AP Calc Notes: ID – 4 Differentiable/Non-differentiable ...

AP Calc Notes: ID – 4 Differentiable/Non-differentiable functions

A function f is differentiable at x = a only if lim f(x) - f(a) exists.
x→a x - a

lim f(x) - f(a)
As x → a, this limit will be of the form x→a . The only way it can exist is if
0

lim f(x) - f(a) = 0. In other words, we need

x→a

lim f(x) = f(a)

x→a

This is the definition of continuity of f at a.

Theorem: If f is differentiable at x = a, then f is continuous at x = a.

Note: The converse is NOT necessarily true. It is possible for f to be continuous at a point
but NOT differentiable.

A function, f(x), is differentiable at x = c if f(x) is continuous at x = c AND

lim f '( x) = lim f '( x) [basically, the slope of the tangent line at x = c exists.]
x→c− x→c+

A continuous function can fail to be differentiable at a point where
1) f has a “corner"

f does not have a unique tangent line at x = a.

The slope “just to the left” of a is not the a
same as the slope “just to the right” of a.

2) f has a vertical tangent

If the tangent line is vertical, then its slope is
undefined.

a

(Note: these are not the only ways continuous functions can fail to have a derivative. They are the only ones
you have to know in this course.)

Summary

A function f will fail to be differentiable at a point x = c if

1. f is discontinuous y
a. f(c) not defined

c x y x
y x c
x
b. lim f(x) DNE c
x→c

c. lim f(x) ≠ f(c) y
x→c c

2. f has a corner y

cx

3. f has a vertical tangent y

cx

Fill in the blanks.

If f(x) is differentiable, then f(x) _________________ continuous.
If f(x) is continuous, then f(x) _________________ differentiable.
If f(x) is not differentiable, then f(x) _________________ continuous.
If f(x) is not continuous, then f(x) _________________ differentiable.
If f(x) is differentiable, then f’(x) _________________ continuous.

Numerical Derivatives on your calculator
Derivatives on the graph

Must write on paper correctly.
To evaluate a derivative on the home screen

To graph derivatives on your calculator

Ex: The derivative, f ', of a continuous function f is given by f '( x) = 1 − −2 .

x3 3x 3

a. On what interval(s) is f increasing? Decreasing?

b. Find the relative extrema of f.