Differentiability Day 2 (Piecewise funcns & graphs) 2024 updated

Front

What does it mean for a function to be differentiable at a point?

Back

A function is differentiable at a point if it has a defined derivative at that point, meaning the function is smooth and has no sharp corners or discontinuities at that point.

Front

What is a piecewise function?

Back

A piecewise function is a function that is defined by different expressions or formulas for different intervals of its domain.

Front

What is a vertical tangent?

Back

A vertical tangent occurs when the slope of the tangent line to the curve approaches infinity, indicating that the function is not differentiable at that point.

Front

What is a removable discontinuity?

Back

A removable discontinuity is a point at which a function is not defined, but can be made continuous by appropriately defining the function at that point.

Front

What is a non-removable discontinuity?

Back

A non-removable discontinuity is a point at which a function is not continuous and cannot be made continuous by redefining the function at that point.

Front

What is a cusp in a function?

Back

A cusp is a point on a curve where the curve comes to a sharp point, causing the function to not be differentiable at that point.

Front

What does it mean for a function to be continuous at a point?

Back

A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point.

Front

How can you determine if a function is differentiable at a point?

Back

To determine if a function is differentiable at a point, check for continuity at that point and ensure there are no sharp corners or vertical tangents.

Front

What is the relationship between continuity and differentiability?

Back

If a function is differentiable at a point, it must also be continuous at that point, but a continuous function is not necessarily differentiable.

Front

What does it mean for a function to be continuous only at a point?

Back

A function is continuous only at a point if it is defined and continuous at that point, but has discontinuities elsewhere.