Differentiability

Differentiability, a cornerstone concept in Calculus AB, essentially describes whether a function has a well-defined derivative at a specific point. It’s closely tied to the idea of a smooth, continuous curve. A function is differentiable at a point if its graph has a tangent line at that point. This means the function must be both continuous and smooth at that point. Let’s break down the key aspects:

1. The Definition of the Derivative

The derivative of a function $ f(x) $ at a point $ x=a $ , denoted as $ f’(a) $ , is defined as:

$ f’(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} $

This limit represents the slope of the tangent line to the graph of $ f(x) $ at $ x=a $ . If this limit exists, the function is differentiable at $ x=a $ .

2. Relationship to Continuity

Continuity

A function must be continuous at a point to be differentiable at that point. However, continuity alone is not sufficient for differentiability. A function can be continuous at a point but not differentiable there.

3. Conditions for Non-Differentiability

A function is not differentiable at a point if any of the following occur:

Sharp Corner/Cusp: The function has a sharp turn or cusp at the point. The slope of the tangent line approaches different values from the left and right.

y = abs(x)

Vertical Tangent: The function has a vertical tangent line at the point. The slope of the tangent line approaches infinity.
Discontinuity: The function is discontinuous (has a jump, hole, or asymptote) at the point.

y = 1/(x-1)
{x>1}
y = 2
{x=1}
y = x
{x<1}

Oscillating Function: The function oscillates infinitely rapidly near the point.

4. Differentiability and Smoothness

Differentiability implies smoothness. A differentiable function will have a smooth, continuous curve without any sharp corners, cusps, or vertical tangents. However, the converse is not always true (a function can be smooth but not differentiable everywhere).

Higher Order Derivatives

5. Practical Applications

Differentiability is crucial for many applications in Calculus AB, including:

Finding instantaneous rates of change: The derivative gives the instantaneous rate of change of a function at a specific point.
Optimization problems: Finding maximum and minimum values of a function often involves analyzing its derivative.
Related rates problems: Using derivatives to relate the rates of change of different variables.
Curve sketching: The derivative helps determine the increasing/decreasing intervals and concavity of a function.

6. Checking for Differentiability

To determine if a function is differentiable at a point, you typically:

Check for continuity at the point.
Calculate the left-hand and right-hand derivatives using the limit definition.
If the left-hand and right-hand derivatives are equal and finite, the function is differentiable at the point.