### Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist

#### Introduction

Welcome back to AP Calculus, mathletes! Ready to dive into the world of differentiability and continuity? Let's navigate these mathematical waters and understand when a derivative exists, doesn't exist, and why it matters. We're going to break it all down, sprinkle in some humor, and hopefully make calculus as easy as pie—well, calculus pie. 🥧

#### Continuity and Differentiability: Besties or Frenemies?

First up, let’s talk about continuity and differentiability. Imagine they’re friends strolling through the park. A function can be continuous (no breaks, jumps, or holes) and differentiable (smooth with no sharp corners or vertical tangents).

If a graph is continuous, we can draw it without lifting our pen—like those doodles you make in class! However, just because a graph is continuous doesn't mean it's automatically differentiable. Differentiability is like leveling up in smoothness; no sudden jerkiness allowed!

For a curve to be differentiable at every point in its domain, the graph must be smooth as silk. If you zoom in, it should look like a straight line, not a jagged edge. Imagine looking at a mountain from afar, it looks smooth, but up close, you see all the bumpy rocks—those bumps are a no-go for differentiability!

#### Zooming into Derivatives: The Close-Up Magic

When determining the existence of a derivative at a point (x = a), the limit from the left must equal the limit from the right: [ \lim_{{x \to a^-}} f'(x) = \lim_{{x \to a^+}} f'(x) = f'(a) ]

If a curve is differentiable at (a), it's also continuous at (a). But the reverse isn’t always true! If a curve is continuous, it might still sport a few non-differentiable quirks. Let's explore some scenarios:

##### Scenario 1: Discontinuous Flops

If your function decides to take a sudden jump (like a kangaroo), it won’t be differentiable at that point. For example, ( g(x) = \frac{1}{x+2} ). The graph is all over the map at (x = -2), making it discontinuous and thus non-differentiable at that point. It’s like trying to dance across a broken bridge—just don’t!

##### Scenario 2: Jump Discontinuity

A function like a kangaroo on a pogo stick, such as: [ f(x) = \begin{cases} 1 & \text{if } x < 2 \ -1 & \text{if } x \geq 2 \end{cases} ] jumps at (x = 2). It’s continuous only in spirit but not on paper. The abrupt change (jump) makes it non-differentiable at (x = 2). Think of it as trying to skateboard over a ramp that just abruptly ends—not a smooth move!

##### Scenario 3: Removable Discontinuity

Consider something like ( h(x) = \frac{x^2 - 4}{x - 2} ). This shape-shifter has a hole at (x = 2). Even though ( h(x) ) looks smooth everywhere else, the missing piece disqualifies it from being differentiable at (x = 2). It’s like a beautiful puzzle with one piece missing!

##### Scenario 4: Vertical Tangents

Sometimes, a graph seems to shoot straight up, like ( f(x) = 2(x+2)^{1/3} ). At (x = -2), the slope would be infinite, which means... no defined derivative! It’s like trying to run straight up a wall—not really happening.

##### Scenario 5: Corners and Cusps

Imagine the graph of ( g(x) = |x| ). At (x = 0), it forms a sharp V. The predator-prey analogy here is like making a quick U-turn at high speed—sharp corners where derivatives just can’t keep up.

#### Practice Makes Perfect!

##### Identifying Differentiability from a Graph

Look at a piecewise function and identify where the function says, “Nah, I don’t feel like being smooth here!” Corners, cusps, jumps, and vertical tangents need to be spotted. If the function has more drama than a soap opera at certain points, it's not differentiable there.

##### Algebraic Checkpoint

For algebraic functions, like the one below, verifying differentiability means checking if derivatives from the left and the right match nicely. If they shake hands, you're good!

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

Step 1: Differentiate each part. [ f'(x) = \frac{4}{5\sqrt{x+1}} \quad x < 3 ] [ f'(x) = \frac{2}{5} \quad x \geq 3 ]

Step 2: Check at (x = 3). [ \lim_{{x \to 3^-}} f'(x) = \frac{4}{5\sqrt{4}} = \frac{2}{5} ] [ \lim_{{x \to 3^+}} f'(x) = \frac{2}{5} ]

Both match! Therefore, differentiable it is. 🚀

#### Conclusion

And there you have it! Differentiating between differentiability (pun intended 🤓) and continuity is crucial for mastering calculus. As you tackle exam problems, keep an eye on sudden jumps, sharp corners, holes, and steep climbs. They’re the clues you need to understand when your derivative friend has left the chat.

Remember: smooth sailing, no breaks in the road, and keep it cool. Mathematics may not always be easy, but we'll make it fun one laugh at a time!