Differentiability A function is differentiable at point c if and only if the derivative from the left of c equals the derivative from the right of c AND if c is in the domain of ID: 264687
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Slide1
3.2 DifferentiabilitySlide2
Differentiability
A function is differentiable at point
c
if and only if
the derivative from the left of
c
equals the derivative from the right of
c
.
AND
if
c
is in the domain of
f’
.Slide3
Differentiability
Find the derivative of at
x = 0
.
f
is not differentiable at 0.Slide4
DIFFERENTIABILITY
A function is
differentiable
if it has a derivative everywhere in its domain. It must be
continuous
and smooth.No “sudden change” in slope.Slide5
DIFFERENTIABILITY
Derivatives will fail to exist at:
corner
cusp
vertical tangent
any discontinuitySlide6
Using the calculator
The numerical derivative of
f
at a point
a
can be found using NDER on the calculator.Syntax: NDER (f(x), a)Note:
The calculator uses h = 0.001 to compute the numerical derivative, so it is a close approximation to the actual derivative.
Example:
Compute NDER of
f(x) = x
3 at
x = 2.Slide7
differentiability
THEOREM:
If
f
has a derivative at x = a
, then f is continuous at x = a.
Differentiability implies continuity.