Differentiability and Absolute Value Functions

The video tutorial explores the concept of differentiability in functions, focusing on the absolute value function as an example. It discusses the continuity of functions and the conditions under which they are differentiable. The tutorial also delves into the concept of tangents, explaining how they relate to gradients and the conditions for a line to be a tangent. Finally, it provides a visual representation of derivatives, emphasizing the importance of continuity in determining differentiability.

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

• The function is continuous at that point.

• The function has a tangent at that point.

• The function is increasing at that point.

• The function has a maximum at that point.

Why is the absolute value function not differentiable at the origin?

• It is not continuous at the origin.

• It is not defined at the origin.

• It has a sharp corner at the origin.

• It has a maximum at the origin.

What is the gradient of the absolute value function on the right side of the origin?

• -1

• 0

• 2

• 1

Which of the following lines is a tangent to the absolute value function at the origin?

• None of these

• y = 0

• y = -x

• y = x

Why is the y-axis considered the tangent for the absolute value function at the origin?

• Because it is horizontal.

• Because the gradients approach the same value.

• Because it is vertical.

• Because it passes through the origin.

What happens to the gradients of the absolute value function at the origin?

• They approach different values.

• They are zero.

• They are undefined.

• They are equal.

What is necessary for a function to be differentiable at a point?

• The function must be decreasing.

• The function must be increasing.

• The derivative must be continuous.

• The function must have a maximum.

What does the gradient function of the absolute value of x look like on the left side of the origin?

• -1

• 0

• 1

• 2

What is the relationship between continuity and differentiability?

• A function can be differentiable without being continuous.

• Continuity and differentiability are unrelated.

• A function must be differentiable to be continuous.

• A function must be continuous to be differentiable.

What is the derivative of the absolute value function at the origin?

• Undefined

• 0

• -1

• 1