Paee I I
e 20 11 by Janice L. Epstein 2.7 Tangents
Tangents, Velocities, and other Rates of Change
(Section 2.7)
Slope of the secant line PQ?
~=- ~(X')--t-(o.J
<Y\ to1
•
2
3-
~ V.J
--"? -E.\
+a.Vl!lro
x
4
5
The tangent line to the curve y = f (x) at the point P (a, f( a)) is
the line through P with slope
m
= lim f(x) ~a f(a)
x-a
= lim f(a + h) -
t \
•
,
~
J
~..
y~
a...)
I
,)_1
. '"
fY)
)(_/ _\ ")(- (-I
J
)<-7-1
== \lm 11.~")("'-'X:·Hl ,,3
)(-?-\ (
h" 3b"l. 3\\.,(v(
a
_ \,VV'\ (-Pr0 '1(\1\ t
M - ",'-'90
'"
~~D
I
_ \\W\ 'nl~~-3'n+?) :: 3
'
(-''1.:
(J(
-
",,~o
"9<
-\~_ ~("'X-(,t))
I
1-;) ~- ~ \-- VV\ (')( -')(;) ~ ~ ~'6~ 3~~'2-
(
)­
n.
- X ~a. ')(.-0-::- ~VV) f.Qf> - £(\]
Fmd the equation
of~
the tangent
I
. . line t 0 th e )t~1
curve at the gl~n pomt
a) y
I ' (1,1)
'\-W..o I/.)
5Y)
\J
.
2.7 Tonge""
:x,-.,
rt
~ro..~NcJ
I '\Ix ~~ ~
rf\ ( \ 'rn ~ ~ ff ,fl - \\~ \- IT
p~
::
>
.
I
~~ \ --')( _ \
~ - '1<7 \ \f5{ ()(-\")
-;:. .QAm -l.-r~ .~
IwY')
'"-~
'<:4 I .,fl (~tF" \ \) ')I-? 0 .J)( ({\( \- \) L
g._~\ -=- 'M l-x. ---X i) -? ~- \:=. -~ CX -,)
-7
b)
h
h O
.., m- .hl'Y1 +'{1() - f.(Ct'j
~~ -~ 'X- ~~
f(a)
EXAMPLE 1
a) Find the slope ofthe tangent line to the curve y = x 3 at (-1,- 1)
using both definitions given above.
b) Find the equation of the tangent line.
c) Graph JiAe curve and the tangent line
~
~ tJ
?l ~
=- \ \ry) "f.: - ( - , :: lrl\ .
• I J
~MPLE 2 /
't.. - (A.,
rJ--7D-­
J..tt
~Q
Page 21 0 201 1 bv I..... L Epmio
_ x
y - 1- x'
rn -=-.llvV\
~~O
m'".t..m -\J.1)= flc\L
(0, 0)
~
')L ~
"'l ~o..
f=-0
-0 -=- h~ J- ~\ 'I- .-II? 0 \ - '?l ~_~\ -:::- "IYl ('1-- 'X,)
l6::"X­
:::
;1..-0­
(
"~..L ::£ )
)(~O ~\\-")(.
~ ~ -0 -= \ eX -0)
Pac.e 31 C 2011 by Janice L. EpSlein
2.7 Tangen ts
2.7 Tangents
Page 41 f'j l Oll by Janice L. Epstein
Let C be the graph of the vector function ret) = (x(t),y(t)). The
tangent vector to the curve C at the point P corresponding to r( a)
is given by
If the position of an object at time t is given by the function s = l(t), then the velocity of the function at time t = a is v(a) = lim I(a + h) - I(a)
h--+O
h
ret) = lim-I-fret) - rea)] = lim-.!.[r(a + h) - r(a)]
t--+a t - a
h->O h
t:-- t
EXAMPLE 3
Let r(t)=(t 2 -t-2);+4t 2
-::z -: - 2­
/(Jt/Y\d. 4 ~ 2;:: if ~ ~
c.. -l'-- t ~ 0
6A
tfl:-i)-=D ~ t= 1/ 0
.
curve gIven by the graph of r(t) at
tan~
' t;:: J\
•
a) Find a vector
the point (_ 2, 4}.
b) Fmd
the
pointparametric
(_ 2,~. equatI ons or the tangent line to the curve at
V==- 11m
Jt]-"i"C\) .
r(I)=-CI''=-\=-?-)L l-ACI1j
-c-\
= -~ \-2.\S
V==- ~~~ K\:-~ - ~ -}-): l-2-~t +- [4t"-- LtJS
EXAMPLE 4
The displacement (in meters) of a particle moving in a straight line
is given by s = t 2 - 8t + 18 where t is measured in seconds.
a) Find the average velocity over the following time intervals
[3 ,4]
[4,5]
_
\, fY)
t='l\ _ \WV\
~
-
~~\
-\
Ut:-1")'t
V -:;; \ lv'tl t'l
~ ~-/ ,~
b)
~
h~O
='
'
rsrn}S
V ~ro:
n
1
«4)
"\
'E:
=\W'\1 (4+-h)'l.-~4i-'hJ~) __-_
h~O
'tI
_ \ ,~ 1 (W+'&~ ~h1- - 32-wh +~!o1= hl'Y\ h ~ Qfflk
- ~...:1/" ~ ~ "
-; h::;O
~
f\
I'
J
~~~t ~~ II"'" '5~5)-S(j) ':>- ~~?~
~,::, -4I"?
s(~ ~'r0 -S(~)
I~
c) Draw the grap~ of s as a function of t and draw the secant and tangent lines from\parts (a) and (b). '1 Lh \')J = L +- 9>5' '"" .>
f ~
)
fo...
'K- t-~ ~
h(1'\
Ut! = r(\) -tvY "'" ,;;l:L -!-YJ 'rt",l ~
:: (t -'d-) t ~ ('3 t +- ~~
-3/
[4,4.5)
1Y)1
\
b) Find the instantaneous velocity when t = 4
~_~t;~I'l(~IX\ "
t5-~
==:?
~
-I-
\ : A - ~ 'j5
11-0
-= 'S(Lt)~S(~,Sl-:;.. \
:=-. 5
. -" If- "3 I?
. '5
V -:;. ~(~?- .$(£'\2. :::- 3-S._ -::. \ 'fY)1 ~
[3 .5,4] -\(
t-,-"7 \
t-\ (l'7-_ t;")t ~ 1.\ (~'1-_\')j V:::-s(~-S01~ ;1.-~ =- _ \ YY)I~
\
/\
)
,
------------~----~~ -L--
?-
~
r----!
~
S
),.-(
P I;I~"t' ~
I C l Ol l by J lI:l l K~ L. GP<;k'll". 21
T ml ~enl "
~It ~
EXAMPLE 5 ~f an arrow is shot upward on tb
.
Its height (in meters) after t seco::~on(1~lth a velocity of 58 mis,
= 58t _ 083 2
....
S IS oJven by
~
.
t
I C'> 20 lJ i'lyJ lI.n(u: L. Ep~1e m
27 TOrli!eTI IS
We definite the instantaneous velocity v(a) at a time t = a to be
v(a) = lim - 1-[r(l) - rea) ] =
H o
1-
a
lim ~ [r(a + h) - rea)]
10 - 0
h
-_\-\ll
(a)
EXAMPLE 6
b) Fmd
F· d the
h velocity
. 0 f the arrow after 1 s. (
m t e. velOCIty of th e arrow when t­
If
is shot
] 5;an+arrow
40 . mJ
. up~~rd on the moon with a velocity of C) W1len WIll the arrow hit th
- a
((d)
J
s,
Its
posJtion after t seconds is given by W. e moon?
ah what velocity wi1l the arrow 'hit the mo ?
. ret) = 1St; + (40t -0.83( 2) j
(4)
on.
(a) Fmd the velocity oftbe arrow after 1 s
'1J(I)~brtll-l( \~~)-\\{I)
-\\-70
{\O
.
= @'g(lt~L- .D.'il?(l t-it)S-[5<6(I)-, ~('fl
_ _
(b) Find th~ velocity of the arrow when 1 ~ a
(c) W?en WIll the arrow hit the moon? (d)~lth what velocity will the arrow hit the moon? ~
~t5'ilh - .:&3'-1,lWh _,~3~2..- 57"')
:. .Q.wy,
(b)
(=xo, =3~-o,g3~)~ 5h r 3</~JS
V(O-)-~ ~W1I±.(IA.~t . 1-\0­
::.
~ '8~~O
~-t- 5gVI .- , ~~:?:::-h !. '='6. h - ,"63 h'J-. §'is 0. l-, '13 0-1­
~
_ fL'VY\
. _. .
.
(!i '8 - " ('1Dl).., - , '1,3 ~ ) = 5S -I ,('o~o.
~70
c) \·HEY~ 0
UlN y\ \v...ts
o ~ -t. (!?~ - . ~3t) -7 58 == I ~ t.
t -: 5~/, -j,?>
~
'"to s
d) ot t,-;: 58;;S3 Ee6
\) ::: C3 <3 - ' \~lo CS~~3)
";. - 'S~ YVll s
M
~
1-
(\Stt +-('bb-,~~
J
~-)~ t: -0.
_ VSo... t +- (~O 0..... \ ~"3 a.~)j
~ t4YYl ~ (CL5t-\So.)t t-(tJot. -L\O().. -,&3t~-,8
-l-7ox
'V\.-70
(0)
vCa)~1.'('Il
~
J
-''53(-I:~a.~):1'
)
7E-~) -0.­
\-~01-~ ->t.3( t+aJJ'1
=t-~
ilmlTstA
t
+
L.L­
0.-
- ..Q.tVV1. rlC5,'t
L--7~
-=
l.
\tdC
Ijo It--o-.).
t (£\0 - \ I (o&, ~)J
I:'
->
5'l +- 3 '2. ,'3L1 j' "'Is '::: \I (\ J
(c) ve~ ~ -4oi"- ,%3C1-::; '0
40
(c..) .t::- \
-=;>
\
r~) I::-=-~ 4q-s
/,'(,3
d) \J (LJOj S3) '" \~t ~(4o- 1'(.00i< 'f9(~),)'
0=
t( 40 - ,'33\:-)
-7
-;::. 'S'l -tiD
ljO -
'7
J
mls
::red =: Iv I ;::--{T5'" -l--( -1.\01-{==' J"\ ~2.s" o:x 4~'71s
P."r: 1 10
11) 11 !):" ) 11111(1::
L Ep" l~in
1 1 Tn .Jm'f
lex) with respect to x is
The average rate of change of y =
D.y _ l(x z )- l(xl )
fu
..... r:8 10 :Wl l by Jllni(;cL C: J"I! ~tt.
EXAMPLE 8
The population P (in thousands) of a city from 1990 to 1996 is
given in the following table:
X 2 - XI
The instantaneous rate of change of y = I(x) with respect to X at
the point X = X I' which is interpreted as the slope of the tangent to
the curve at the point p(xl,f().J) is
lim fu = lim
6y
b,.T- O
l (x2 ) x2
Xl~XI
-
60
-
­
Find the rate at which the water is flowing out of the tank after 20
2.~..h\'1- t"'rI~
.
~f'(\ \J(.z..O +~- V('2.Dl =-}\rY) \Ooca.)(I - ~_<:. ./ _I_\: _LO
R-9D
'",~u
:: )OO(tOO JJ;W\
.
~O
•
'.
::- \oo,ODl"
flI
~
VV)
u
",.... -7
~.
1. (<?1;(,~ _
1-\
.
71~
1--
~
:: \ COl 00 D
¥-
ttls) ~ - ~ gov/mAr-:- ?-n
,... ~r'¥tJU'
(b'-\ - \n
"ii-
h4LP
?lG?-
_ Lf ~ -:..
II" ~
-
-l"'lT
\~--, - \l1/ V'At.\- I~'Z.=
(iv) from 1992 to 1993
\~~-=-U1.
~C;3-lq~'2..-- ~\
=.
_
~
I
''t)
33J3 -- )
.0/7- -:: . )0
q
P
(b) Estimate the instantaneous rate of growth in 1992 by measuring
the slope of a tangent. 1-bY\ A'r&,
II'
~~
r'i!>e
fl.'"' .i.
6\up>
=0
~ ~ ~1 ~~
::0
rUn
4
I{I?
W
I ccJ cvJofvY fj,j( S 8
1.U1-th Q.uDd. m ht .f;-\-)
!CP '"
\~o qI £\1- ~ /; lJoA
q"
J?
';l.~1MLVl
1996
164
150
(iii) from 1992 to 1994
~
-= \~C) toO &.lm ~~!t.go ~~~(~OJ
1995
1994
137
\50 -- n}{0fl~-lqq2 -=-
.
(n(loO) -,
]993
]26
(ii) from] 992 to J 995
0
':). ,Qh
')' l/~O
..,~
.~ -:+"
1992
117
(a) Find the average rate of growth
~ ("-~)"-
(L- Vo'_=Ifi;:L.- I~
(II
.yJ;
~.
1991
110
XI V(t)=100 ,000(1 - _ )2 0 < 1<60
~
1990
105
(I) from 1992 to 1996
J )
1
--
yeaJ
P
-
f(x
EXAMPLE 7
If a cylindrical tank holds 100,000 gallons of water, which can be
drained from the bottom of the tank in I hour, then Torricelli 's
Law gives the volume Vofwater remaining in the tank after t
minutes as
minutes.
2 7 Tntt'nu
t\Dk \C{<-\\ to 'Aq2.. ~ 5 t}{JIW