calculus module 8

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4

3

Exercises.la?fisnicaaingiffSoonthatnitenal,and
hreainggfcoonuiund.no
mugaphfoonlaDandcsinfoonlbDandc

z

87.hu

,

fnineaeaingmlanandcspfndcnhrgon.cl/51andlI8).lb)Zffchangestssign.fiaidtohuelgalmuimunorgdmiwimum
Fomthegmph.tn pontsoflogdmaximumǎueuandx-7

Zhepōmtofhgaluùnimunisx-5

nnomthegraph.hu
achanthelow

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X 1 014) 4 (4 67 6 16187 8 18,⽐)

檆 品 正品 下品 i
(a)2ffsoonannitend.fincomgmvuintendhu.fmthepaph.fismwanigonlo4NIG8i.hn
uhenfio.xistcuialnu.hr
fomthechanix-4andx-8anethelocdmaimms.ro617

theloealmiuinum.cc?foconeaveupwarduhenf”>0,
andciconaedounuarduhenfko.whenfso.fi
ismweaniguhenfao.fidareaing.Fomthepaph.fiioingonlaDUlzsvcsipfndoeaingmlnzs.vn,51017191
hnunfisconeaveupwardoncanvn.sunpfiscouavedownwand m (1, 21 0 1 3 151 0 17 9)
d) pomcsinflectionponuxi.ro, 如了, x_x-7

fhexoordinatesof.ae

11 .f) =

x

4
-2ㄨ年3 X ⽐,-1) – 1 (-1 1 01 0 10, 1) 1 (1 , a)

⽨)
= 4 -4✗
“”

2-g-oto.co ⼗
⼆ 4✗ (x-1 1) (划

fantminfmaxtminfletfixr-o.ix-o.tn/xco,-g#gro)fl0)=3.⽨) t – +
f-1)= (⼀)1

4
-2(⼗)年 3 = 2 f) 0

2 0

conauecomaeconaue.fi1 1 = 1 4- 2 +3 : 2 fupd.am up

f
”

⽐)= 1⽐三千

⼆ 413✗

2

-1)

hetfixiio.i.net#(T)4t2n2t3zT-Tt3Tfl-
驻图4-1 2岒年 了 ⼆年

Haue , (a)

fismaaingonH.nu/o)fisdecreangonl-o,-DUco,Dcbhoeal
maimumisfchocalminimwmsare.fi-1) = 2

andfn-2.cc?fisconaeupuardonl-ri1ugto)fiscoucaredowwww.ong#7hecoordinatesofpointofinflatnnae gjud内分

x

1 2 . fx) = x2+ 1NC-diD-IHDIC-7_x-fixklx4D-x.toy – o + 0 –
(奸⻔

2

fnnlminimaxdl-x2-cxz.li
)
2

✗ 也, ⼀万 ) (疗,0710,可(vi)

Lltf ix) = 0 , ⼈ x- f

‘

⼼ ⼀ ⼗ ⼀ +

f)==
‘

f 1)= – ⼠ fixt.tt
9

concaveconcaveconcaveconcavc.fi) = 2必化到2 – ( l-xj.zx.tn/pfMd0wnwardupwarddownwadupuad到 4
2ㄨ (如 3)

⼆

氹
2
-1 1)3

Let 批上0.ci✗ ⼆0.x疗 ,如疗
f 1 07 = 0

.

f后)= T
f (加 -_-#
Hence

, (a) fisincreasingonltnfisdeweaingonc-oivu.to)
(b)

todmaimumisfD-hoedminimumisfD.tl
fiscouweupward.mn 1万 , 070 仃了 ,⽐)
fisconeavedonnwadml-o.nu (0,可 )
Thetutionpontsare.li, 有 and (⼀万,年)

萨帕和
14 fxE.cos2x-2sinocoaeza~COEY-c-3-gofxkzmx.co-2 cosxfixi-oto-z.cn

✗ 1的 九 ⼗1)

fnntminlluhetfln-o.i.zcoyxeoixc.to
,
⼤7

i. ✗ = Ǐorx-_-
的九 ⼗1=0,0 E [01 2元] ✗ 1 0, 划法别

⼀⼀

陪, 2ㄤ)
i . 0 = # f

‘

⼼ ⼀ ⼗ ⼀

f周⼆ 0 – 2 = -2 fix t T t
所 䈘⿏auaueconau.fi#)=O-2x(-D=2

upwanddounwadfxkzsnxcosx-zcosxt-snzx-2s.ro
f
”
以 ⼆ -2coszxtz.mx

Letf
”
必 ⼆ 0 .

i. -2oszxtz.sn✗ = 0

-(1 – 2 的刘

tsnx-ozsixtsnx-koczsnx-DcsnxtD-oixc-T.co/zn7:.snx==or
的 九 ⼆ ⼗

⼋ 九 ⼆ 哥 ,或orx-f-oft-fcttle.me
, (a) fisincreasingmc-fisdeweaingonlo.SU#,zrD
(b) todmaimumishoedminimum.si灓⾮z
(c) fisconaueupward.mn#)fisconeavedonnwadonccou惨叫
Thetutimpontsare.gl andg_

38 . fN)= 360+ 3 X
⾄20 X 1-0)-2) -2 1-2,373 (3, -10)

⽨) = 36
-1 60- 60 fx)

– 0 -1 0

Letfx) =0 .fxtmintmaxti.GEx40-1 6)=0
(⼀九⼗了) (x-1 2) =0 X (加,⼠) ⽣ , ⼗

。)

x= 3 orx-2 ⽨) t
–

93 )= 36 ×3 -1 3 × 9 – 2吓-81 f) 0lvconaue.comave
f -2)=72-1 121 16 = -44

fupwarddounuadfxk6-hetfoko.d.kz
fl⽣) = 184 3×本- 2字==

37

Heue, (a)

fisinueaingon.tn?fisdaongon(-no,z)Ul3,to).lb)7he
loealmaoimumisfc31-817leloedmiwimumisfli-44.ca
fisconeareupwadoncx-1.fisconauedounuadonhta.hu
inflatàapoug -_-

d) , ^
80 .

LetfM-140.fi36xtn

zx

了
。6 0 .

-2奸 3.x-1 36 =0

1 20
. i X= -3 ±1 9+ 4112X36

-4iii.焱-20 . = 3 ±1年
4

年 etoo
~~等

⼈ X1 = 5 , 加= -3 . 5

X3 = 0 .

40.gg200+8✗
3
+ ✗4

.

0 ⽐,
⼀6) -6 g) 0 1 0, +

g)⼆ 2402+4 fix)
– 0 t O t

hetglxko ,ytminixfi.ae2 (6+0)

=0 ,

⼈ ✗= o or x= – 6 X (加,

-4) (-4,0) (o,⽐)

f 1 0) =20 0 ,f-6)=
-232 f

“必 ⼗ ⼀ ⼗

g
”
如480-1120

flxltttconcavecouareoowehetg.MN:0,
fxlupwarddounwadupwardiilzxkttxl-oi.no

-0

orx-4fl-4F-56.Heucaccnfisiuangon.lt, ⼗。)
fisdaungonc-o.tl

(b) 7heloealmaoimumdoeswteoistneloedm.inìmumìsf6) -_-232
(c) fisconeareupwadonc-ag-41UCO.to)

fisconavedounuadonl-41077heiuflutionpoiwts.aev4, ⼀561 and 1012以
(d) hetfxl-o.at715洁20 0t-j6-5f-X22-316.to
i -200

:
(-6,2了2)

Eoercn 4,4

✗-3cosX8.gr-9 笳哭, 13 . 点, tsìnx
– sinx

⼆点7点 ⼆点个⾯✗-3) ⼆ 怨的⼗ _osx
= i = 然 右 ⼆ ⼗ ⼆ 点䟞

tanx-o16.hn”
21 . 息红华

⼼ 0 l-cosxAsx-ot.hxt-o.ae
= tm 2ㄨ s 。LHgiulkulecannotbeapphd.no

snxAsotisavegmadpontnenumher-lim2.nocosx ” 箱笼 = -a
= 2

27, 㖌
‘普

24.hn
845t

t→ o
t ⼆点妛

= 1㖌
8-8-5ㄠ5

1 ⼆点 坒
= m8-h5 = hf = E

3 2 、 㖌等 3 4, hmcosmxtsnx
✗→ 0 ✗

2

⼆ 感。
2幾 ⼆ 㖌 -mimmxtn.snnx

zx

⼆点咬
⼆ līm _musmxtnisnx

= 0
X-0 z

=
nEm

2
z

guniei @ limx.PT0
✗3

X_X

⼆点% 。如 ⼆点%
eT

=
ūm

3ˇ
= e㗊等

xsozx.ci
= e

‘
品-

x
= Ì㖲 eㄨ2

= e
0

==
3 limlxsozx.li = 1
-_-。
⼆ 0

4. 5

ExercisesCEF.G.tl/2,y=2t3xEx3E.g’=6x-3×2
i.ysowhenoa.kz

= 3✗(2) ufowhenx.co orx> 2
hety

‘
=0 ,

soyisnncreaingonloi.to
or ✗=2

, anddareaingonl-NPUK.to) 。
t.From.E.hhalmaximumisy-z.tn ✗4- 8 = 6

Theloealminimumisyccg-Gy-6-6x.iyisca.ae dounwadonll.to)
lety

”
⼆0.ie1 giconcaueupwardonl.no, 1)

y
”
> 0 whenxel y

1 1) = 2 -1 3 – 1 = 4

gkouhenxsiihepointfnflectimislh47H.li
←
i
”

y (
-1)= 2-1 3 + 1-6

H,
6)

y (37
= 2-1 9- 27= -165-

4-渊
㓝

i i o i i 中

4
. y

=0
4
– 80+8

E. y
‘

= 4吹 160 x (s-2) z (-2,070 (0,2) 2 k,⼀

⼆秕红4) y
‘
– 0 -1 0 – o t

hetyia tm.int

maxtminfgax-OOrx-tzsayishreasngonl-DUk.to)
yisdeaeaingong-uc-FFwmETheloealmaim.cm

īsy 10)
= 8

zheloealmiuimumsareyl-z.FI6
– 32 -1 8 = – 8

andy 127= 16-32 -1 8=
-8

Gg
”

= 121
2
– 16 y__

– 8年+8⼆⼀年
– 413✗

2
-4) p y

1 ⼀⼆年6-8件+8三号
Letg

”
=0

.
i . 0= ⼟⼦

✗ 1-_- 畛 野那样。)
y

-1 0
– o +

concave 8

conaefconecueyupward-gdounwad-qupwaduep.intsofntanarel-g-andgg.tl
。 g.IO, 8)

亨 2.1.2
4.

i-i-4.it
(⼀点

-8 –

6
. y = 2

5
-5✗

E. y
‘
= 5✗4- 5 y

‘
70 whenktor 071

= 5⽐
4
– 1) ykouhen.to-1

Lety
‘
⼆0 ,Sogninncreaingon.cat?Ua,tMi.x-l

gisdeueamgonl-hDF.FromE.7heloealmaimumi.gl-1)= -1 -1 5= 4
7heloealminimu.in is y

( 1) = 1-5= -4
.

G y
”
= 200

3

hety
”
⼆0 , ⼭ 九 ⼆0.gl 07= 0 .

y
”
>

owhenoo.ykowhenxai.yiscouaeupwadonlo.to7
yìscomauedownuadonl-o77hepmttnationiio.cn

H , y 1
2)= (2)

5年10 = -22

fi
乆” 2

5
-10 =22

:

– ii.fi
-4
ii)

42 .

gaxleXE.yE-extwnpiwhenx.co-_-xdykowhenoo.hu
go , ⼭gnnuneanngonci.no=0 . 1

yisdueaingonla-F.FamE.IN
loealmaoimuuisyioFU-lzuh.at

miuimumdoesuotexist.G.g-e.to圳 i.gs ouǎweupwardonbo,⼀)
hety

”
⼆ 0yiconeavedownwadonhtrni.nl
⼆ ⼗ ,

gc-D-zet-2-jzowhenxc.ly” -0 whenxnihemntdiuftation.in。
H . 2 .

(⼀阔 ”

落
“”

-1 .