Connecting Differentiability and Continuity: Determining When Derivatives Do and .... Chapter Notes | Calculus AB - Grade 9 PDF Download

Welcome to your guide on connecting differentiability and continuity in AP Calculus! This topic is essential for mastering derivatives by exploring when they exist and how they relate to continuity. We'll combine our understanding of continuity over intervals with derivative concepts to strengthen your skills.

Continuity and Differentiability: The Connection

In AP Calculus, most functions you'll encounter are both continuous and differentiable. However, issues arise when a function fails to meet one or both conditions. To review continuity, check out this guide: 
Confirming Continuity over an Interval.
A function is differentiable if it has a derivative at every point in its domain, meaning the graph is smooth and resembles a straight line when zoomed in closely (not necessarily horizontal). For example, consider the graph of cos(x). Zooming in at the point (0, 1), the curve appears linear, indicating differentiability.
For a function to be differentiable at a specific point, the left-hand and right-hand derivatives must match the derivative at that point:
lim_x→a⁻ f'(x) = lim_x→a⁺ f'(x) = f'(a)
Key rule: If a function is differentiable, it must be continuous. However, continuity does not guarantee differentiability. A function may fail to be differentiable if it is discontinuous, has a sharp change in slope, or features a vertical tangent.

When Is a Function Not Differentiable?

Let's explore scenarios where a function lacks differentiability, using graphs to illustrate each case.

1. Discontinuous Graphs

Discontinuity often leads to non-differentiability because the derivatives from the left and right sides of a point don't align.
Example: Unequal Derivatives
For most discontinuous graphs, we can see that the derivatives approaching the point from either side will not be equal.
For example,  looks like the following in the graph below. Since the denominator cannot equal 0, x ≠ 2. The domain of this graph is (−∞,−2)∪(−2,∞).

2. Vertical Tangents

Consider f(x) = 2(x + 2)^(1/3) below.

3. Corners and Cusps

We can determine that a curve will not be differentiable at a corner or a cusp because the derivative will not approach the same value on both sides of the point. Additionally, there may be a vertical tangent! Let's observe the graphs below.
This is the graph of We can see that this graph would have a vertical tangent line as we approach x = 3, so it cannot be differentiable.
Let’s look at another graph! This is the graph of g(x) = ∣x∣. This function has a corner, and is not diffferentiable because lim_x→0−g′(x)<0 while lim_x→0−g′(x) > 0, and cannot be equal.

Practice Problems

Problem 1: Graphically Identifying Differentiability
Determine the number of points where the following piecewise function is not differentiable.

Problem 2: Determining Differentiability Algebraically

From the 2003 AP Calculus AB Exam, courtesy of College Board.
Given the piecewise function:

Assuming f(x) is continuous at x = 3, is it differentiable?
Solution: Step 1: Left-hand derivative (x < 3)

Therefore,

Step 2: Right-hand derivative (x ≥ 3)
For f(x) = (2/5)x + 2, the derivative is:

Therefore,

Step 3: Derivative at x = 3
To check the derivative at the point, we need to take the derivative of the equation representing the function when x = 3, which is the same as when x ≥ 3.
Therefore,

Now we check that all of the limits equal each other, and find that the function $f(x)$ is differentiable at x = 3 ! Check the graph below to confirm our work.

As we can see from the graph, the function f(x) is smooth at x=3, and is therefore differentiable.

Closing

You're mastering these concepts, and with practice, you'll navigate derivatives and continuity with confidence. Determining when derivatives exist and ensuring continuity is a crucial skill in AP Calculus. As you encounter questions on the exam, remember to check for domain restrictions and assess piecewise continuity.

Key Terms to Know

• Continuous Function: A function with no breaks, holes, or jumps, drawable without lifting the pen.
• Differentiability: A function has a derivative at every point in its domain, indicating a smooth graph with a defined slope.
• Differentiable Function: A function with derivatives throughout its domain, allowing slope calculation at any point.
• Function: A relation where each input has exactly one output.
• Limit Definition of the Derivative: A formula to find a function’s instantaneous rate of change as the change in x approaches zero.
• Limit: The value a function approaches as the input nears a specific point or infinity.
• Piecewise Function: A function defined by different rules for different intervals of the input.
• Point of Discontinuity: A point where the function’s graph breaks or jumps, losing continuity.
• Step Function: A piecewise function that jumps between constant values, resembling a staircase.