3.2: Continuity

3.2: DifferentiabilityObjective: Students will be able to…• Find points where the derivative

fails to exist.

There are 4 instances where the derivative fails to exist…

Derivative

Where f’(x) Fails to Exist

O Is the function f(x) = |x-2| differentiable at x=2?

O Find the left and right hand derivative:

O Look at the graph…what does it look like at x=2?

DNExfxf

x

2)2()(lim

2

1. CORNER:O One sided derivatives exist, but

differ

Given: f(x)=x2/3. Find f ’(0 ).

0)0()(lim

0 xfxf

x

• What is the domain of f(x)? What is the domain of f’(x)?

• Graph the function. What do you see at x=0?

2. A CUSP: (rational exponent—numerator even, denominator odd)

O an extreme cornerO The slopes of the secant lines

approach ∞ from one side and -∞ from the other side

Find the derivative of f(x)=x1/3 at x =0.

Look closer ….

Look at graph of function.

00lim

31

0 xx

x

3. VERTICAL TANGENT(rational exponent—numerator and denominator are odd)

O Slopes of the secant lines approach either ∞ or -∞ from both sides

Find f’(0) if

Graph f(x). What do you see?

0,20,2

)(xx

xf

4.DISCONTINUITYO Will cause one or both of the one

sided derivatives to be ±∞

Find all the points in the domain where f is not differentiable. Justify your answer.

1. f(x)= |x +3| -1

2. 3 62)( xxf

Find all points where f is not differentiable. Justify your answer.

1. 2.

1,

1,)(

2 xx

xxxf 3

23)( xxg

Most functions are differentiable wherever they are defined. Look for corners, cusps, PODs, and vertical tangents.

O Polynomials→ differentiableO Rational, Trig, Exponential, Log

functions→ differentiable in domains

Given:

O Where is y not differentiable?

Think of the graph….check endpoint of interval

1 xy

Differentiability Implies Continuity

O If f has a derivative at x=a, then f is continuous at x=a

O Is the converse true?

Differentiability Implies Local Linearity

O A function that is differentiable at ‘a’ closely resembles its own tangent line very close to ‘a’

O p.107: Exploration

Is f(x)=sin x differentiable at x=л? If so, what is f’(л)?

Intermediate Value Theorem for Derivatives

O 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).