3.2 Differentiability

Arches National Park

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 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-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 when called for.

Remember that half the test is no calculator.

3y xExample: Find at x = 2.dy

dx

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

This is the derivative symbol, which is .8Math

12

This can also be done by viewing the graph of 3y x

Select CALC 6: dy/dx

Either move the flashing cursor or enter the point.

Just enter 2

This can also be done by viewing the graph of 3y x

Select CALC 6: dy/dx

Either move the flashing cursor or enter the point.

We can also DRAW a TANGENT( line to the function at x = 2

The display will be the equation iny=mx+b form.

3y xExample: Find at x = 2.dy

dx

nderiv(

ENTER returns

This is the derivative symbol, which is .6Math

12

which then appears as𝑑𝑑 ()

()|𝑥=()

Take the derivative with respect to x

of the function x³

solve it whenwhen x = 2

appears slightly differently on the TI-84

Warning:

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

Examples:

nderiv(1/x,x,0)

nderiv(abs(x),x,0)

returns 1000000

returns 0

Graphing Derivatives

Graph: y = nderiv(lnx,x,x) What does the graph look like?

This looks like:1

yx

Use your calculator to evaluate: ln ,d x x1

x

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

ln x 0x

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, if it has a derivative then it is continuous.

1

2f a

3f b

Intermediate Value Theorem for Derivatives

Between a and b, must take

on every value between and .

f 1

23

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

p