Problem 21 - Calculus

Question

explain what is wrong with the statement. A function \(f\) that is not differentiable at \(x=0\) has a graph with a sharp corner at \(x=0\)

Step-by-Step Solution

Answer

The statement is incorrect; it assumes a corner is the only cause of non-differentiability.

Step 1: Understanding Differentiability

For a function to be differentiable at a point, it must be smooth at that point. This means there is no sharp corner or cusp and the slope (or derivative) from the left and right of the point must agree. If the function has a sharp corner at a point, it cannot be differentiable there.

Step 2: Evaluating the Statement

The statement claims that a function not differentiable at a specific point must have a sharp corner there. While a sharp corner is one reason for a function to not be differentiable, it is not the only reason. A point of non-differentiability could also occur where there is a vertical tangent, a discontinuity, or any other type of undefined derivative.

Step 3: Identifying Other Causes

A vertical tangent line, where the slope becomes infinite, or a discontinuity, where the function jumps, are also reasons for non-differentiability. Therefore, these situations don't involve a corner but still result in a function not being differentiable at that point.

Step 4: Conclusion on Statement Validity

The statement is incorrect because it incorrectly implies that the only reason for non-differentiability is a sharp corner. There are multiple other scenarios that can lead to the same conclusion.

Key Concepts

Sharp Corner

When we talk about sharp corners in the context of differentiability, we refer to places on a graph where there is a sudden change in direction. Imagine drawing a graph line without lifting your pencil. A sharp corner implies that you'll abruptly change direction at some point, forming a jagged edge.

• Mathematically, a sharp corner occurs when the left-hand and right-hand derivatives at a point do not match.
• This discrepancy means there's no smooth transition through the point, breaking the continuous flow of the derivative.
• As a result, the function is not differentiable at that point.

Sharp corners are common examples of non-differentiability, but remember, they aren't the only culprit. Many students might think a sharp corner is the only reason for a function not to have a derivative, but other factors could also cause nondifferentiability.

Vertical Tangent

Vertical tangents are another fascinating scenario where differentiability fails. Imagine a graph where the line shoots almost straight up or down at some point. This happens when the slope or the rate of change becomes infinitely steep.

• In mathematical terms, this means the derivative at that point becomes infinite.
• A vertical tangent line at any point implies undefined behavior for the function’s derivative.
• Hence, the function is not differentiable wherever a vertical tangent exists.

This differs from a sharp corner since both left-hand and right-hand limits of the derivative approach the same infinite value. However, the infinite slope results in no defined derivative, causing differentiability to break down.

Discontinuity

Discontinuities can be real sticking points in understanding differentiability. When a function has a gap, jump, or break at a certain point, this is described as a discontinuity.

• A function must be continuous at a point to possess a derivative there.
• If there is a break, the limits from the left and right of the discontinuity do not meet.
• Without continuous behavior, the derivative does not exist, leading to non-differentiability.

Discontinuities might be less visually intuitive than sharp corners or vertical tangents. However, they signify a complete lack of smooth flow in the function’s graph at certain points, which critically affects differentiability.