Differentiability, Local Linearity

Differentiability, Local LinearitySection 3.2aA 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

fail to approach a limit as x approaches a. Four instanceswhere this occurs:1. A corner, where the one-sided derivatives differ.

Example:There is a corner at x = 0A 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

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.

Example:There is a cusp at x = 0A 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

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.

Example:There is a vertical tangent at x = 0A 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

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).

Example: The Unit Step FunctionThere is a discontinuity at x = 0Relating Differentiability and ContinuityTheorem: If has a derivative at x = a, then iscontinuous at x = a.

Intermediate Value Theorem for Derivatives

If a and b are any two points in an interval on whichis differentiable, then takes on every valuebetween and .

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

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. 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?

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

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

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

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

Left-hand derivative:

Right-hand derivative:

Guided Practice

For the given function, compare the right-hand and left-handderivatives to show that it is not differentiable at point P.Left-hand derivative: 0Right-hand derivative: 2

Since , the function isnot differentiable at point P.

Graph the function:

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?

(a) All points in [2,3] except x = 1, 0, 2(b) x = 1(c) x = 0, x = 2Guided 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.

Check the one-sided derivatives!!!

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.

Check the two-sided derivative!!!

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