1. 3.2 Differentiability Arches National Park
Created by Greg Kelly, Hanford High School, Richland, Washington
Revised by Terry Luskin, Dover-Sherborn HS, Dover, Massachusetts
Photo by Vickie Kelly, 2003

2. Arches National Park
Greg Kelly, Hanford High School, Richland, Washington
Photo by Vickie Kelly, 2003

3. What function is this?
Does the window setting matter?

4. ZoomTrig view of the function!

5. 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!)…

6. A derivative will fail to exist at any x-value where
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
discontinuity at x = 0
vertical tangent at x = 0

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

8. Derivatives on the TI-83/84:
You must be able to calculate derivatives
with the calculator and without (using limits) .

9. Find at x = 2
Example:
From your home screen, the MATH 8 command
calculates the derivative of y1 “at” a point; the syntax is:
nDeriv(function, independent variable, coordinate)
nDeriv( x ^ 3, x, 2 )
ENTER 12
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
y1 = x^3
y2 = nDeriv(y1, x, x)

10. function is not differentiable (at a discontinuity,
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

11. Graphing Derivatives
You may recognize the patterns of some derivative graphs!
Graph:
Y1=nDeriv(lnx, x, x)
This graph looks like:

12. 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 already
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!!!)

13. p 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
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. p