Differentiability and Its Implications

Which of the following is NOT a condition where a function is non-differentiable?

• Point of discontinuity

• Horizontal tangent

• Cusp

• Corner

What is a common example of a function with a corner?

• Logarithmic function

• Exponential function

• Absolute value function

• Quadratic function

In a cusp, what happens to the slopes of the tangent lines?

• They approach zero

• They remain constant

• They become undefined

• They approach infinity from opposite directions

Which type of function often has a vertical tangent?

• Linear function

• Cubic root function

• Quadratic function

• Exponential function

What is a characteristic of a point of discontinuity?

• The function is linear

• The function is continuous

• The function has a jump

• The function is differentiable

How can you show that a function is not differentiable at a point using limits?

• By showing the left and right limits are equal

• By showing the left and right derivatives are not equal

• By showing the function is continuous

• By showing the function is linear

What does differentiability imply about a function at a point?

• The function is quadratic

• The function is linear

• The function is continuous

• The function is discontinuous

What does the Intermediate Value Theorem for derivatives state?

• A function is continuous if it is differentiable

• A function is linear between two points

• A function takes on every value between two points

• A function is always differentiable

If a function is differentiable on an interval, what can be said about its derivative?

• It is always positive

• It takes on every value between the derivatives at the endpoints

• It is always zero

• It is always negative

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

• The function must be quadratic

• The function must be discontinuous

• The function must be linear

• The function must be continuous