#AP Calculus AB/BC: Differentiability and Continuity - The Ultimate Review 🚀

Hey there, future calculus conqueror! Let's make sure you're totally prepped for anything the AP exam throws your way. This guide is designed to be your go-to resource for understanding differentiability and continuity, especially when time is tight. Let’s dive in!

#🔗 2.4 Connecting Differentiability and Continuity

#↗️ Continuity and Differentiability

Most functions you'll see in AP Calc are both continuous and differentiable, but it's crucial to know when things go sideways. Remember, for a function to be differentiable at a point, it must be smooth and look like a straight line when you zoom in super close. This line doesn't have to be horizontal!

Check out the graph of $\cos(x)$ below. Notice how zooming in at $(0,1)$ makes it look like a straight line. That’s differentiability in action!


Zooming into graph of $\cos(x)$

Graph created with Desmos

For a function to be differentiable at a point 'a', the left-hand limit of the derivative must equal the right-hand limit of the derivative, and both must equal the value of the derivative at that point:

$$\lim_{x\to a^{-}} f'(x) = \lim_{x\to a^{+}} f'(x) = f'(a)$$

#🚧 When is a Function NOT Differentiable?

A function is NOT differentiable if it has any of the following:

• Discontinuities (jumps, holes, or vertical asymptotes)
• Sharp Turns (corners or cusps)
• Vertical Tangents

Let's break down each scenario:

#🕳 Discontinuous Graphs

#Unequal Derivatives

If a graph has a discontinuity, the derivatives from the left and right sides will not match. Take a look at $g(x) = \frac{1}{x+2}$.


Graph of $g(x) = \frac{1}{x+2}$

Graph created with Desmos

Notice that $\lim_{x\to -2^{-}} g'(x) < 0$ and $\lim_{x\to -2^{+}} g'(x) > 0$. Since the derivative values are different, $g(x)$ is not differentiable at $x = -2$.

#Jump Discontinuity

Similarly, with a jump discontinuity, the derivatives from the left and right sides will not be equal. Observe the graph below:


Graph of jump discontinuity

Graph created with Desmos

Here, $\lim_{x\to 2^{-}} f'(x) > 0$ and $\lim_{x\to 2^{+}} f'(x) < 0$, so the function is not differentiable at $x = 2$.

#Removable Discontinuity

Even with a removable discontinuity, like in $h(x) = \frac{x^2 - 4}{x - 2}$, the function is not differentiable at the point of discontinuity. Even though the left and right limits of the derivative are equal, $h'(2)$ does not exist.


Graph of removable discontinuity

Graph created with Desmos

#⬆️ Vertical Tangents

Vertical tangents occur when the derivative approaches infinity (or negative infinity). For example, look at $f(x) = 2(x+2)^{\frac{1}{3}}$.


Graph with vertical tangent

Graph created with Desmos

At $x = -2$, the tangent line is vertical, meaning the derivative is undefined. Therefore, the function is not differentiable at $x = -2$.

#📐 Corners and Cusps

Corners and cusps are points where the derivative changes abruptly, making the function non-differentiable.


Graph with a vertical tangent

Graph created with Desmos

For example, in $f(x) = 2(x+2)^{\frac{1}{3}}$, there is a cusp at $x=-2$. Also, in $g(x) = |x|$, there's a sharp corner at $x=0$.


Graph of piecewise function with a corner

Graph created with Desmos

#🧮 Practice Problems

Let's solidify your understanding with some practice!

#1) Graphically Identifying Differentiability at a Point

How many points on the graph of $f(x)$ below are not differentiable?


Graph of piecewise function with a number of points that are not differentiable

Graph created with Desmos

Answer: There are 6 points where the function is not differentiable:

1. Infinite discontinuity at $x = -1$
2. Jump discontinuity at $x = 2$
3. Cusp at $(1, 2)$
4. Cusp at $(9, 3)$
5. Corner at $(6, 0)$
6. Vertical tangent at $(10, 4)$

Practice Question

Multiple Choice Questions:

1. At which of the following values of x is the function $f(x) = |x-3|$ not differentiable? (A) -3 (B) 0 (C) 3 (D) All real numbers

2. Which of the following statements is true about a function that is continuous at x=c? (A) The function must be differentiable at x=c. (B) The function must have a limit at x=c. (C) The function must be defined at x=c. (D) The function must be both differentiable and have a limit at x=c.

Free Response Question:

Consider the function $f(x)$ defined as follows:

$$f(x) = \begin{cases} x^2 + 1, & x \leq 1 \\ 3x - 1, & x > 1 \end{cases}$$

(a) Is $f(x)$ continuous at $x=1$? Justify your answer. (b) Is $f(x)$ differentiable at $x=1$? Show your work.

Scoring Rubric:

(a) Continuity at x=1 (3 points)

• 1 point: Checks the left-hand limit
• 1 point: Checks the right-hand limit
• 1 point: Concludes continuity with justification

(b) Differentiability at x=1 (6 points)

• 2 points: Finds the derivative of $x^2 + 1$
• 1 point: Evaluates the left-hand derivative at x=1
• 2 points: Finds the derivative of $3x-1$
• 1 point: Evaluates the right-hand derivative at x=1 and concludes differentiability (or lack thereof) with justification.

#2) Determining Differentiability at a Point

Let's tackle a free-response question (FRQ) from the 2003 AP Calculus AB exam. The function $f(x)$ is defined as:

$$f(x) = \begin{cases} \frac{8}{5}\sqrt{x+1}, & x < 3 \\ \frac{2}{5}x + 2, & x \geq 3 \end{cases}$$

Given that $f(x)$ is continuous at $x = 3$, is it also differentiable?

#👉 Checking the Left-hand Side

Find the derivative of the left-hand side:

$$f'(x) = \frac{d}{dx} \left( \frac{8}{5}\sqrt{x+1} \right) = \frac{4}{5\sqrt{x+1}}$$

Then, find the limit as $x$ approaches 3 from the left:

$$\lim_{x\to 3^{-}} f'(x) = \frac{4}{5\sqrt{3+1}} = \frac{4}{5(2)} = \frac{2}{5}$$

#👉 Checking the Right-hand Side

Find the derivative of the right-hand side:

$$f'(x) = \frac{d}{dx} \left( \frac{2}{5}x + 2 \right) = \frac{2}{5}$$

Then, find the limit as $x$ approaches 3 from the right:

$$\lim_{x\to 3^{+}} f'(x) = \frac{2}{5}$$

#👆 Checking $f'(x)$ exists at $x = 3$

Since the derivative of the right-hand side is $\frac{2}{5}$, the derivative at $x=3$ is also $\frac{2}{5}$.

Since $\lim_{x\to 3^{-}} f'(x) = \lim_{x\to 3^{+}} f'(x) = f'(3) = \frac{2}{5}$, the function $f(x)$ is differentiable at $x = 3$!


Graph of piecewise function

Graph created with Desmos

#🌟 Final Exam Focus

High-Priority Topics:

• Understanding the relationship between continuity and differentiability
• Identifying points of non-differentiability (discontinuities, corners, cusps, vertical tangents)
• Using limits to determine differentiability

#🎉 Closing

You’ve got this! With a solid grasp of these concepts and a bit more practice, you'll be ready to tackle any differentiability and continuity question that comes your way. Keep up the great work, and remember, you're doing amazing! 🚀

Encouraging GIF with animated ice cream

Image Courtesy of Giphy