Midterm 2 Review teeth tox Slope ofLiartangent wieh o hat Ga y

f function I Derivative of

y fee t Csc IT 7cal did 7 with

respect to a

Core Derivatives Power Rule

xr roar da b l lulls b

dda gb x ddxcsiuc.ci cos x

Idg es x Siu Cal Kanai see Cx

ddacarcsiucxllfad larccosc.ci 7d

arctanex1

I 1 2

Zules of Differentiation

Sm DiH HE g 7 Ig constant

Constant Multiple Rate Cct EIDroductRate tog 7 g t t g

Quotientrate Eg t'gGZ

Chaint da 9 CgCx 7 Cgcsa g coal

Important Special Cases

da leg es g ex ddacgc.cir

rcgcx Y g'cx

Hulgan 9g g cxi gcxsddxuucg.cn9

LogarithmicWorst Case Scenario Differentiationancx k

g x Mtl Kucxg x 2

1g Caus

T 3kt 11

ilacgcx tan X1 tu Eti t aclu luca

Iz tu 3kt l

I LucgGc Sec 264 tu x2ti tan x12 1

tu Clube al u Gy

Et32 1

tank x 944g'Get Citi Kuch

TT3kt It

ysee Calla 24 1 tank 7Tu Hubei a I

I

CencgayIf you see products quotients at functions Like

gcse as this method

Determining deDX

L from an equationOverview of Implicit Differentials'minvolving se and Y

Y Differentiate both sides of equation with respect to a

2 Using Laws of differentiation Product Chain Sam

expand both sides untileverything is expressed

in terms of x y ddda

Solve in deda

Example Find slope of tangent to y x2 y2

at 2,2

yes at day 2tyChain Rule

dy 22L3 yr date 2x did ada 3y2 Ideda l 4

x z l Gustanty zk ogrowthco decay

Natwalarowl.hr

DecayytCt7 experiences natural growth decay dat yconstant y co

Fact dyTt Ky yet Cent

Main Examples

Natural Population Growth Radioactive Decay Newton'sLaw of cooling

E A radioactive material has mass

10kg on the 1st of January 2006 and 3kgon the 1st of January 2015 When will the

mass be 2kgact Mass at time t in years after 01 01 2006

Mco 10 and MCE Cert Mct lo etatTkMlt to e 3 k tu Io

Luc IMitt so e Tt

infoMlt 2go e g t Z

The mass is 2kgt ahu E cthis many years after

l U Z 01 01 2006

Intermediate Value Theorem I V T

f cts ou Carb There exists C such that7cal Edf Fcb Fcc D

orT

b Ed E 7cal II V T tells us about values at F

dUsed to prove 7 takes specific

Id

ValuesC b

Mean Value Theorem I V T

f cts ou Carb There exists c such that1 diff ou arb I tCb 7

T b a

IFM V T tells us about values at

a C b IUsed to prove 7 takes specificvalues

E Show that FCx e t 2 7 2

7has exactly 2 roots differentiable on Rl

Y Use I V T to show at least 2 roots

4 C l e z 7 e t s o

1 o 3 o

f l e 12 7 COf lol e

02 700 70

I V T There are rants in C l o co 1 CI lo

21 Use M V T to show at most 3 roots

Assume 7 Cal has 4 roots A B C D

Observation a bf

M.V.T It 9cal g b o g'Cc o Forsome c in Ca.nl

A B C Dx

7

7

MNT 7

IF

conclusion7 has 4 roots 7 has a root

4 x e z 7 2 7 x e t O

7 cannot have 4 roots

7 has exactly 3 roots

Sign Analysis Determining when gas o and gencogcc o

and B9 discontinuous ate

y find all type A1 points

4 Draw number brie mark A1 By points and

test sign around them

PictureC t Cz Cz g

i la b C d

g ca o g Cb o SC co gcd 0

Derivatives and Curve sketching

Central Facts

F'Cx o ou Ca b I i g on Carb

F x co ou ca b 7 decreasing on Ca b

7 Cx 0 on Ca b I on acb

F Cx co ou ca b 7 dwn on Ca b

1 cts ate

c 7 ex 7 Cc Local max7 X1 o 4 Gal co

1 cts ateee 7 CN 7 Cc Local min

7 74 Co 4 Gal o

1 cts at cY I'enC Ce 7 c Inflection

1I X1 o 7 Gal co

1 cts at cn Tu

7 C Cc 7 Cc InflectionF Mco 7 Gal o

Cwvesktedy Domain

4 Odd Even3 Vertical Asymptotes c L Hospital4g Behavior at a Horizontal start asymptotes

5 sign Analysis of 7 c Increasing 1Decreasing catMax 114in

Gg sign Analysis of 7 Concavity Inflections

7g Mark key points and put everything together

Domain Ca bFind critical number

7 and evaluate at 1

Finding Absolute Maximinof 7 Cx Domain a b

Pray there is one criticalnumber where there isa local max min

C

edopcimiwwistyobg.euity to be maximized 1 minimized

4 Draw Picture and Label unknowns

Give objective in terms of unknowns

Give constraint in terms of unknowns

Solve constraint ui one unknown and sub into objective

giving 7 a single variable Function

Gigantean and Find abs maximin