Differentiability at a point: algebraic (function is differentiable) | AP Calculus AB | Khan Academy

TL;DR

The given function is continuous and differentiable at x = 3.

Transcript

• [Voiceover] Is the function given below continuous slash differentiable at x equals three? They've defined it piece-wise, and we have some choices. Continuous, not differentiable. Differentiable, not continuous. Both continuous and differentiable. Neither continuous not differentiable. Now one of these we can knock out right from the get go. In o... Read More

Key Insights

• ❓ Differentiability implies continuity in functions.
• 😥 The limit of a function as it approaches a specific point can determine its continuity at that point.
• 😥 To evaluate the derivative at a specific point, the limit of the difference quotient needs to be evaluated.

Questions & Answers

Q: Is it possible for a function to be differentiable but not continuous?

No, in order to be differentiable, a function must also be continuous. Differentiability implies continuity.

Q: How do we determine if a function is continuous at a specific point?

To determine continuity at a specific point, we need to check if the function's value at that point matches the limit of the function as it approaches that point.

Q: What does it mean for a function to be differentiable?

A function is differentiable at a point if the limit of the difference quotient exists as x approaches that point. This implies that the function has a well-defined derivative at that point.

Q: How do we find the derivative of a piece-wise defined function?

To find the derivative of a piece-wise defined function, we need to find the derivatives of each piece separately and ensure they match at the point where they intersect.

Summary & Key Takeaways

• The function needs to be both continuous and differentiable to determine its behavior at x = 3.

• By evaluating the limits from the left and right-hand sides, it is confirmed that the function is continuous at x = 3.

• By evaluating the difference quotient, it is confirmed that the function is differentiable at x = 3.