3.2 Differentiability

Photo by Vickie Kelly, 2003

Arches National ParkCreated by Greg Kelly, Hanford High School, Richland, WashingtonRevised by Terry Luskin, Dover-Sherborn HS, Dover, Massachusetts

Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003

Arches National Park

What function is this?

Does the window setting matter?

ZoomTrig view of the function!

For a function like y = x^5 – 6x…

Zooming in makes the curve straighten out…

“Differentiable functions are continuous and locally linear” (but not vertical!)…

A derivative will fail to exist wherever the slope of f(x) changes drastically or is undefined, or at an x-value where f(x) is discontinuous:

corner at x = 0 cusp at x = 0

vertical tangent at x = 0 discontinuity at x = 0

f x x 2

3f x x

3f x x 1, 0

1, 0

xf x

x

Most of the functions we study in calculus will be differentiable!

Derivatives on the TI-83/84:

You must be able to calculate derivatives with the calculator and without (using limits .

31y xExample: Find at x = 2.dy

dx

nDeriv( x ^ 3, x, 2 ) ENTER returns

12

From your home screen, the MATH 8 command calculates the derivative of y1 “at” a point; the syntax is:nDeriv(function, independent variable, coordinate)

y1 = x^3

y2 = nDeriv(y1, x, x)

From your y= screen, the MATH 8 command calculates and plots the derivative of function y1 at all x values in the window: the SLOPES along y1 are graphed as

the HEIGHTS on y2

W a r n I n g:

The calculator can return an incorrect value if you try to evaluate a derivative at a point where the function is not differentiable (at a discontinuity, a cusp, a corner, or a vertical tangent location!). This is known as “grapher failure.”

Examples:

nDeriv(1/x,x,0) returns 1,000,000 (or some other large number!)

nDeriv(abs(x),x,0) returns 0

Graphing Derivatives

Graph:

This graph looks like:

1the right branch of y

x

Y1=nDeriv(lnx, x, x)

You may recognize the patterns of some derivative graphs!

There are two theorems on page 110:

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

Since a function must be continuous to have a derivative: limit = f(a) = limit x →a- x→a+ then each function that has a derivative is continuous on its domain.

A typical logic error by beginning calculus students is to try to switch the “if” and the “then”, thereby creating the converse (which may or MAY NOT be true!!!)

1

2f a

3f b

Intermediate Value Theorem for Derivatives

If a and b are any two x-values in an interval on which f is

differentiable, then takes on every value between

and .

'f

'f a 'f b

The slope takes on every value between the slope at a and

the slope at b

…for this function, every slope between ½ and 3.