George B. Thomas, Ross L. Finney-Calculus and Analytic Geometry

27 Pages·2016·56.49 MB·English

1.6 Tangent Lines 97

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Tangent Lines

iy rns moans the discus of acts an tgs heen in Sesto
We cacalte has of secant slopes tod ayes cures.

What fs a Tangent to a Curve?

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ou P hens le the bn ef the wasnt slopes nn ether sie

How do you find 4 tangent to-& EXAMPLE? Jt he slope of We parabola y= a Us pot Piso. le
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Finding a Tangent to the Graph of a Function
“To find a tangs to an airy carve

fa) aba solar Ply, fags 8 se
fe came dyn epost. We ean shape fe sin aay Pa
wind Diy ta Fh We th iment le He a he $e a Bn

(Win, LST te ei exit, el i the shige a Fe canoe a lee te
angen? o be be fine argh hinge ope,

Definitions

. ‘he slope of tas worve yy a be pnt Pe. F.3 the mnber
7 Fea 1h firm
FTE sri he it ex

12 the angene sane
fe LS Hd

oh with bi slope,

When
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“tooo ha th re ey iow nf ape gees wil tbe wl deli ote aps}
Few roxvetc oes,

rE wake Avo Sail HL A yas se by a a

How to Find the Tangent to the 7 7
Curve y= fd at fc 3) EXAMPLE 2 Testing the definition
4 Cae Joa man

fig hfs Solution Ws kl fe

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1 and ning Fr wor a tse te
Step tial fenced 1265 AY

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100 chapter 1 Lets and Continuity

Piase e Fermat (1607-1665)

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hat ft we a en Yo
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Celie Sng aa Fc D>

he two tangent Ines aye
hosing she

Step 2 Find we sop ling (414g 481 Foe

Lite bt = fost

py WME A fz AY

step 3 Fide sangen sine uns te points equation. ‘Ih tzagent oe al
fe pit Cran, P85
Fo fae Bate gh

veme 4

EXAMPLE 2

ay ial slope of be curve yew ae
1) Where dows te slope eal 4?
1) What happens Wo Ue Len ie cans a pit 1 we cme

‘elution
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ntl the stage wire we sou evaluat te Ha: by svn f=
1) Ihe sloze of y= Lex at the Fone where xa is La ew be LE
rrimidea

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it athe te pe 2. 123 and 12, 12 Fig, 1.

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This wgica wie > Ae roves avy Ira Ue iin Ue slope ppm
fund he engentivel ft: 7

Rates of Change
‘he expres
flow foes

7
Gt tangensy moves ay

All ofthese afer to the same thing

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ref cig with respec ea We ged ©. Utero ine of dhe
Uc ovat eprint oles cere i ales Se ie 8
‘torah sd oft in Caper 2

EXAMPLE 4 Instn:
Fasmpler } and?)

rieous speed (Cantinuatian of Section 1.1,

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fiom feet sea th eutues ofthe ear We hoe Bite De cock Gel 3 = 167 Tet
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7

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Exercises 1.6

Slopas and Tangent Lines

1. ces gid ada ai ge sa ag ti

be Be ere i ets os ame Ds
102 chapter tLe and cantioally

Intsacn $V nd qn eng ca a

(2-3)

2a ris Uo, te elope be ste ap te gn

ak <u
waned ma
a

Tangent Lines with Specified Stopes

2 He sal of st sg a2

BWaieat

Rates of Change
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Fates le ia tr

29, Wa idhe se of cogs of he ano cae 4

sith

M. Winch inaconge he meas ARI

idepete de aus Fiche aes 26 = 2

Testing for Tangants
fs. [sat on oe

se wt

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a sug sate fv po

Vertical Tangemts

nf
Questions to Guide Your Review 108,

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‘The Derivative of a Function

[Ac the end of Chapter 1, we dened ue tze of a curve 9s fiat a be si
Shots =e bE

Fw Soe)
‘sll nit chon “exis, ds dsceave Of Fat ay thi sein
iota Za derisive a fsaie diced fon Py eonssag de hie
fast pins Ps nea,

Definition

The devvatw ofa fantin, Fi taser he viable fanetion
Pxbiae valve ale

rtd etm BEE I)
FO) =f SE

noc he lev,

‘The domaa of j* the soo pont in he demain of for whic the Smit eas,
ony be svete a the doinin off 16 J") exist. We say has Fis dereaoe
(Ge dttrentable)

108
190 chaptes 2: Derkatives

Why all these notations?

‘The “i” enon <a com fan
duaict at None od Ue Uso,

‘Steps for Caleuating 0) fom the
Definition of Dervative
2 ipod anny Ue elas

Notation
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os rsh sas nd ef

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We ateo ead Aya es he dtiates of» ih rgpest #9 and tf ad
aida) a8 hs arsine of espe tt <P

Caleulating Derivatives from the Definition

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413 Saxton 1. Maat he pwns fn Ee Inte wa = Ban pL
espe 2 shove that

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In Bxarase 3 ws thar

on ms 00 wos acne.

EXAMPLE 1

Diese 8}

oh Where does be sve

fFlyy fave slope 12
Solution

2) Wit ths Se ep ag

‘Step Be Move we hae i

ral
far)

oar)
@-6-1
i|
:

22 yon atx-vand e-2

ieee °

een

Herm fy

2A The Derivative ota Function 111

step 2 a
Lube Boat
TeV
hate aye
, 1
00 3 P= Wh og gy TG
by Lae slope af 3 = Fy will be = posi
Ves ocnaton is muiilenl Woe IP <1ysor = 2 er
EXAMPLE 2
fa) Find the derivative of» = ye =
DL Fin the rane ice ihewtive SE ae
Solution
wo Supt Fo1= ve aad fut VEER
a a)
ieoives

Wetie

Uatig, 22,

4

23) The rap

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