Problem 55 - Calculus - AP Edition

Question

Writing Write a paragraph that explains what it means for a function to be differentiable. Include examples of functions that are not differentiable as well as examples of functions that are differentiable.

Step-by-Step Solution

Answer

A differentiable function has a derivative at each point, being smooth without sharp angles, unlike functions like \(|x|\) at \(x=0\) which are not differentiable due to corners.

Step 1: Define Differentiability

A function is differentiable at a point if it has a derivative at that point, meaning it can be locally approximated by a linear function around that point. This implies the function must be smooth, without any sharp corners or cusps at that point.

Step 2: Continuous and Differentiable

For a function to be differentiable, it must be continuous at the point of interest. However, a continuous function is not necessarily differentiable. Differentiability implies continuity, but not vice versa.

Step 3: Discuss Examples of Differentiable Functions

Polynomials like \(f(x) = x^2\) or \(f(x) = x^3\) are differentiable everywhere because they are smooth and continuous across their entire domains. Similarly, \(f(x) = \sin(x)\) and \(f(x) = \cos(x)\) are also differentiable at all points.

Step 4: Discuss Examples of Non-Differentiable Functions

Functions such as \(f(x) = |x|\) are not differentiable at certain points, like \(x = 0\), because they exhibit a sharp corner or cusp at that point. The function \(f(x) = \sqrt[3]{x}\) is also not differentiable at \(x = 0\), as it has a vertical tangent line at this point.

Step 5: Summarize Differentiability Concept

To summarize, differentiability concerns the ability of a function to have a well-defined tangent at any given point. Functions with sharp angles or discontinuities at a point will not be differentiable there, while smooth and continuous functions like polynomials and trigonometric functions often are.

Key Concepts

Continuous Functions

In the world of calculus, continuous functions play a fundamental role. A function is termed continuous at a specific point if there is no sudden jump or gap at that point. Essentially, it means that you can draw the graph of the function without lifting your pen from the paper. For a function to be differentiable, it must first be continuous.
But not every continuous function is differentiable. This is because continuity ensures no breaks, but differentiability requires smoothness as well. An example of a continuous function that is not differentiable is the absolute value function, \( f(x) = |x| \), at \( x = 0 \). At this point, the function is continuous, but there's a sharp corner which means it is not smooth enough to have a derivative.
Continuous functions help establish the foundational requirement for differentiability, emphasizing the need for uninterrupted graphs.

Derivative

The derivative is a core concept in calculus and is essentially the "slope" of a function at a given point. It reflects the rate at which a function is changing at that particular point. This is critical in understanding how functions behave and is used to determine whether a function is increasing or decreasing at any given point.
A function is differentiable at a point if a derivative exists there. Several tools from calculus, like the power rule or chain rule, can help find derivatives of functions like polynomials. For instance, the derivative of \( f(x) = x^2 \) is \( f'(x) = 2x \), indicating that the rate of change at any point \( x \) is twice the value of \( x \).

• Differentiability implies smoothness at every point within an interval.
• A lack of differentiability often suggests an abrupt change or corner at that point.

Thus, differentiability and derivative not only reveal the changes in functions but unleash numerous applications in various fields.

Non-Differentiable Functions

Non-differentiable functions are those that fail to have a derivative at one or more points on their domain. This can occur due to several reasons such as corners, cusps, or vertical tangents in the function graph. A classic example is the absolute value function \( f(x) = |x| \) at \( x = 0 \), where the graph makes a sharp corner.
Similarly, functions like \( f(x) = \sqrt[3]{x} \) at \( x = 0 \), are non-differentiable because there is a vertical tangent here, meaning the slope becomes infinite. Other examples include functions with discontinuities or those with oscillations that are too wild to define a consistent tangent.

• Sharp corners prevent a single, clear tangent line from touching the curve at just one point.
• Vertical tangents suggest an undefined slope, hence no derivative.

Non-differentiable functions challenge the smoothness required for a clear mathematical understanding of changes and are essential in understanding the complexities of mathematical functions.