TI89 and Differentiability

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

Arches National Park

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

Arches National Park

To be differentiable, a function must be continuous and smooth.

Derivatives will fail to exist at:

corner cusp

vertical tangent discontinuity

f x x 23f 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-89:

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

Today you will be using your calculator, but be sure to do them by hand if possible.

Remember that half the AP test is no calculator.

3y xExample: Find at x = 2.dydx

d ( x ^ 3, x ) ENTER returns23x

This is the derivative symbol, which is .82nd

It is not a lower case letter “d”.

Use the up arrow key to highlight and press .23x ENTER

3 ^ 2 2x x ENTER returns 12

or use: ^ 3, 2d x x x ENTER

Warning:

The calculator may return an incorrect value if you evaluate a derivative at a point where the function is not differentiable.

Examples: 1/ , 0d x x x returns

, 0d abs x x x returns 1

Graphing Derivatives

Graph: ln ,y d x x What does the graph look like?

This looks like:1yx

Use your calculator to evaluate: ln ,d x x 1x

The derivative of is only defined for , even though the calculator graphs negative values of x.

ln x 0x

There are two theorems that you need to remember:

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, if it has a derivative then it is continuous.

12

f a

3f b

Intermediate Value Theorem for Derivatives

Between a and b, must take

on every value between and .

f 12 3

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

differentiable, then takes on every value between

and .

f f a

f b

Assignment: page 114 - 115 Do 1 – 15 odds, 31-37 all, 39