Understanding Differentiability of Piecewise Functions

What is the shape of the graph of the absolute value function, and what does it indicate about differentiability?

• A smooth curve, indicating differentiability everywhere

• A U shape, indicating differentiability at the vertex

• A V shape, indicating a sharp turn and non-differentiability at the vertex

• A straight line, indicating differentiability everywhere

Which method involves rewriting the absolute value function using a square root to find its derivative?

• Definition of absolute value method

• Chain rule method

• Limit definition method

• Graphing method

What is the domain restriction for the derivative of the function f(x) = |x - 4|?

• x = 2

• x = -4

• x = 4

• x = 0

How do you determine if a piecewise function is continuous at a point?

• Check if the derivative exists at that point

• Check if the left and right limits are equal and match the function value

• Check if the graph is a straight line

• Check if the function is defined at that point

What is the derivative of the function g(x) = 4x for x < 2?

• 2x

• 4

• 0

• x^2

Why is it important to check continuity before differentiability?

• Because a function can be differentiable even if it's not continuous

• Because continuity ensures the function is defined everywhere

• Because a function must be continuous to be differentiable

• Because continuity affects the graph's color

What happens if the left and right limits of a function at a point are not equal?

• The function is continuous at that point

• The function is differentiable at that point

• The limit does not exist, indicating discontinuity

• The function is undefined at that point

In the example of h(x), what is the left side limit as x approaches 1?

• Undefined

• 1/2

• 0

• 1

What is the right side limit of h(x) as x approaches 1?

• 1

• Undefined

• 0

• 1/2

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

• The function must be a polynomial

• The function must have a derivative everywhere

• The function must be continuous at that point

• The function must be increasing