·~;;:l1BOOK NOTES - MODULE 7
ffiCREASING AND DECREASING FUNCTIONS
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c.
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the BehavIOI.
Recognize
on an mterval.
d1fterent1able
f 1s
---
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f is increasing on the interval if ~'(X) -:>()
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f is decreasing on the interval if ~I()(:\~ t)
()t\ ),\I(
\o\e~::\[c,.:,,
( ,~, ~ p \ , \"\es \Jo\
Define the Tenn: Complete the following definition.
Partition numbers off I are ... ~~,e., ~l-\J(~\)t,~ W(\ue4
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~ (:/~)~'.,() " '
~ 'C"-'l ~ ~o\- e:,)~\&~o
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Increasing/Decreasing Test: To find the intervals on which a function f is increasing or decreasing, do
the following:
I.
~~~N\\~ ~~ ~O~\C'\ ~ ~.,
2.
T\~ ~Q, ~~>n\\OC\ ~·-c cF, \'\_ ~u.ol\ ~\- ~ on..
~e, ~\I~,~ ~5:~~ Q'~)~o ()~ \!'(:~) ~ f\O\e.,~:~*~
3. L,{"~'\e. A,
~~ O\CA.~'.\ ,$ t'(i) \Jl5·.~ ~~ '\bS \:\¥,~ ' " ~ ~
(:R ~• ~ ~'"~ ~~ ~~ C ~ \ (~ '.it;\~.Ji \ \ ~ ",~\o - , ~ .
Thei"\.
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~~,Vo..¥\\lt ~i ~~ ~ ~-v~vi r\o ~\~,f\r\\~ w\\e..~~ ~\ (.:J...)•\~
~~,~ (!)\" ~~•;\Jt OC"\· ~~l'~ ;~~J~, \~',CA.,~t.i ~~ ~'~) ·,"
D"I ~ \~~o\ b~ wi\:,~ n+'' °''\-".
• ~,\;ve, oc
~ ~~~ /~eOS~" "\~\ .\c, \'.',~ -\he, \~e11"~s
~ ~ \-o \nU-~\~ ~ ~ • , ~ ~ ~ \\- ~,::, dtt<-~
n¥''\
\\w~
THE FIRST-DERIVATIVE TEST
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-w ~1 ~~ :,~ ws,~Nt.-\ ~~5t ,~·,~~~ . i
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\Q Q. °'"')
I':> ~,\\\1(... 1
~ Q\~ ~~"'~•
Define the Term: Complete the following definition.
A function f has a critical value at x
= c if... e;,~~
f
'(l)-:.O ~ ~'(c,,) ~ ("\O\
e.,~* ~ ~,;.(., \&\(\~ ~" ~ ~-
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f is
First-Derivative Test: Suppose f is continuous on an interval containing the critical value x :::: C and
differentiable near, and on both sides ot: x = c. Then, f(c) satisfies one of the following:
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1
A
f ( c) is a local maximum off if ~ (:i\ (\vmc_ y:-;, ~ ttQTn ~ ',..< ~
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f ( c) is a local minimum or f ir\ (:)4.) < h ~ s ~ 1 , ,,1 fl I
0-\ .,,.~.,,
'-
"t)' Ai'llf ~ox-f f:~....""·
~-.:..C
the
Work the Process: To find the intervals where f is increasing or decreasing and local extrema using
first derivative test. do the following:
~ ·.t
t
THE SECOND DERJVATIVE
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Interpret the Concept: In your own words, write at least one sentence describing how to find the second
~
derivative of a function y = f (x).
To
~J; \\e, ~ ~e..n'1o..-\-\\le, ~ou l'tU1 \o ~~ ~-, vo..\Ne,
& ~,{' (tl ~\;0 1'1 ~ru-. 6;We:,CM\ok ~-
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o ff
.. h. . ,• atn l' l l'~I: Suppose/ i'i twice diffrrcr1rit1
S11Jd
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Tl1e n
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succhJlt1at/'(c)~
:::: o.
{) .
ot x = c if ~ ) I
um
im
ox
m
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:nl
lrn
s
• f ha
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--(L
L-"
,t --\'
nt x = c ..
0
.,_ _ .
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f hns n Joc:1l mm•11•11um
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test fails if
The second derivative
Q\\ (L):::. 0 ·
and secon
and contrast the firstTest ,·s. Test: Compare
text
• fon nation from the
\\1•t I1 m
d-deriv
St
First Derivative Te
Critical Values
Test Points
Failure Potential
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a critical value
- ,. MJ d x _ c, iq
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~ n ,\ \) t~
c:Ji \1
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Q '( )( )- :: :. .D .~
\lo.\\le.5 ~ \
~ \) ,f e
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~
~ \ -\ t~ ~o\~~
( ~ (.C ~C G \.\
\J G .\ ~
s
~ ¥ , ~ ': > ~ ,\
~ ~ '6
ive test, do th e fo llo
o\ e .. r~ U \'
4!.\Jc.\\Jll\-\.t, \ \ o.
, ~1~"-~
1 1~ c-"-('re\
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w in g:
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~ \oc.c..\ e > I - . ~
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l
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w
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o...-\- >me_,~ Gr,~C(U
TetW.
w
e T es t
d Derivat
~iv< ,~ ,~
Se co n
~ ~ w1
\
f
O
derivat
ex trem a using the second
ca
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lo
d
fin
To
s:
es
oc
Work the Pr
3.
in g th e ta bl e be lo
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~ '( "1 -.
,5 0 ~
~ ~a;\ ~~
\Q(\_\)\\C-5 ~ \l i\
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• e tests by co m pl et
ativ
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following table, listing
N_o~c the Notation: Giv~n n diffcl'cntinhlc function y
= I (x). complete the
d1ffercnt ways to dcnnte the second derivntivc off.
-
Below. list 2 varintions of th~ Lagt·nn~c 1rnt11tiot1
. . of the Leibniz notation
Below, list 2 var1at10n_s of y::: / (x).
for the second derivative of y = f(x).
for the second dcrivauve.--r.- _ _ _ __
~" ~'\ (\\\~ ~'
\
----i-
~'\)
-'l.\K
~'1.,
~
\
<>-i, i
~---, ..... ,~
L___Q_lf'_~_0t_~-~-~-~-~-(\~----_L______:{=-::.....:..i_ _
o_ ~_ ~_ -;!
CO~CAVITY
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Concavity Test: Suppose f is twice-differentiable on
an
interval.
• f is concave up on the interval if \: \ \ (~~ •... 4) (){) \\
,C
• f is conca,,e down on the interval if ~\\ (_'f.J '•~0 0
•
Define the Tenn: An inflection point is a point wher
ef
c.,.,r.,\;t\0\)$
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,,
\\
r \
r, __ __ _,
\'I'r. \c<)J'~ ~ - - -
is ...
OJ\O ~ C.c\\C,,,, ii'-~ c,1 <> .~\~
Work the Process: To find the intervals where f is
concave up or concave down and points of inflection
using the second derivative test, do the following: