order derivatives. If a function has a vertical tangent, the first derivative is undefined at that point. Consequently, any higher-order derivatives (second, third, etc.) will also be undefined at that point. This is because the higher-order derivatives are themselves derivatives of lower-order derivatives.

Applications and Significance

The concept of differentiability at vertical tangents is crucial in various areas of mathematics and its applications. For example:

• Optimization Problems: Finding maxima and minima of a function often involves analyzing the derivative. Knowing that a function is not differentiable at points with vertical tangents is crucial to avoid errors in optimization analysis. Points of non-differentiability could be critical points.

• Curve Sketching: Understanding differentiability helps in accurately sketching the graph of a function. Knowing that a vertical tangent exists helps define the behavior of the function at that point.

• Physics and Engineering: Many physical phenomena are modeled by functions, and understanding the behavior of these functions, including their differentiability, is essential for accurate predictions and modeling.

• Numerical Analysis: Numerical methods for solving differential equations and approximating functions often require careful consideration of points of non-differentiability.

Conclusion: Differentiability and Vertical Tangents—A Summary

A function is not differentiable at a point where it possesses a vertical tangent. This is because the limit defining the derivative does not exist at such points. The slope of the tangent line is undefined, and the difference quotient becomes arbitrarily large as we approach the point in question. While the function may be continuous at such a point, it lacks the smoothness necessary for differentiability. Understanding this distinction between continuity and differentiability, and its implication in the presence of vertical tangents, is crucial for mastering calculus and its applications across numerous scientific disciplines. Remember, the existence of a vertical tangent fundamentally alters the behavior of the function at that specific point, rendering it non-differentiable regardless of its behavior elsewhere.