Page 1
110 2011 bV Janice L
3,7
Epstl!ln
Dertvatl ': ~ofl/I!ctor
PaSI! 2 I (12011 by Janice L. Epstll!ln
Functions
EXAMPLE 1
Sketch the following vector equations and include the direction of
the curve. Find the equation of a unit tan¥ent vector to the curve at
the given value of t.
-~.
".1_ \
r(t) = (t,t2),
,
\J
l-?' I
J\
t.
'Y~t ILL
~
-
t=1
I
-\
I
o
"Z.
,.~
it= ~
I
t..J
,
I
\:/
.1
.
,
-.I..
I
-\
-1.-
\
.
_&:­
I'"v~
'V'3
L
1-1. \~
_\
U
I
I, \
i~
\ -
3
I-J- L\ \
0
0 \ D
J
d\ I
~ ) 1~
.jj
+---
­ l
L\
' f'(f)~<'2~f\e->
(1) ~~
~
'I ':LV"J ­
EXAMPLE 2
0-­
t :. a.tQl t _3£\
o · L.
1jh :l(~/t)
- ..,
r'(I)~<2., q)
1$
/
.')
r;:::.<...,2)
r(t ) = (2 cost,3sint ), t=1T1 2 'j..
')_
'lA. ':a
2"/.,
.l
\
1- .v -
of V,etc>< Functlon,
t,2-­
< > \ \ \'3
'f \(t)~ \:tt.
f\{\J~ <\,2..)
:1-)..
')
I
3-
3.70., , 1
ret) = (t 2 ,3t 3 ), t = 1
Derivatives of Vector Functions (Section 3.7)
.tl-
~
~
3Cl}
'Yt1 l~ \fih-') 3 \~~')
I:r..o.
~ ! ~ (~) 3({3~
fl (t) =.. <.- 2S)'<\t \ 3 cos t
r \(11/-;..) -:. <-d-, ') 0 '? 1fi£ii'
(/2 -4, .J9- t)
O-01V'W.1(\ o-}?" l t J --'>
ret) =
t ~9
~' l t \) -- <---) ';L Ll-) - c9.-- J1
\ --~ "
.. t
lfh-,1 (0) '3l\)
11": .(- ~ )0)
Find r'(/) and the domain ofr(t) and r '(t) for
-a.
=- (-')0)
JJJY\~l\
t.(~
Page 3
I0
3,113
2011 by JJnice L EpsteIn
3.7 Derlvauve1 of Vector functIons
EXAMPLE 3
The position (in feet) of an object at time t (in seconds) is given by
<
r'
r(t)=(t, 25t-5t ) )
/-t)=IDt>
(a) Find the position, velocity, and speed at the time t= 1
2
I) d 5 -
="i)(t)
f U)-=- <\)d5 -5'; -: :. ( \ 12... 0 '>
13 (\") -: :. <I) 1-,15-)0') -::. <\) \<; >
\13 (\ ) \ -= 'I.!i . . H 52.' :::.J !).?-lo
Pap"
I 02011 by Janice L
Epstein 011 , if
3.7 Denvatlvn of Vector Functlons
EXAMPLE 4
Find the angle of intersection of the curves ret) = <1-t,3+r> and
s(u)=<u-2, u2>.
. ~~
CD \j{)lPJ\OP [.,~ /JJ?\1Y\ 'Y.~J ;; :,.~K M~
,'2'2... (J
(J.,
J 1-_'(, :::.
-2) \ - t
Y-- ~
~
I..:-:.:=-> ~ 31- t'2- ~ (3_t)7..­
-~.. 1.A.=3- t
I
7;.
b \) 1):-2.
f( ' ):::<O)"\)
3'" t -z..
~
V
-::
3 - !uI::A t 7.­
lot:=-I,o
"'7t:::.j
~ (\)-= <0 ~'>
().JV<J. 'll-:::' 3 - \:::: ;:}...
CDI{"(t)'= i.-\)~t)/'(\)= <-1)2..)
:s'lid , , ~ 1) '). 1)..) 1 -S' (aY ~ <.. \ ) Lf '>
I
(b) When does the item strike the ground and with what speed?
srr \!R~
~ ttJM.n ~:= 0 5 t. = 0 -:) 5t ('5 - t.') -;:: 0
~ -::. 'J.'J -1;-
"2-
"~ ;
-:;..') t ~ '5 1J(~) -=- (\ ) d-5-IO(~»
\ \5ll?)\ ~'"'/)
Z
~S\(\)
. :_, -'\,
fl~~
\
\}'
. :
l 0)£..\) -::'<I)-2S; 1-(_25)2-'- ~
1%ieo
~G--~~(t) "~' (1--)}
~ L- \ ) L /
-::. - \ Jr ()
e- :::
(,Os-t
f'.;
L.\ \
°'(\)\ \
'f1(2-)\
.( \ \'-\ '; /~-;;,..' •
/4IiS'
('1/ J'i~') Jr-'z.-,l iZ
~ ~/~