calculus module 8

4 3

Exercises.la?fisnicaaingiffSoonthatnitenal,and hreainggfcoonuiund.no mugaphfoonlaDandcsinfoonlbDandcz87.hu

,

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

Zhepōmtofhgaluùnimunisx-5

nnomthegraph.hu achanthelow :

X 1 01 4) 4 (4 67 6 161 87 8 18,⽐)

檆 品 正品 下品 i

(a) 2ffsoonannitend.fincomgmvuintendhu.fmthepaph.fismwanigonlo4NIG8i.hnuhenfio.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

1 1 . 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 ) = x 2 + 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 , -1 0)

⽨) = 3 6 -1 60- 60 fx) - 0 -1 0

Letfx) = 0 .fxtmintmaxti.GEx4 0 -1 6) = 0

(⼀九⼗了) ( x -1 2) = 0 X (加,⼠ ) ⽣ , ⼗。)

x = 3 orx-2 ⽨) t -

9 3 ) = 3 6 ×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.huinflatàapoug -_-

d) ,

^ 8 0 .

LetfM-140.fi36xtnzx 了 。6 0 .

-2奸 3.x-1 36 =0

1 2 0 . 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.gg 200+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 o r x = - 6 X (加, -4) (-4,0) ( o,⽐)

f 1 0) =20 0 , f-6)= -232 f "必 ⼗

⼀

⼗

g " 如 480-1120flxltttconcavecouareoowehetg.MN:0,

fxlupwarddounwadupwardiilzxkttxl-oi.no -0

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

(b) 7heloealmaoimumdoeswteoistneloedm.inìmumìsf 6) -_-232 (c) fisconeareupwadonc-ag-41UCO.to)

fisconavedounuadonl-41077heiuflutionpoiwts.aev4, ⼀561 and 1012以

(d) hetfxl-o.at715洁20 0

t-j6-5f-X22-316.to i -20 0 :

( -6,2了 2)

Eoercn 4,4

✗ -3cosX8.gr-9 笳哭, 1 3 .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" 2 1 . 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 845 t

t→ o t ⼆点妛 = 1㖌

8-8-5ㄠ 5

1 ⼆点 坒 = m 8-h 5 = 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% e

T

= ūm

3ˇ = e㗊等 xsozx.ci

= e '品-

x = Ì 㖲 eㄨ

2

= e 0

== 3 limlxsozx.li = 1

-_-。 ⼆ 0

4. 5

ExercisesCEF.G.tl/2,y=2t3xEx3E.g'=6x-3x2 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- 3 2 -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

' 7 0 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 " = 20 0

3

hety " ⼆ 0 ,

⼭ 九 ⼆0.gl 07 = 0 .

y " >owhenoo.ykowhenxai.yiscouaeupwadonlo.to 7 yìscomauedownuadonl-o77hepmttnationiio.cn

H , y 12) = (2) 5年10 = - 2 2

fi 乆" 2

5 - 10 = 22

:

- ii.fi -4 ii)

42 .

gaxleXE.yE-extwnpiwhenx.co-_-xdykowhenoo.hugo , 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 .