Calculus HandWritten Notes

MATHEMATICS
CALCULUS


cAT-_1

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fcn)=2x3
Max -18k ,




¥¥Ñ¥¥ima
get -
A collection of well defined "
"" ⇐ Abs . Q .




examine
µa✗ critical pts and
.




"" Find
a.

"" "
"""
"
too
function F- bodies fincueases
.
i
in which interval
function - A
function is a
binary inter
(a) upward on an i decreases .

sets that concave on
ovulation between duo real "

Afn!ÉÑ critical pts → -1/-53
-




associates each element of the
z if f inuuasinoniz:
'
is i. : i. ! ,
.




6 x2 18--7 6th 3)
"
set to exactly one element of down on
first (b) open int pecond derivative dust
fan
-


concave a . -




second set %)
decreasing 0h2
the • '


if fl 1^-3
.




2 is . '

values
of function Local fnteumua
Extreme .

Ulta)
function absolute A. econd derivative test fan choke diff j-zv-fg-z.gg
'


suppose firs Igg
'
has
A f
.
an .




maximum) Concavity
maximum ( global is continuous 0W
.
''
and
'
C' it feel 2- fcc) for all
-
f that contains
find absolute max & absolute
at its d-will
differentiate interval min
ATPase f
.




✗ in ☐, where ☐ is due domain .ee open '
of flock ¥
-
s
,
-
zsasz
interval 'z :
'
on an c
"
(c) is called
.

- /
(c) =O& f (c) < 0
of f ! The number
f. qffiicx) > ov-x.CI
then the
Daf f
-
Liz '
f ( n) 2-3
, :
1=0
'


maximum value
'
in'D has local maxima
off is concave
upward f
'

graph off then
.
.




( convex Downward) does not
'

absolute min at 'c { f
has at ;
-


f :
an , c -




c- I Then the " critical
RED and Lii ) 9f f" (a) Lot ✗ fie )=o& f @) > 0 I any
if f( c) ⇐ feast
have
/ 2) af
the number
fcc ) in called the graph of f'
'
is concave downward
has dual minim points .




'

find!
then f. a
f (3) =-3
( convene upward) on 2) =
-19g
a
min of fl
-


absolute value z .
-




at c.
minimum and maximum of fl'd cruaph Example y ✗
3

-2=-19-3
=

=o &
min at
fl (c)
/
one called extreme values of 3) qf
'
f (c) = o f has abs

→ concave
( o, so ) at


g) downward
""
b abs
fails 3=-3
has Max
upward .
Then the best .
f
Local Maxima/ Minima as -3×2+1
function f
.
have a
&" " flu) -1-21×14
=
The
( may
.



Max
A
function f has local concave
-
.




local maxima or minimal
^
↳ •




at e' if there is
f' Cx) = 3×2-6✗
open
'
an




¥-8 ( further invest:
.¥*
"
neither b'" " °

containgci foeipi
such "
interval I' on
'
concave .
¢ >
x=o&n=2
that
ftk) Efcc)v- ✗ c-I , , Meg
. .




upwau
-




critical
inflection Find cuitical
points of
'

Point of → A point 'D f has
similarly f
minimum at .ci
, has a local
exists curve is called point
"

pls at 0 & 2
if mum on a
4+822+16 and
.
a
,
fee)= ✗

of inflection if fl b) I
'
interval Ii
contained dhe curve
' - =

open an
ate the interval on
fcz) =-3
upward Depue
much that if (c) £ floe)t • c-Z changes from concave
which
f is increasing and lo ) 1
flu)= 17

÷÷Ydn;¥÷;n f
=

to
Guiaph Abs Max f is decreasing
'
versa
which
' '
.
on
.


4 17
Abs Max at
=

16=42-432
> convex
Max upward 4- great
Local
. .




foe, ✗
=

derivative debt fan
←
y=fcx) finest
Abs pain at 2 =-3

Abs / local Min .
Local Extrema
'



f' (a) = 2 ( x2 4) (2×3=0
-




consider fln )=Ñ( 1- ×) ?
M""
'a
'
I d
' ' suppose dhatciuacuitical
continuous fume
=
"✗ G- ↳ ( " + when abb are
positive
e
point of a .
b
Find
is differentiable
critical pts { 0,2 ,
-2
} meal numbers . the



¥¥¥÷Éf Function of of final]
'

y and f maximum value
:
a
in some interval
at
containgc
.ci
if
fer) flew
flak seal 1- xD see is
a-
variable ifcxl , is
except possibly C- • ' -4 t ve

single
value Inis interval
moving
-

a

fli)
across

flown self fl = 0 =

pudhe domain of
'



beom left iboeuignt
ie .
- zcseco T tve
'm its
it is not differentiable
1) af if changes ith sign b' Into
.

'
drive -0
stationary points f
→
•<
.


< z ne
)b
✗ -

a- '

stationary inpoints are cuitical
from neg to pas
ate them . = a ✗ (I -
×


it has local minima
'
l
points 'd me domain
of't > 2 T + M
bgca ( 1- xjb
-
.
se .




( convexity).→4f the graph 2) 96 f changes align
-




foes which f @ 1=0
'
'
from pas .



b is "
" Masint " the
Concavity '
dies above all of its
tang
do neg . at then
f has
local
gb-lfay.gg?bzginteuval(-2,o)U(2io)
, = ✗
a -1
, - ✗


af f
'
. .




maxima
in an interval 'I ! Then it is
]
.




↳(
in the
called concave upward (convex 3 .
y f
'
does not changes sign f
is
decreasing " = critical
interval C- •if vfo , 2) pt
f is positive
'
downward) on 2 ate i. e or
.



. .



sides of at
negative f
both c. max
has abs
graph of f' lies below
on
af
'
ihre
has local extremism
it is Then f no
all
of
called
its
tangents on

downward concave
,

(converse at
'

C' .
✗ =

¥
upward) on z. a9bb/@ + b)
9th