Section 42a Definition IncreasingDecreasing Functions A function that is always increasing or decreasing on a particular interval is monotonic on that interval Let be a function defined on an interval ID: 473015
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Slide1
Increasing/ Decreasing Functions
Section 4.2aSlide2
Definition: Increasing/Decreasing Functions
A function that is always increasing or decreasing
on
a particular interval is monotonic on that interval
Let be a function defined on an interval I andlet and be any two points in I.
(a)
f
increases
on
I
if
As
x
gets bigger,
y
gets bigger…
(b)
f
decreases on I if
As
x
gets bigger,
y
gets smaller…Slide3
The “Do Now”
Writing: How can the derivative help in identifying
w
hen a function is increasing or decreasing?
Let be continuous on and differentiableon .
If
at
each point of
, then
increases on .
(b)
If
at
each point of
, then decreases on
.Slide4
“Seeing” this new tool…
Consider the
function
First, look at the graph:
x
y
Next, the derivative:
on , so
the function is
decreasing
on
.
on , so
the function is
in
creasing
on
.Slide5
Guided Practice
Use analytic methods to find (a) the local
extrema
, (b) theintervals on which the function is increasing, and (c) the intervalson which the function is decreasing.
The derivative:
There are no endpoints, and the only critical point occurs at:
Minimum of atSlide6
Guided Practice
Use analytic methods to find (a) the local
extrema
, (b) theintervals on which the function is increasing, and (c) the intervalson which the function is decreasing.
The derivative:
is increasing on .
(b) Sinc
e on ,
is decreasing on .
(c) Sinc
e on , Slide7
Guided Practice
Use analytic methods to find (a) the local
extrema
, (b) theintervals on which the function is increasing, and (c) the intervalson which the function is decreasing.
The derivative:
Since the derivative is never zero and is undefined only
where
is undefined, there are no critical points. Also, the
domain has no endpoints. Therefore
has no local extrema.Slide8
Guided Practice
Use analytic methods to find (a) the local
extrema
, (b) theintervals on which the function is increasing, and (c) the intervalson which the function is decreasing.
The derivative:
(b) on .
So, the function is increasing on .
(c) on .
So, the function is decreasing on .Slide9
Guided Practice
Find (a) the local
extrema
, (b) the intervals on which the functionis increasing, and (c) the intervals on which the function isdecreasing.
The derivative:
First, graph
the function…
Critical Points:
x
= –2,
x
= 0
The local extrema can occur at the critical points, but the
graph shows that no extrema occurs at
x
= 0. There is a local
(and absolute) minimum atSlide10
Guided Practice
Find (a) the local extrema, (b) the intervals on which the function
is increasing, and (c) the intervals on which the function is
decreasing.
(b) Since on the intervals: …
And since is continuous at
x
= 0…
is increasing on .
(c) Since on the interval: …
is decreasing on .Slide11
Guided Practice
Find (a) the local extrema, (b) the intervals on which the function
is increasing, and (c) the intervals on which the function is
decreasing.
The derivative:
First, graph
the function…Slide12
Guided Practice
Find (a) the local extrema, (b) the intervals on which the function
is increasing, and (c) the intervals on which the function is
decreasing.
Since the derivative is never zero and is undefined only wherethe function is undefined, there are no critical points. Since thereare no critical points and the domain includes no endpoints, thefunction has no local extrema.
(b) Since the derivative is never positive, the function is not
increasing on any interval.
(c) Since the derivative is whenever it is defined, the function is
decreasing on: