Differentiability, Local Linearity

Section 3.2a

A function will not have a derivative at a point P (a, f(a)) wherethe slopes of the secant lines,

How f (a) Might Fail to Exist

f x f a

x a

fail to approach a limit as x approaches a. Four instanceswhere this occurs:

1. A corner, where the one-sided derivatives differ.

f x xExample:

There is a corner at x = 0

A function will not have a derivative at a point P (a, f(a)) wherethe slopes of the secant lines,

How f (a) Might Fail to Exist

f x f a

x a

fail to approach a limit as x approaches a. Four instanceswhere this occurs:

2. A cusp, where the slopes of the secant lines approachinfinity from one side and negative infinity from the other.

2 3f x xExample:

There is a cusp at x = 0

A function will not have a derivative at a point P (a, f(a)) wherethe slopes of the secant lines,

How f (a) Might Fail to Exist

f x f a

x a

fail to approach a limit as x approaches a. Four instanceswhere this occurs:

3. A vertical tangent, where the slopes of the secant linesapproach either pos. or neg. infinity from both sides.

3f x xExample:

There is a vertical tangent at x = 0

A function will not have a derivative at a point P (a, f(a)) wherethe slopes of the secant lines,

How f (a) Might Fail to Exist

f x f a

x a

fail to approach a limit as x approaches a. Four instanceswhere this occurs:

4. A discontinuity (which will cause one or both of the one-sided derivatives to be nonexistent).

1, 0

1, 0

xU x

x

Example: The Unit Step Function

There is a discontinuity at x = 0

Relating Differentiability and Continuity

Theorem: If has a derivative at x = a, then iscontinuous at x = a.

f f

Intermediate Value Theorem for Derivatives

f fIf a and b are any two points in an interval on which

is differentiable, then takes on every valuebetween and . f a f b

Ex: Does any function have the Unit Step Function as itsderivative?

NO!!! Choose some a < 0 and some b > 0. Then U(a) = –1and U(b) = 1, but U does not take on any value between –1and 1 can we see this graphically?

Differentiability Implies Local Linearity

Locally linear function – a function that isdifferentiable at a closely resembles its own tangentline very close to a.

Differentiable curves will “straighten out” when wezoom in on them at a point of differentiability…

Differentiability Implies Local LinearityIs either of these functions differentiable at x = 0?

2 0.0001 0.99g x x 1f x x 1. We already know that f is not differentiable at x = 0; its graphhas a corner there. Graph f and zoom in at the point (0,1)several times. Does the corner show signs of straightening out?

Continued zooming in at the given point (assuming asquare viewing window) always yields a graph with theexact same shape there is never any “straightening.”

Differentiability Implies Local LinearityIs either of these functions differentiable at x = 0?

2 0.0001 0.99g x x 1f x x

2. Now do the same thing with g. Does the graph of g show signsof straightening out?

Try starting with the window [–0.0625, 0.0625] by[0.959, 1.041], and then zooming in on the point (0,1).Such zooming begins to reveal a smooth turning point!!!

Differentiability Implies Local LinearityIs either of these functions differentiable at x = 0?

2 0.0001 0.99g x x 1f x x

3. How many zooms does it take before the graph of g looksexactly like a horizontal line?

After about 4 or 5 zooms from our previous window, thegraph of g looks just like a horizontal line.

This function has a horizontal tangent at x = 0, meaningthat its derivative is equal to zero at x = 0…

Differentiability Implies Local LinearityIs either of these functions differentiable at x = 0?

2 0.0001 0.99g x x 1f x x

4. Now graph f and g together in a standard square viewingwindow. They appear to be identical until you start zooming in.The differentiable function eventually straightens out, while thenondifferentiable function remains impressively unchanged.

Try the window [–0.03125, 0.03125] by [0.9795, 1.0205]

How does all of this relate toour topic of local linearity???

Guided Practice

2 3f x x

2,3

Find all the points in the domain of wherethe function is not differentiable.

Think about this problem graphically. We have the graph of theabsolute value function, translated right 2 and up 3:

There is a corner at (2,3),so this function is notdifferentiable at x = 2.

Guided Practice

2, 1

2 , 1

xf x

x x

1,2P

For the given function, compare the right-hand and left-handderivatives to show that it is not differentiable at point P.

Graph the function:

0

1 1limh

f h f

h

Left-hand derivative:

0

2 2limh h

0lim 0h

0

0

1 1limh

f h f

h

Right-hand derivative:

0

2 1 2limh

h

h

0lim 2h

2

1,2P

Guided Practice

2, 1

2 , 1

xf x

x x

For the given function, compare the right-hand and left-handderivatives to show that it is not differentiable at point P.

Left-hand derivative: 0

Right-hand derivative: 2

0 2Since , the function isnot differentiable at point P. 1,2P

Graph the function:

1,2P

Guided PracticeThe graph of a function over a closed interval D is given below.At what points does the function appear to be(a) differentiable?

(b) continuous but not differentiable?

(c) neither continuous nor differentiable?

y f x : 2 3D x (a) All points in [–2,3] except x = –1, 0, 2

(b) x = –1

(c) x = 0, x = 2

Guided PracticeThe given function fails to be differentiable at x = 0. Tell whetherthe problem is a corner, a cusp, a vertical tangent, or adiscontinuity. Support your answer analytically.

4 5y x Check the one-sided derivatives!!!

0

0 0limh

y h y

h

4 5

0limh

h

h 1 50

1limh h

0

0 0limh

y h y

h

4 5

0limh

h

h 1 50

1limh h

The problem is a cusp!!!

(support with a graph???)

Guided PracticeThe given function fails to be differentiable at x = 0. Tell whetherthe problem is a corner, a cusp, a vertical tangent, or adiscontinuity. Support your answer analytically.

1 33y x Check the two-sided derivative!!!

0

0 0limh

y h y

h

1 3

0

3 3limh

h

h

1 3

0limh

h

h

2 30

1limh h

The problem is a vertical tangent!!!(support with a graph???)