Page 12
•
The Calculus of Parametric Curves
1.
dy/dx and also d2y/dx2 for each of the following parametrically described curves
at the given value of the parameter. You may wish to check your results by
eliminating the parameter, if possible, and re-working the problem as a non­
parametric problem. You should also use your graphing calculator to examine the
graphs and judge whether your calculated information looks graphically plausible.
(a)
x = t + l,y = t 2 + 3t : t = 3
(b)
x= Ji,y=.Jt::I
(c)
2.
x = cos
3
: t= 2
e, y = sin e
3
Find the slope ofthe tangent line to the curves below at the given points:
(a) x= 2cote,y= 2sin 2 e at (0,2) and(2.J3, 1/2)
•
(b) x
3.
=2 -
3cose, y = 3 + 2 sine at (2,5)
Find all points of horizontal and vertical tangency to the curves below:
(a) x= 1-
t, y=
t
2
(b) x= I-t,y= t 3 -3t
(c) x
•
=cost + t sint,
y
= sint - t cost
on [0, 3!l']
Page 13
•
4.
Find the arc-length of the following parametric curves over the given intervals:
(a) x = eO' cost, y = e- t sint : [0,2Jr]
(b) x = t 2 , Y
= 2t : [0,2]
t~
(c) x
5.
1
= t, Y = 10 + 6t 3 :
[1,2]
Find the surface area generated by revolving the given curve about the given axis:
(a) x:= t, y:= 2t, revolved about x - axis, 0 ~ t
~
4
(b) same curve as in (a) revloved about y-axis
(c)
6.
•
x:= 4cost, y:= 4 sint, 0::; t::;
Find the area of the region in the first quadrant bounded by the given parametric
curve, and the line x=2:
x = 2 sin 2 t, Y = 2 sin 2 t tan t, 0::; t ~
7.
/2, revolved about y-axis
1t
1t /
Find the area ofthe region bounded by the given parametric curve and the entire
x-axts:
x
•
2
=2 cot t,
Y
= 2 sin 2 t,
0 < t < 1l
..,10­
aDII!I.
'.
'
Page 14
~
(c)
X=tAI)
j~ t':o. -t3t "f=;),
JJ _ d~/04:- :::.
elf:- +3
JP{ - C;;;/d!e-
dj
a;x
-=-:1(
-
J.t + 3
ISo
JJh~ t
I
3) -t J ~ 4 ·
It
V-t-t
•
:=
j)
Page 15
.
. L
~ :::: - 3 C.--0 S tJ-r-'
11:• e·
dl:r
'-'
so~
.1\
I
:
:
.
+0\
d/X & =- -rr/f
, dJ
"
-
crLVS
TIlL(
e­
- 3S'~~s~
=- Y"\
t.
'L~
.
3~
+V\~
e-­
-=- -I
:. J 1-J
eYx 1-
I
-=
()
=- rrJLI
-
yJL
3
®
(4)
Iu.-f e- (o)d.)
()((\"~/J-
•
whew.. if
3
-,2CoscrS:"" e-
OCj'M/l-
fIJv'
Tl/fo .
d.~
-==-
tT = "/ L
-=-
cPi
:. d'j"
tJ:X
-=
(pJ Y
t9- =- ·1\jL
(2.13)
11'2_)
~ ~~(T4~Er _
-
0
,,~~
L c. ~c '1-..e­
t~~L '!::t I
cLx
=.
tr =- ff
.
l/
-{3/V
Page 16
Jb) x ~ a- 3f.vStr) j='
•
lJate--
(2.) C;)
,JtCMh-
3+ .1.~tr
wk""
f}-
=-
a{Ti/'L .
dJ/~e- ~ LG~f5r } Jx/dtf -=- 3~e-
:, Jq/h
J
-=-
l-CeJ<){;)
-=- ~Cvte-
3~C9-:\
(J./5) ,
Page 17
-( c..)
•
JtyJ f::I'w-t-e-
-::c
0
~ -=-
etc
tJkM -t
g -=-
=
D
0)
OV\k,
~
It ).J T<;
LJk"",
~l\
-c =
0
.
Page 18
(~) ;x =:- -t )
'L..
•
d- =- J t- ) (0) ~J
-:::- J t"i,
) y-ytx't -== ~
tit<
(Jf
:,
Is
1
A-<L
)
J LJ I./t
L.iL. =:-
-t '(
L
df
D
- ' S, 41b
=
LJ 'lIe- -1--1
eft:
6
(bd'-
[/"c{vzm G l..kjYhi-i OV\ tJ-,
I,)
J
Dv D"e_ C. ,,-_
- i~.1t-JrtJe.. .PXtl'~vit>L HL. -/-r ~;uJ.d- . t -=- +a" e-- J e +c..
. \.
(,1
c1F-::o I) ~ =
•
I+--: [
t
._"1.(
~
..
;C - \ - ,
f'LiQ. J ~
L
I
+
~)1.+-~)L =
:.
e--
J..
+t
--
~
=- -tr -I-.?1-;;9­
9
Lf
5 ()
A-rc
'Jf-4:::'
-L (L
J)
_
•
1
L
J
L-
J
r--=---~.,.----
J~<;-H+£F)jy
~ 0'1 +E~)
rAt" =­
Jt- =--tJ tYff''f)J.I:--­
'L--
L
j
[~J- +£J 'L...:.
S
/
- 3
I
3.
I
J.~to
..
Page 19
j
S4.=
~
vr(Jt)!I'LfL'- oi~ = 3J7\J?
L
D
(Ll
';,1\ .
-=
j ~"t J? olt- =
IPl~
()
'X :::. ~ ~1- -t~ J..~L-t-fp~t-
0
J
<!::-
t
/2-
<.. T/
11/-....-­
!tN'L =
S~Jr: ~ Sl~~1-t-f";,,f-)(4S"~Gst-)olf---
'Y. -::::, D
•
-=c
C+o
C
-='-·-D
~ r'/~.) -Ie 01 t-- -=:;' 311/1- ,~
l.(. 7/ L-
v
I 5yt).-+t:-II V\~
I
~
-:::.
.)("' I Y\"-h....
(7
I -. LC0
:.
J-f)
<'" _ , .
t
:J(.-.;-(r1"...J..
I]
T1w",£,,-
J
Page 20
y =-
~Cotf
I
d=Jr Lt ) OLt < I l
t
'\.....
eLy -=- -- 'Leu-
=
1[/1­
$;--- )
o
•
•
J t-
-=- '( r
\1