3.2 Differentiability

ObjectivesStudents will be able to:

1) Find where a function is not differentiable2) Distinguish between corners, cusps,

discontinuities, and vertical tangents3) Approximate derivatives numerically and

graphically

Recall:

Cases Where a Function is Not Differentiable

1) A corner2) A cusp3) A vertical tangent4) A discontinuity

A Corner

• A corner is a location where the one-sided derivatives differ.

• Example:

Absolute Value 1Absolute Value 2

A Cusp

• A cusp is where the slopes of the secant lines approach infinity from one side and negative infinity from the other side.

• It is an extreme case of a corner.• Example:

Cusp

A Vertical Tangent

• A vertical tangent occurs when the slopes of the secant lines approach either infinity or negative infinity from both sides.

• Example:

Vertical Tangent

A Discontinuity

• A discontinuity will cause one or both of the one-sided derivatives to be nonexistent.

• Example:

Jump Discontinuity (FF to 3:30)

Review: Types of Discontinuity• Types of Discontinuity

• Another type of discontinuity is oscillating discontinuity.

• Oscillating discontinuity occurs when the function oscillates and the values of the function appear to be approaching two or more values simultaneously.

• Example: f(x)=sin(1/x)

Local Linearity

• Differentiability implies local linearity.• A function that is differentiable at a point

(a,f(a)) is locally linear at that point, meaning that as you “zoom” in on that point, the function will resemble a straight line.

• Local Linearity

Derivatives on a Calculator

Luckily, the nSpire can calculate derivatives, as well as a derivative at a point.

Derivatives on the nSpire

• Try a few to see if you got it!– Find the following derivatives on your nSpire:

Continuity

• Very important note: Differentiability implies continuity.

• Another important note: the converse is NOT true!!! A function may be continuous at x=a, but not differentiable there.• Example: the absolute value function is

continuous at x=0 but it is not differentiable there.

Intermediate Value Theorem for Derivatives

• If a and b are any two points in an interval on which f is differentiable, then f’ takes on every value between f’(a) and f’(b).

• What does that even mean???– Intermediate Value Theorem