Local LinearityThe idea is that as you zoom in on a graph at a

specific point, the graph should eventually look linear.

This linear portion approximates the tangent line to the curve at the specific point.

Most of the functions you will encounter are "nearly linear" over very small intervals; that is most functions are locally linear.

All functions that are differentiable at some point where x=c are well-modeled by a unique tangent line in a neighborhood of c and are thus considered locally linear.

A function is differentiable if it has a derivative everywhere in its domain. It must be continuous and smooth. Functions on closed intervals must have one-sided derivatives defined at the end points.

p

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

Differentiability vs ContinuityTheorem:If f is differentiable at a, then f is

continuous at a.

*The converse to this theorem is false. A continuous function is not necessarily

differentiable.

y f x

y f x

The derivative is the slope of the original function.

The derivative is defined at the end points of a function on a closed interval.

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) returns 1000000 on a TI-84, but -∞on a TI-89

nDeriv(abs(x),x,0) returns 0 on a TI-84, but +/-1 on a TI-89

Graphers calculate numerical derivatives by using difference quotients (the same way you’ve been doing it).

Forward difference quotient:

Backward difference quotient:

Symmetric difference quotient:

( ) ( )f x h f x

h

( ) ( )f x f x h

h

( ) ( )

2

f x h f x h

h

1( , , )y nDeriv y x x

21 3y x

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

Given f(x) = 2x3–3x2 – 6x+2Plot the derivativeWhat type of function is the derivative?

Quadratic (one degree less than f(x))When the derivative is 0, what is happening

to the graph of the function?There is either a local max or local minAt x = -1 (local max: f(x) changes from increasing to

decreasing)At x = 2 (local min : f(x) changes from decreasing to

increasing)

Let g(x) = f(x) + 5Graph in Y3

Graph the derivative of g in Y4

How does the derivative of f compare to the derivative of g?The derivative is the same. The vertical shift

does not alter the rate of change. Thus, the derivative remains the same.

Increasing/Decreasing TestIf f '(x) is > 0 on an interval, then f is

increasing on that interval.

If f '(x) is < 0 on an interval, then f is decreasing on that interval.

Suppose that c is a critical number of a continuous function f:If f ' changes from positive to negative at c,

then f has a local maximum at c.If f ' changes from negative to positive at c,

then f has a local minimum at c.

(x = c is a critical number of the function f(x) if f ' (c) = 0 or f ' (c) does not exist.