x dx

Transcription

x dx

Integration by Parts - 1lasswork
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dy
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dy = u " dv $ v " du !
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# dy = # u " dv $ # v " du
# u " dv = # dy $ # v " du
# u " dv = y $ # v " du
# u " dv = uv $ # v " du
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Technique9 Integration by Parts
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# u " dv = uv $ # v " du
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Technique9 Integration by Parts
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# u " dv = uv $ # v " du
PQ6@0&2!-)!G)*C!
u = ________
# x " cos x dx !!H%!C%!(:)+!79!)*(2;36()%*!79!063(+<!+2(!'0!6!5:63(!! du = _______
v = ________
dv = ________
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!
u= x
du = dx
v = sin x
dv = cos x dx We can now complete the chart
u = x'''''''''v = %*9 x
''du = dx'''''dv = ;$% x''dx
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# x " ;$% x'dx = x " %*9 x $ # %*9 x'dx
u = x''''''''''''''''v = %*9 x
''du = dx'''''''''''dv = ;$% x''dx
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=:)5:!:%02?'&&9!=2!56*!(6I2E
= x " sin x $ cos x $ C
L2(D+!5:25I!)(!%'(R!(6I2!(:2!C23)86()82!%?!! = x " sin x $ cos x $ C
1
PQ6@0&2!/)!G)*C!
x
x
# x " e dx !6*C! # xe dx E
0
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u= x
v = ex
du = dx
dv = e x dx
x
x
$
x
x
$ ex % = 1
# x " e dx = xe
x
# e dx = xe
1
# x " e dx = xe
0
#
x 2 ln x dx
u = ln x
du =
PQ6@0&2!O)!G)*C!!!!!!!
#
x 2 ln x dx =
v=
1
dx
x
x3
ln x $
3
#
x
$ ex $ C
1
0
x3
3
dv = x 2 dx
x2
dx
3
x3
x3
= ln x $ $ C
3
9
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# ln x dx
u = ln x
v=x
1
dx dv = dx
x
x 2 ln x dx = x ln x $ # 1 dx = x ln x $ x $ C
PQ6@0&2!T)!G)*C!!!!!!!!!!
du =
#
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x2
2
dv = x dx
u = ex
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v=
x
du = e dx
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x< x
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e $
e dx
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# < x'e
H39!(6I)*;!
u = ex
x<
v = x2
2
du = e x "2 x # dx
x2
#
dx 79!)*(2;36()%*!79!063(+E!!H:232!632!(=%!0%++)7)&)()2+E!H39!265:E
2
# 2x e
#
x " e x dx =
dx = e
x2
dv = 2 x dx
x2
" x # $ # e "2 x # dx
2
3
u = 2x
v =?
du = 2 dx
dv = e x dx
2
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#
<
<
x
x
79!%3C)*639!u,+'7+()('()%*E!H39!)(E!!! '' < x'e dx = e $ C
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# arctan x dx !!!
u = tan$1 x
1
du =
dx
1$ x2
v=x
dv = 1 dx
x
1
dx = x tan$1 x $ ln"1 $ x 2 # $ c
2
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$1
# tan
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!
$%!
#
#
x dx = x tan$1 x $
x 2e x dx !!!!$2(!'0!9%'3!5:63(R!!!!
u = x2
# 1$ x
2
v = ex
du = 2 x dx
dv = e x dx
x 2e x dx = x 2e x $ # 2 xe x dx = x 2e x $ 2 # xe x dx !!!V)C!=2!;2(!6*9=:232M
4:6(!)+!(:2!?)*6&!6*+=23M
# x 2e x dx = x 2e x $ # 2 xe x dx = x 2e x $ 2 # xe x dxx 2e x $ 2" xe x $ e x # = x 2e x $ 2 xe x $ 2e x $ C
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PQ6@0&2![)!G)*C!
!$%!
#e
x
# e x sin x dx !E!u,+'7+()('()%*!C%2+*D(!=%3IE!H39!063(+E!!
x
sin x dx = sin xe $
#e
x
u = sin x
v = ex
du = cos x dx
dv = e x dx
cos x dx 42!+()&&!:682!6!(%';:!)*(2;36&E!S63(+!6;6)*E!
#
x
x
x
!!!$%! ' e %*9 x'dx = e %*9 x' $ e ;$% x $
2 # e x sin x dx = e x sin x $ e x cos x
!!!!
e x sin x $ e x cos x
x
e
sin
x
dx
=
$C
#
2
# sin
2
#e
x
u = cos x
v = ex
du = $ sin x dx
dv = e x dx
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x dx
u = sin x
du = cos x dx
v = $ cos x
dv = sin x dx
# sin x dx = $ sin x cos x $ # cos x dx = $ sin x cos x $ # "1 $ sin x # dx
PQ6@0&2!])!H%';:23!%*2E!!!!!! # sin x dx = $ sin x cos x $ x $ # sin x dx
2 # sin x dx = $ sin x cos x $ x
2
2
2
2
2
2
# sin
#x
2
2
x dx =
cos 4 x dx
u = x2
du = 2 x dx
#
1
sin 4 x
4
dv = cos 4 x dx
v=
x2
1
sin 4 x $
4
2
1
v = $ cos 4 x
4
dv = sin 4 x dx
x 2 cos 4 x dx =
u= x
PQ6@0&2!^)!H%';:23!%*2!!!!E!
x $ sin x cos x
$C
2
du = 1 dx
# x sin 4 x dx
#x
2
cos 4 x dx =
x2
1 %$ x
sin 4 x $ ' cos 4 x $
4
2& 4
#x
2
cos 4 x dx =
x2
x
1
sin 4 x $ cos 4 x $ sin 4 x
4
8
32
1
(
# 4 cos 4 x*)
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Integration by Parts - >omework
# x cos"5 x # dx
# x sin x dx
u= x
-E!! du = dx
v = $ cos x
dv = sin x dx
# x sin x dx = $ x cos x $ # cos x dx
# x sin x dx = $ x cos x $ sin x $ C
# xe
8x
1
v = sin 5 x
5
du = dx
dv = cos 5 x dx
/E!
1
x
# x cos"5 x # dx = 5 sin x $ 5 # sin 5 x dx
1
x
# x cos"5 x # dx = 5 sin 5 x $ 25 cos 5 x $ C
u= x
dx
1
v = e 8x
8
du = dx
dv = e 8 x dx
OE!!
x
1
# xe 8x dx = 8 e 8x $ 8 # e 8x dx
x
1
# xe 8x dx = 8 e 8x $ 64 e 8x $ C
# 6 xe
u= x
# 6 xe
$3 x
# 6 xe
$3 x
2
dx = $2 xe$3 x $ e$3 x $ C
3
x ln x dx
v = tan x
dv = sec 2 x dx
1
dx
x
du =
ZE!
#
#x
# sin
2
2
1 $ x dx
#
$1
2
32
" x $ 1#
3
dv = 1 $ x dx
[E! du = dx
2x
2
32
32
# x 1 $ x dx = 3 " x $ 1# $ # 3 " x $ 1# dx
2x
4
32
52
# x 1 $ x dx = 3 " x $ 1# $ 15 " x $ 1# $ C
v=
v=
2 1.5
x
3
dv = x .5 dx
+2
.
2
x ln x dx = - x 3 2 0 ln x $ # x .5 dx
,3
/
3
+2
.
4
x ln x dx = - x 3 2 0 ln x $ x 3 2 $ C
,3
/
9
# x sec x dx = x tan x $ # tan x dx
# x sec x dx = x tan x $ ln cos x $ C
u= x
dx = $2 xe$3 x $ 2 # e$3 x dx
u = ln x
# x sec x dx
$1 $3 x
e
3
dv = e$3 x dx
v=
TE!! du = 6 dx
2
YE! du = dx
dx
u = 6x
#
u= x
$3 x
x dx
u = sin$1 x
1
du =
dx
]E!
1$ x2
$1
# sin
$1
# sin
v=x
dv = 1 dx
x dx = x sin$1 x $
#
x
1$ x2
dx
x dx = x sin$1 x $ 1 $ x 2 $ C
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#x
2x
# " x $ 4# e dx
Ê!!
du = dx
# " x $ 4# e
2x
xA
A
B
'dx''''''''dv = x @ 'dx
x
-.E!
x A 69 x B
$
x @ 69 x'dx =
x @ 'dx
A
A
x A 69 x x A
$ $C
x @ 69 x'dx =
A
BC
''
<
x %*9 x'dx
du =
x $ 4 #e 2 x 1
"
$
dx =
#e
2
2x
dx
2
" x $ 4#e 2x $ 1 e 2x = xe 2x $ 7 e 2x $ C
2x
4
$
=
x
e
dx
#" #
2
4
2
4
# x e 'dx
@ x
#
#
#
#
@
u = x ''''''''''''''''''v = e
x
du = @ x < 'dx''''''''dv = e x 'dx
--E!
69 x'dx
u = 69 x''''''''''''v =
1
v = e 2x
2
dv = e 2 x dx
u= x$4
@
-/E!!
# x e 'dx = x e $ @ # x e 'dx
# x e 'dx = x e $ @" x e $ < xe $ <e #
# x e 'dx = e " x $ @x $ C x $ C# $ C
@ x
@ x
@ x
@ x
@ x
x
< x
< x
@
x
x
<
u = x < ''''''''''''''''''v = $ ;$% x
du = < x'dx''''''''dv = %*9 x'dx
# x %*9 x'dx = $ x
# x %*9 x'dx = $ x
# x %*9 x'dx = $ x
# e ;$% x'dx
#
<
<
;$% x $ < x ;$% x'dx
<
<
;$% x $ <" x " %*9 x $ ;$% x #
<
<
;$% x $ < x %*9 x $ < ;$% x $ C
x
u = ;$% x''''''''''''''''v = e x
#x
-OE!!
<
du = $ %*9 x'dx''''''''dv = e x 'dx
;$% x'dx
#e
u = x < ''''''''''''''''''v = %*9 x
du = < x'dx''''''''dv = ;$% x'dx
#x
#x
x
''#
x
;$% x'dx = e x ;$% x' $
#e
x
%*9 x'dx
u = %*9 x''''''''''''''''v = e x
-TE! du = ;$% x'dx''''''''dv = e x 'dx
#
<
;$% x'dx = x < %*9 x $ < x %*9 x'dx
<
;$% x'dx = x < %*9 x $ <"$ x ;$% x $ %*9 x #
<
;$% x'dx = x < %*9 x $ < x ;$% x $ < %*9 x $ C
#e
#e
x
;$% x'dx = e x ;$% x $ e x %*9 x $
x
;$% x'dx =
#e
x
;$% x'dx
e x ;$% x $ e x %*9 x
$C
<
D*9:',+&'5$6"7&')+&9'y = 69 x'*%'-$,.,&:'./$",
e
e
,+&'x 0 .1*%'/&,)&&9'x = B'.9:'x = e 1 V = !
# 4 x ln x dx
B
1
u = ln x
1
du = dx
x
v = 2x
u = 69 x'''''''''''''v = x 69 x $ x
du = B x 'dx''''''''dv = 69 x'dx
2
dv = 4 x dx
e
!
e
-YE!
# 4 x ln x dx = 2 x
2
ln x $
1
# 2 x dx
# 4 x ln x dx = 2 x
2
ln x $ x 2 %
2e 2 $ e 2 $ 1 = e 2 $ 1
e
1
<
# "69 x # dx = !&69 x " x 69 x $ x# $ # "69 x $ B#'dx%
B
!!!-ZE!!
e
!
e
1
<
# "69 x# dx
e
<
# "69 x # dx = ! %&69 x " x 69 x $ x # $ x 69 x $ x $ x% ()
B
B
e
<
! %!x " 69 x # $ < x 69 x $ < x ( = !"e $ <e $ <e $ <# = !"e $ <#
)B
'' &
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Integration of Powers of Sine and 1osine - 1lasswork
$'00%+2!9%'!=6*(2C!(%!?)*C!
# sin
4
x dx E!_%'!=%'&C!+2(!'0!9%'3!5:63(!
u = "sin x #
3
v = $ cos x
2
du = 3"sin x # cos x dx dv = sin x dx
2
# $ cos x " 3"sin x # cos x dx !!!!!!#&26*)*;!(:)+!'0<!=2!;2(
!!!!!!!!!!!!=!!! $ sin x cos x $ 3 # "sin x # cos x dx
!!!!U%=!'+2!(:2!?65(!(:6(! cos x = 1 $ sin x <!=2!;2(
!!!!!!!!!!!!=!! $ sin x cos x $ 3 # "sin x #"1- sin x # dx
!!!\;6)*<!!6!&)((&2!5&26*'0
!!!!!!!!!!=!! $ sin x cos x $ 3 # "sin x # dx $ 3 # sin x dx
!!!a%82! $3 # sin x dx (%!(:2!&2?(!+)C2<!=2!;2(
!! 4 # sin x dx = $ sin x cos x $ 3 # "sin x # dx
!!!
$%&82!?%3! # sin x dx
1
3
# sin x dx = $ 4 sin x cos x $ 4 # "sin x # dx
42!:682!*%=!32C'52C!%'3!%3);)*6&! # sin x dx !(%!%*2!)*8%&8)*;! # sin x dx E!42!?%'*C! # sin x dx )*!6*
$%!
# sin
4
x dx !=! $ sin 3 x cos x $
2
3
3
2
3
4
2
4
4
2
3
2
2
2
3
4
2
4
2
4
2
2
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=2!:682!=:6(!632!32?2332C!(%!6+!32C'5()%*!?%3@'&6+R
Guidelines for evaluating integrals involving powers of sine and cosine
G)3+(<!9%'!ac$H!32@2@723!(:6(!+)*52! sin 2 x $ cos2 x = 1,
sin 2 x = 1 $ cos2 x,
cos2 x = 1 $ sin 2 x
-E!1?!(:2!0%=23!%?!(:2!+)*2!)+!%CC!6*C!0%+)()82<!+682!%*2!+)*2!?65(%3!6*C!5%*823(!(:2!32@6)*)*;!?65(%3+!(%!5%+)*2E
!!!!H:2*!2Q06*C!6*C!)*(2;36(2E
/E!1?!(:2!0%=23!%?!(:2!5%+)*2!)+!%CC!6*C!0%+)()82<!+682!%*2!5%+)*2!?65(%3!6*C!5%*823(!(:2!32@6)*)*;!?65(%3+!(%!+)*2E
!!!!H:2*!2Q06*C!6*C!)*(2;36(2E
OE!1?!(:2!0%=23+!%?!both!(:2!+)*2!6*C!5%+)*2!632!282*!6*C!*%**2;6()82<!@6I2!32026(2C!'+2!%?!(:2!)C2*()()2+
1 $ cos 2 x
1 $ cos 2 x
and cos2 x =
2
2
!!!!!(%!5%*823(!(:2!)*(2;36*C!(%!%CC!0%=23+!%?!(:2!5%+)*2E!H:2*!03%522C!6+!)*!;')C2&)*2!/E
sin 2 x =
# %*9 ;$%
@
A
x'dx
#
G'J%&' %*9 x = B $ ;$% '.9:'-&)-*,&H''''''''''''''''''''''''' # %*9 x "B $ ;$% x # ;$% x'dx
G'K$)'7"6,*#6('$",H'''''''''''''''''''''''''''''''''''''''''''''''' # "%*9 x $ %*9 x ;$% x # ;$% x'dx
PQ6@0&2!-)! G'L-*,&'*,'.%',)$'*9,&2-.6%H'''''''''''''''''''''''''''''''''''''' ;$% x %*9 x'dx $ ;$% x %*9 x'dx
#
#
G'4+*%'*%'2"*:&6*9&'BH'I&)-*,&'*,'"%*92'$9&'%*9&H''''' %*9 x %*9 < x ;$%A x'dx
<
<
<
A
<
A
G'M.;+'*9,&2-.6';.9'/&',.>&9'/('u 0 %"/%,*,",*$9H'''''
A
C
0;$%E x ;$%F x
$C
$
E
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Improper Integrals - 1lasswork
$'00%+2!9%'!632!C3)8)*;!6&%*;!(:2!:);:=69!6(!-..!?(f+25!(67%'(!Z]!@0:)E!\(!()@2!t!=!.!+25E<!9%'!(6I2!9%'3!?%%(!%??
(:2!6552&236(%3!6*C!&2(!(:2!563!+(63(!(%!+&%=!C%=*E!!_%'3!82&%5)(9!)+!;)82*!79
v " t # = 100e$0.1t !!=:232! v " t # !)+!)*!?22(!023!+25%*CE
\55%3C)*;!(%!(:)+!@%C2&<!(:2!82&%5)(9!6003%65:2+!B23%!6+!()@2!)*5326+2+!7'(!)+!*2823!2>'6&!(%!B23%E
100
= 0 = 100e$0.1t
0
.
1
t
t 23 e
lim100e$0.1t = lim
t 23
4%'&C!(:2!C)+(6*52!9%'!;%!6003%65:!6!&)@)()*;!86&'2<!%3!=%'&C!)(!)*5326+2!=)(:%'(!7%'*CM!1*!(:)+!+25()%*<!9%'!=)&&
&263*!:%=!(%!6*+=23!+'5:!>'2+()%*+!79!286&'6()*;!)@03%023!)*(2;36&+E
g)82*!6*!)@03%023!)*(2;36&<!=2!=)+:!(%!C2(23@)*2!=:2(:23!%3!*%(!)(!converges!((:6(!)+<!6003%65:2+
6!?)*)(2!*'@723!6+!6!&)@)()E!1?!)(!C%2+<!=2!=)+:!(%!?)*C!(:2!*'@723!(%!=:)5:!)(!5%*823;2+E
H:2!0)5('32!72&%=!+:%=+!(:2!82&%5)(9!?'*5()%*!@2*()%*2C!67%82E!H:2!C)+(6*52!(:2!563!;%2+!72(=22*!t!=!.!6*C
t!=!b!)+!2>'6&!(%!(:2!6326!'*C23!(:2!;360:E
B==
S*%,.9;&'
Q-&.'=':*%,.9;&',-.5&6&:
b
= # 100e$0.1t dt
0
b
= $1000e$0.1t |0
= $1000e$.1b $ 1000
b
1?!b!=!-.!+25%*C+<!(:2!C)+(6*52!)+!ZO/E-!?(E!\+!b!6003%65:2+!)*?)*)(9<!(:2!1000e$0.1t !(23@!6003%65:2+!B23%E!H:'+!(:2
C)+(6*52!6003%65:2+!-<...!?(E!H:2!@6(:2@6()56&!@%C2&!(2&&+!9%'!(:6(!(:2!563!*2823!06++2+!6!0%)*(!-<...!?22(!?3%@
3
=:232!9%'!+(63(2C!+&%=)*;E!H:2!)*(2;36&! # 100e$0.1t dt !)+!56&&2C!6*!)@03%023!)*(2;36&!7256'+2!%*2!%?!)(+!&)@)(+!%?
0
)*(2;36()%*!)+!*%(!?)*)(2E!H:2!)*(2;36&!converges!(%!-...!7256'+2!(:2!)*(2;36&!?3%@!.!(%!b!6003%65:2+!-<...!6+
b!6003%65:2+!)*?)*)(9E
h'2+()%*R!4:6(!)+!(:2!()@2!&6;!72(=22*!(:2!563!(3682&)*;!^Y.!?22(!6*C!^^^!?22(M
950 = 1000 $ 1000e$.1t
999 = 1000 $ 1000e$.1t
1000e.$1t = 50
1000e.$1t = 10
e.$1t =
+1.
1
1 $.1t = ln- 0
, 20 /
20
t = 29.957 sec1000e$0.1t
+ 1 . !!!! time lag is 39.121 sec.
1
1 $.1t = ln0
,1000 /
1000
t = 69.078 sec
e.$1t =
! www.MasterMathMentor.com!! "#!$%&'()%*+!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!,!--[!,!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Illegal! (%! 0%+(! %*! 1*(23*2(
$1
$'00%+2!(:2!82&%5)(9!?'*5()%*!:6C!722*!! 400" t $ 4 # E!!H:2!;360:!72&%=!&%%I+!6&@%+(!(:2!+6@2E!H:2!82&%5)(9!+()&&
6003%65:2+!B23%!6+!()@2!)*5326+2+E
b
S*%,.9;&' =
# A=="t $ A#
$B
B==
dt
=
b
=
#
=
Q-&.'=':*%,.9;&',-.5&6&:
A==
'dt
"t $ A#
= A== 69 t $ A
b
T
=
= A== 69 b $ A $ A== 69 A
''
\+!b!6003%65:2+!)*?)*)(9<!+%!C%2+! 69 b $ A E!H:2!)*(2;36&!divergesE!c*&)I2!(:2!?)3+(!@6(:2@6()56&!@%C2&<!(:)+!%*2!(2&&+
9%'!(:2!563!=%'&C!;%!)*?)*)(2&9!?63!?3%@!(:2!+(63()*;!0%)*(!)?!9%'!=6)(2C!&%*;!2*%';:E
A definite integral is improper if the following hold.
e!H:2!'0023!%3!&%=23!&)@)(!%?!)*(2;36()%*!)+!)*?)*)(2E
b
3
!!!!!!!!!!
#
a
b
f " x # dx = lim # f " x # dx
b 23
#
or
a
b
f " x # dx = lim # f " x # dx
a 23
3
a
e!H:2!)*(2;36*C!)+!C)+5%*()*'%'+!?%3!6(!&26+(!%*2!86&'2!%?!x!6(!%3!72(=22*!(:2!&)@)(+!%?!)*(2;36()%*E
b
!!!!!!!
b
k
# f " x # dx = lim # f " x # dx $ lim # f " x # dx <! f )+!C)+5%*()*'%'+!6(! x' = c'*9'&a8 b%
a
k 2c $
k 2c $
k
a
\*!)@03%023!)*(2;36&!converges!(%!6!523(6)*!*'@723!)?!265:!600&)567&2!&)@)(!+:%=*!67%82!)+!?)*)(2E
W(:23=)+2<!(:2!)*(2;36&!divergesE!U%(2!(:6(!6*!)@03%023!)*(2;36&!=)(:!6*!)*?)*)(2!&)@)(!%?
)*(2;36()%*!6&=69+!C)823;2+!)?!(:2!)*(2;36*C!:6+!6!&)@)(!%(:23!(:6*!B23%!6+!(:2!863)67&2!%?!)*(2;36()%*
6003%65:2+!)*?)*)(9E
3
PQ6@0&2!-)!G%3!(:2!)@03%023!)*(2;36&!
#xe
< $x
dx
=
.)'V-.#+',+&'*9,&2-.9:'.9:',&66')+&,+&''''$-'9$,',+&'*9,&2-.6'7*2+,';$95&-2&H
!!!!!!!!!!!!!!!!!!!
W&-+.#%
!!!!!!!
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7)!1?!(:2!)*(2;36&!@);:(!5%*823;2<!?)*C!%'(!=:2(:23!%3!*%(!)(!C%2+<!6*C!)?!+%<!(%!=:6(!&)@)(
!!!!!)(!5%*823;2+E
b
!!!!e!i20&652!!!=)(:!JbK6*C!&2(!b!6003%65:!)*?)*)(9E!!!!!! lim
# x 2e$ x dx
b 23
0
!!!!e!42!*22C!(%!)*(2;36(2!(:)+!2Q032++)%*E!!4:6(!3'&2!@'+(!=2!'+2M
x 2e x $ 2 # xe$ x dx
2 x
&
$x
$x
#e
$ 2e %
x e $ 2 $ xe $
!!!!!!
&x e
2 x
$ 2 xe$ x
dx
u = x2
%
du = 2 xdx dv = e$ x dx
!!!!!
$x b
v = $e$ x
0
$b 2 2b 2
$ $ $ "0 $ 0 $ 2#
eb eb eb
u= x
v = $e$ x
du = dx
dv = e$ x dx
!!!!e!42!*%=!*22C!(%!(6I2!6!&)@)(E!d%=!C%!=2!C%!(:)+M
%$b 2 2b 2
(
$
$
$
0
$
0
$
2
"
#
!!!!!!!!!! lim
'
*
b 23 e b
eb eb
&
)
3
!!!!e!S'(!)(!6&&!(%;2(:23E!4:6(!)+!
2 $x
#xe
dx = 2
0
3
PQ6@0&2!/)!!G%3!(:2!)@03%023!)*(2;36&!
#x
0.3
dx
0
.)'V-.#+',+&'*9,&2-.9:'.9:',&66')+&,+&''''$-'9$,',+&'*9,&2-.6'7*2+,';$95&-2&H
X,';.99$,'.%',+&'.;;"7"6.,&:'.-&.'
2&,%'/*22&-')*,+$",'/$"9:
7)!1?!(:2!)*(2;36&!@);:(!5%*823;2<!?)*C!%'(!=:2(:23!%3!*%(!)(!C%2+<!6*C!)?!+%<!(%!=:6(!&)@)(
!!!!!)(!5%*823;2+E
b
3
% x1.3 (
+ b1.3 .
.2
= lim- 0 = 3 so the integral is divergent.
!!!!!!!!!!!!! # x dx = lim
'
*
b 23 1.3
& )0 b 23, 1.3 /
0
! www.MasterMathMentor.com!! "#!$%&'()%*+!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!,!--^!,!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Illegal! (%! 0%+(! %*! 1*(23*2(
2
PQ6@0&2!O)!!G%3!(:2!)@03%023!)*(2;36&!
2
# x dx
$2
.)'V-.#+',+&'*9,&2-.9:'.9:',&66')+&,+&''''$-'9$,',+&'*9,&2-.6'7*2+,';$95&-2&H
W&-+.#%
!!!!!
!!!!7)!1?!(:2!)*(2;36&!@);:(!5%*823;2<!?)*C!%'(!=:2(:23!%3!*%(!)(!C%2+<!6*C!)?!+%<!(%!=:6(!&)@)(!)(!5%*823;2+E
0
2
# x dx $
$2
2
2
#x
dx
0
"
b
2
lim 2&ln x %$2 $ &2 ln x % b
b 20
#
lim"2 ln b $ 2 ln 2 $ 2 ln 2 $ 2 ln b#
b 20
DNE
3
3
#e
$x
#
dx
0
=
PQ6@0&2!T)!P86&'6(2!
&$e %
$x
3
1
dx and
2
x $1
&tan x %
$1
PQ6@0&2!Y)!P86&'6(2!
=
$e$3 $ e=
$B
$B=B
e3
dx
#3x
0
(improper - 0 in denominator)
1
PQ6@0&2!Z)!P86&'6(2! % 3
2 3(
x
'&2
*)
0
3
2
#x
$3
1
dx
$1
2
3
0
$1
tan 3 $ tan$1 0
!
1
4
3
#x
$3
1
1
3
1
!
dx =
2
$1
2
dx
#x
3
0
(improper - 0 in denominator)
1
% x $2 (
PQ6@0&2![)!P86&'6(2! '& $2 *)
0
1
% 1 (
'&2 x 2 *)
0
DNE
! www.MasterMathMentor.com!! "#!$%&'()%*+!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!,!-/.!,!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Illegal! (%! 0%+(! %*! 1*(23*2(
Improper Integrals - >omework
G%3!265:!%?!(:2!03%7&2@+!72&%=!,!6)!;360:!)(!?)3+(!(%!C2(23@)*2!=:2(:23!%3!*%(!(:2!)*(2;36*C!@);:(!5%*823;2!6*C
7)!)?!(:2!)*(2;36&!@);:(!5%*823;2<!?)*C!%'(!=:2(:23!%3!*%(!)(!C%2+<!6*C!)?!+%<!(:2!&)@)(!(%!=:)5:!)(!5%*823;2+E
3
1
dx
x2
#
-)!
2
3
%$1( 1
'& x *) = 2
2
3
#
TE!!
1
1
dx
x 0.2
3
% x .8 (
' * = Divergent
& .8 )1
1
[E!!
1
#x
1.2
3
/E!
#
1
1
% x $.2 (
' * = Divergent
& $.2 )0
1
1
ZE!
3
% x $.2 (
1
' * = =5
& $.2 )1 .2
]E!!
0
&ln x % 0 = Divergent
1
dx
x1.2
3
1
# x dx
OE!
3
% $1 (
1
'& 3 x 3 *) = 81
3
dx
0
1
dx
4
3
3
YE!
1
#x
dx
1
% x .8 (
1 5
' * = =
& .8 )0 .8 4
1
Ê!!
0
0.2
0
1
# x $ 1 dx
1
#x
1
# x ln x dx
0
1
3
&ln x $ 1%0 = Divergent
&ln ln x %0 = Divergent
3
11
# " x $ 3#
$2
3
# xe
dx
-.E!!
#
3
x dx
$1
DNE
# " x $ 3#
--E!!!
$2
11
3
dx $
&
$2
# " x $ 3#
3
3
3
2
13 3
3" x $ 3#
dx
0
2
2
$x
% &
1
1 3 11
$ 3" x $ 3#
3&0 $ "$1#% $ 3&2 $ 0% = 9
%
3
dx
-/E!!
&$ xe$ x % $
0
3
3
#e
$x
dx
0
3
%$ x ( % 1 (
'& e x *) $ '&e x *)
0
0
0 $ 0 $ "0 $ 1# = 1
! www.MasterMathMentor.com!! "#!$%&'()%*+!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!,!-/-!,!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Illegal! (%! 0%+(! %*! 1*(23*2(
3
# "1 $ x #e
-x
dx
1
3
3
# e -x dx $
# xe
1
3
#
-OE!!!
1
-x
dx
3
1
+
1
$x 3
dx
$
-&$ xe %1 $
x
e
,
$x 3
& xe %
1
ex
# 1+e
.
# e$ x dx 0/
1
3
2x
u = e x , du = e x dx
dx
$3
3
-TE!
3
%x(
=' x*
&e )1
du
# 1+u
x = -3, u = 0, x = 3, u = 3
2
$3
&tan u%
$1
$1
e
3
0
=
!
2
2
!?3%@!x!=!-!(%!x!=!b!6*C!(:2!x,6Q)+E
x
!!!!!!6E!$:%=!=%3I!(%!C2(23@)*2!=:2(:23!(:2!32;)%*D+
7)!1?!(:2!32;)%*!)+!3%(6(2C!67%'(!(:2!x,6Q)+<!C2(23@)*2
!!!!!!!!!6326!6003%65:2+!6!?)*)(2!&)@)(!6+!b!6003%65:2+!!E!!!!!!!!!!!!!!)?!(:2!8%&'@2!%?!(:2!+%&)C!6003%65:2+!6!&)@)(!6+!
!!!!!!!!!!!!!!! b 2 3E
-YE!V36=!(:2!;360:!%?!(:2!32;)%*!'*C23!(:2!;360:! y =
!!!!!!!!!!!!!!
!!!!!!!!!!
3
3
A=
#
1
V =!#
2
dx
x
1
4
dx
x2
3
%$4 (
!!!!! !' *
& x )1
3
!!!!!!!!!!!!!!!! &ln x % 2
DNE
4!
!!!!!5E!H:2!32;)%*!)+!3%(6(2C!67%'(!(:2!y,6Q)+!(%!?%3@!!!!!!!!!!!CE!H3'2!%3!?6&+2R!J1?!6!32;)%*!:6+!)*?)*)(2!6326<!(:2*
!!!!!!!!!6!+%&)CE!V%2+!(:)+!8%&'@2!6003%65:!6!?)*)(2!
!!!!!!!!!!!!(:2!+%&)C!?%3@2C!79!3%(6()*;!(:6(!%7X25(!67%'(
!!!!!!!!!&)@)(!6+! b 2 3M
!!!!!!!!!!!!6*!6Q)+!:6+!)*?)*)(2!8%&'@2EK
3
+2.
V = 2! # x- 0 dx
, x/
1
3
!!!!! 2!&2 x % 2
False...a# and b# contradict that.
3
! www.MasterMathMentor.com!! "#!$%&'()%*+!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!,!-//!,!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Illegal! (%! 0%+(! %*! 1*(23*2(
-ZE!!P63&)23<!9%'!?%'*C!(:6(!)(!=%'&C!32>')32!-.<...!@)&2,(%*+!(%!03%02&!6!-Y,(%*!+0652!@%C'&2!(%!6!:2);:(!%?!]..
@)&2+!67%82!(:2!263(:E!d%=!@'5:!=%3I!)+!32>')32C!(%!03%02&!(:2!@%C'&2!6*!'*&)@)(2C!C)+(6*52!?3%@!(:2
263(:D+!+'3?652M
C
C
F " x # = 2 1 15 =
1 C = 240, 000, 000
x
4000 2
3
3
%$240000000 (
240000000
W = #
dx = '
*)
&
x2
x
4000
4000
W = 60000 mile - tons = 6.336 " 1011 ft - lbs
-[E!42!I*%=!(:2!(:2!5)35'@?232*52!%?!6!5)35&2!=)(:!36C)'+!=!-!)+!/"E!c+2!(:2!635!&2*;(:!?%3@'&6!(%!+:%=!(:6(!(:2
5)35'@?232*52!%?!(:2!5)35&2!%?! x 2 $ y 2 = 1!)+!/"E
x2 $ y2 = 1 1 y2 = 1$ x2
$x
$2 x
y = 1 $ x 2 1 y4 =
=
2
2 1$ x
1$ x2
1
1
x2
1
1
dx
4
dx
4
dx
=
=
#
#
2
2
2
1
x
1
x
$
$
1
x
$
0
0
0
1
+
.
!
L = & 4 sin$1 x % = 4- 0 = 2!
0
,2/
1
L = 4 # 1$
-]E!!W*!9%'3!56&5'&6(%3<!;360:! y = e x $
x $2
!%*!(:2!=)*C%=![-<!O]<![.<!/.]E!!$'00%+2!9%'!632!(%!286&'6(2
x $2
+
x $2.
!!!!!! # -e x $
0 dx E!\&(:%';:!(:2!)*(2;36*C!)+!C)+5%*()*'%'+!%*!(:2!5&%+2C!)*(2386&![-<!O]<!(:232!)+!%*&9!6!+(20
x $2/
1 ,
!!!!!!C)+5%*()*')(9!6(!x!=!/E!H:)+!)+!56&&2C!piecewise-continuous!%*!(:6(!)*(2386&E
3
6E!!43)(2!(:2!)*(2;36&!6+!(:2!+'@!%?!(=%!)*(2;36&+<!%*2!?3%@!x!=!-!(%!x!=!/!6*C!(:2!%(:23!?3%@!x!=!/!(%!x =!OE
+
x $2.
# -,e x $ x $ 2 0/ dx =
1
3
+
x $2.
# -,e x $ x $ 2 0/ dx $
1
2
3
+
# -,e
x
$
2
x $2.
0 dx
x $2/
7E!!"%(:!)*(2;36&+!632!)@03%023E!43)(2!(:2@!=)(:!(:2!5%3325(!&)@)(!(23@)*%&%;9E
3 +
%b + x x $2. (
% 3 + x x $2. (
x $2.
x
' # -e $
' # -e $
0 *dx $ lim
0 *dx
# -,e $ x $ 2 0/ dx = lim
b 22
b 22
x $2/ )
x $2/ )
&1 ,
&b ,
1
x$<
!2>'6&+!6!5%*+(6*(!(%!(:2!&2?(!%?
x$<
!!!!!!!!!!!!!!!x!=!/!6*C!6*%(:23!5%*+(6*(!(%!(:2!3);:(!%?!x!=!/E!G)*C!(:2!86&'2!(%!=:)5:!(:2!%3);)*6&!)*(2;36&!5%*823;2+E
5E!!$:%=!(:6(!7%(:!)*(2;36&+!5%*823;2E!W7+2382!(:6(!(:2!2Q032++)%*!
+
x $2.
# -,e x $ x $ 2 0/ dx =
1
3
&e
x
2
3
1
2
2
3
# "e x $ 1# dx $
# "e
1
2
x
$ 1# dx
$ x % $ &e x $ x % = e 2 $ 2 $ e $ 1 $ e 3 $ 3 $ e 2 $ 2 = e 3 $ e
!!!!
! www.MasterMathMentor.com!! "#!$%&'()%*+!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!,!-/O!,!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Illegal! (%! 0%+(! %*! 1*(23*2(
Finding HImpossibleI Integrals - 1lasswork
x
E!1?! F "0# = 2 <!?)*C! F "3# E
x $1
x
$%&'()%*R!!42!*22C!(%!(6I2!(:2!)*(2;36&!%?! 3 E!!"'(!56*!=2M!!u,+'7+()('()%*!C%2+!*%(!=%3IE!42!56**%(!+0&)(
x $1
!!!!!!!!!!!!!!!!!(:2!?'*5()%*E!H:232!+22@+!(%!72!*%!0%++)7)&)(9!?%3!)*823+2,(3);E!!$%!:%=!56*!=2!C%!(:2!03%7&2@M
PQ6@0&2R!!L2(! F " x # !72!6*!6*()C23)86()82!%?!
3
!!!!!!!!!!!!!!!!!42!632!*%(!72)*;!6+I2C!(%!?)*C!(:2!)*C2?)*)(2!)*(2;36&!%?!(:2!2Q032++)%*E!42!X'+(!*22C!(%!?)*C!(:2!86&'2
!!!!!!!!!!!!!!!!!%?!(:2!)*(2;36&!?'*5()%*!%?!(:)+!2Q032++)%*!6(!OE
3
3
x
x
x
dx <!(:2*! F " 3# $ F "0# = # 3
dx !6*C!(:'+! F " 3# = 2 $ # 3
dx E
!!!!!!!!!!!!!!!!!!$%<!)?! F " x # = # 3
x $1
0 x $1
0 x $1
!!!!!!!!!!!!!!!!!c+)*;!(:2!56&5'&6(%3R! F " 3# = 2 $ .879 = 2.879 E
/)!!L2(! F " x # !72!6*!6*()C23)86()82!%?! 4 x 2 $ 5 E!1?! F "1# = 3 <!?)*C! F "5# E
5
F "5# $ F "1# =
#
4 x 2 $ 5 dx
1
!!!!!!!!!!!!!!!
5
F "5# = F "1# $
#
4 x 2 $ 5 dx = 3 $ 25.866 = 28.866
1
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Numerical Solutions of Differential Tquations
Using TulerEs Method - 1lasswork
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Numerical Solutions of Differential Tquations
using TulerEs Method - >omework
dy
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!!!!!+:%=!(:6(!)(!)+!(:2!+%&'()%*!(%!(:2!C)??232*()6&!2>'6()%*!67%82E
dy
dy
= $Ce$ t $ 1 1
$ 1 = $Ce$ t
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dy
1$
= Ce$ t 1 1 $
= y$t 1
= 1$ t $ y
dt
dt
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5E!G)*C!(:2!86&'2!%?!C!?%3!(:)+!063()5'&63!2>'6()%*!(:6(!?)(+! y "0# = 0
y = Ce$ t $ t
0 = Ce 0 $ 0
C=0
y=t
!!
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dy
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YE!!#%*+)C23!(:2!C)??232*()6&!2>'6()%*!
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6E!c+2!P'&23D+!@2(:%C!=)(:!Y!+(20+!%?!+)B2!E/!(%!2+()@6(2! y "1#
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dx
2
2
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=
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x
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# y #
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1.6487 $ 1.4593 = 0.1894
dy
ZE!!#%*+)C23!(:2!C)??232*()6&!2>'6()%*!
= 1 $ 3 x $ 2 y, y "0# = 2 E
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6E!c+2!P'&23D+!@2(:%C!=)(:!Y!+(20+!%?!+)B2!E/!(%!2+()@6(2! y "1# E
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2 4
3
3 dy
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$ $2Ce$2 x $ 1 2Ce$2 x = $
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!!!!6;6)*+(!(:2!0%)*(+!9%'!:682!67%82E!G)*C!(:2!C)??232*52!72(=22*!(:2!2+()@6(2!6*C!65('6&!86&'2+!%?!! y "1# E
1
9
2 = Ce 0 $ 0 $ 1 C =
1.5545 $ 1.4250 = .1295
4
4
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Logistic Growth - 1lasswork
c+)*;!(:2!2Q0%*2*()6&!;3%=(:!@%C2&<!(:2!;3%=(:!%?!6!0%0'&6()%*!)+!03%0%3()%*6&!(%!)(+!5'332*(!+)B2E!!H:2
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= kP 6*C! P = Ce kt E!H:)+!@%C2&+!;3%=(:!(:6(!5%*()*'2+!?%32823!,! lim P " t # = 3E
t 23
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;3%=(:!%?!6&;62!%*!6!&6I2<!C!@);:(!32032+2*(!(:2!@6Q)@'@!6@%'*(!%?!6&;62!(:6(!(:2!&6I2!56*!+'+(6)*E
L%;)+()5!;3%=(:!)+!(:2!(23@!(:6(!@%C2&+!+'5:!;3%=(:E!!H:2!=%3C!logistics!32?23+!(%!(:2!63(!%?!0&6**)*;!6*C
5%%3C)*6()*;!(:2!C2(6)&+!%?!6*!%0236()%*E!1(!5%@2!?3%@!(:2!g322I!=%3C!logistikos<!@26*)*;!J325I%*)*;K!%3!J326+%*EK
1*!&%;)+()5!;3%=(:<!(:2!>'6*()(9!P!;3%=+!6(!6!36(2!(:6(!)+!03%0%3()%*6&!(%!)(+2&?!6*C!(%!C ,!P<!=:)5:!)+!(:2!J3%%@
686)&67&2K!?%3!?'3(:23!;3%=(:E
dP
= kP "C $ P # !!!!!L2(D+!+%&82!)(E
!!!
dt
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e!c+2!063()6&!?365()%*+!?%3!(:2!&2?(!+)C2
+1
1 .
dP
C
# P "C $ P # = # k dt
# - P $ C $CP 00 dt = # k dt
,
/
e!1*(2;36(2!265:!+)C2
1
&ln P $ ln"C $ P #% = kt $ D
C
e!43)(2!(:2!&2?(!+)C2!6+!6!&*!+(6(2@2*(
+ P .
ln0 = Ckt $ D
,C - P/
e!$%&82!?%3!P
P
C$P
C
C
C
= eCkt $ D 1
= e$Ckt $ D 1 $ 1 = de$Ckt 1 = 1 $ de$Ckt 1 P =
1 $ de$Ckt
C-P
P
P
P
Logistic growth
$%!=2!2*C!'0!=)(:!(:2!32+'&(R!!!!
P "t # =
dP
= kP "C $ P # has solution
dt
C
where k and d are constants and C is the carrying capacity
1 $ de$Ckt
Rumors and logistic growth9 !\!3'@%3!+0326C!6+!J(2&&23+K!06++!)(!%*!(%!J:26323+EK!W*52!(%&C<!6!:26323!725%@2+!6
(2&&23E!H:2!3'@%3!+0326C+!+&%=&9!6(!?)3+(!=:2*!(2&&23+!632!?2=E!1(!+0326C+!?6+(23!=:2*!7%(:!(2&&23+!6*C!:26323+!632
0&2*()?'&!7'(!+&%=+!C%=*!6;6)*!6+!:26323+!725%@2!+56352!6*C!)(!+(%0+!=:2*!28239%*2!I*%=+!(:2!3'@%3E!H:2
*'@723!%?!02%0&2!I*%=)*;!(:2!3'@%3!((2&&23+)!;3%=+!6+!+:%=*E
,&66&-%
=
=
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! www.MasterMathMentor.com!! "#!$%&'()%*+!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!,!-OO!,!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Illegal! (%! 0%+(! %*! 1*(23*2(
PQ6@0&2!-)!4)++6:)5I%*!d);:!:6+!-O..!+('C2*(+E!H:2!T!d%*%3!$%5)2(9!%??)523+<!a6((<!$5%((<!i2*6E!6*C!L)+6
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!!!!!!!!!!!!!!!!!!!!725%@2!03)*5)06&E!\!C69!&6(23<!Z/!+('C2*(+!I*%=!(:2!3'@%3E!4:6(!:6002*+!%823!-.!C69+M!4:2*!)+
!!!!!!!!!!!!!!!!!!!!(:2!3'@%3!+0326C)*;!(:2!?6+(2+(M
dP
= kP "1300 $ P #
e!G)3+(<!=3)(2!(:2!C)??232*()6&!2>'6()%*!(:6(!C2+53)72+!)(E!!
dt
1300
e!H:)+!9)2&C+!(:2!&%;)+()5!2>'6()%*R!! P =
1 $ de$1300 kt
e!42!I*%=!(:6(!=:2*!t!=!.<!P =!/YE!S&';!(:)+!)*!6*C!+%&82!?%3!dE
1300
25 =
1 d = 51
1$ d
e!42!I*%=!(:6(!=:2*!t!=!-<!P B Z/E!S&';!(:)+!)*!6*C!+%&82!?%3!kE!g2(!6!&%(!%?!655'3659E
e!$%!=:6(!)+!(:2!+%&'()%*!?'*5()%*M!!g360:!)(E 1(!+:%'&C!&%%I!&)I2!(:)+R
1300
1300
1300
1 62 =
1 1 $ 51e$1300 k =
$1300 kt
$1300 k
1 $ 51e
1 $ 51e
62
+ 1300 .
1300
$1
- 62 $ 10
1300
$1300 k
$1300 k
62
51e
=
$1 1 e
=
1 k = ln0
62
51
- 51 0
,
/
P=
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$1300
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e!H%!C2(23@)*2!=:2*!(:2!3'@%3!)+!growing fastest<!=2!*22C!(%!@6Q)@)B2!(:2!;3%=(:!?'*5()%*!%?
!!!PE!!G)3+(<!=2!*22C!(%!?)*C!(:2!C23)86()82!%?!P!6*C!(:2*!=2!*22C!(%!@6Q)@)B2!)(E!!c+)*;!(:2
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!!
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e!PQ520(!?%3!(:2!*2=!)*)()6&!5%*C)()%*+<!(:2!+)('6()%*!)+!2Q65(&9!6+!)*!(:2!03252C)*;!2Q6@0&2E
!!!!!!!!!!!!!!!$%<!+%&82!?%3!d!6*C!=3)(2!(:2!*2=!&%;)+()5!2>'6()%*E!!H:2*!;360:!)(!)*!5%@063)+%*!(%!(:2!&6+(
!!!2Q6@0&2E!!
!!!
P=
1300
1300
14=
1 1 $ d = 325 1 d = 324
$1300 kt
1 $ de
1$ d
!!!!
e!V2(23@)*2!=:6(!C69!)+!(:)+!3'@%3!+0326C)*;!(:2!?6+(2+(E
!!!!
!!!
!!!!
! www.MasterMathMentor.com!! "#!$%&'()%*+!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!,!-OT!,!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Illegal! (%! 0%+(! %*! 1*(23*2(
Logistic Growth - >omework
-E!!"256'+2!%?!&)@)(2C!?%%C!6*C!+0652<!6!+>')332&!0%0'&6()%*!56**%(!2Q522C!-/..E!1(!;3%=+!6(!6!36(2!03%0%3()%*6&
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dP
= kP "1200 $ P #
6)!43)(2!6!C)??232*()6&!2>'6()%*!(:6(!C2+53)72+!(:)+!+)('6()%*E!!
dt
1200
7)!43)(2!(:2!+%&'()%*!(%!(:)+!C)??232*()6&!2>'6()%*E!!! P =
1 $ de$1200 kt
5)!1?!(:232!632!-..!+>')332&+!(=%!9263+!6;%!6*C!T..!%*2!9263!6;%<!:%=!@6*9!+>')332&+!632!(:232!*%=M
!!!!!(d)*(!,!'+2!P(.)!=!-..<!6*C!P(-)!=!T..E!_%'!=6*(!P(/)E!!$:%=!=%3IE
1200
1 1 $ d = 12 1 d = 11
1$ d
+2.
1200
2
400 =
1 1 $ 11e$1200 k = 3 1 e$1200 k = 1 k = ln- 0 $1200 = .0014 1 P "2# = 880
$1200 k
,11/
1 $ 11e
11
100 =
C)!g360:!(:2!?)3+(!Y!9263+!%?!+>')332&!0%0'&6()%*E
2)!c+2!9%'3!56&5'&6(%3!(%!?)*C!=:2*!(:2!+>')332&
!!!0%0'&6()%*!)+!;3%=)*;!(:2!?6+(2+(E!!!!V69!-ET.ZZ
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=:2*!(:2!0%0'&6()%*!=!Z..!6*C!?)*C)*;!=:2*!S!=!Z..E!!!!
/E!!$'00%+2!6!?&',&)I2!8)3'+!)+!+0326C)*;!(:3%';:!6!0%0'&6()%*!%?!Y.<...!6(!6!36(2!03%0%3()%*6&!7%(:!(%!(:2!*'@723
%?!02%0&2!6&326C9!)*?25(2C!6*C!(%!(:2!*'@723!+()&&!'*6??25(2CE
dP
= kP "50000 $ P #
dt
50000
7)!43)(2!(:2!+%&'()%*!(%!(:)+!C)??232*()6&!2>'6()%*E!! P =
1 $ de$5000 kt
6)!43)(2!6!C)??232*()6&!2>'6()%*!(:6(!C2+53)72+!(:)+!+)('6()%*E!!
5)!1?!-..!02%0&2!=232!)*?25(2C!92+(23C69!6*C!-/Y!632!)*?25(2C!(%C69<!C2(23@)*2!:%=!@6*9!=)&&!72
!!!)*?25(2C!6!=22I!?3%@!(%C69E
50000
50000
100 =
1 1 $ d = 500 1 d = 499 1 125 =
1$ d
1 $ 499e$50000 k
+ 399 .
399
1 $ 499e$50000 k = 400 1 e$50000 k =
1 k = ln0 $50000 1 P "8# = 592
, 499 /
499
C)!g360:!(:2!?)3+(!Y.!C69+!%?!?&'!)*?25()%*E
2)!c+2!9%'3!56&5'&6(%3!(%!?)*C!=:2*!(:2!?&'
!!!!)+!;3%=)*;!(:2!?6+(2+(E!!!!V69!/[E[[^
!!!!!!!!!
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dt
=:2*!(:2!?&'!*'@723!=!/Y...!6*C!?)*C)*;!=:2*!S!=!/Y...E!!!!
! www.MasterMathMentor.com!! "#!$%&'()%*+!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!,!-OY!,!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Illegal! (%! 0%+(! %*! 1*(23*2(
OE!!!\!*2=!7%%I!:)(+!(:2!+(6*C+!6*C!725%@2+!>')(2!0%0'&63E!!H:2!;3%=(:!%?!326C23+:)0!%?!(:2!7%%I!56*!72
+
dR
R .
C2+53)72C!79!(:2!VPh!
= 4 R-2 $
0 !!=:232!C!32032+2*(+!(:2!*'@723!%?!02%0&2!:68)*;!326C!(:2!7%%I!)*
, 200 /
dt
6!063()5'&63!%??)52!7')&C)*;!6*C!t!)+!@26+'32C!)*!C69+E!!1?! R"0# = 3 <!?)*C
6)!! lim R" t #
7)!(:2!86&'2!%?!C!(:6(!(:2!326C23+:)0!)+!;3%=)*;!(:2!?6+(2+(E
t 23
!
TE
+
R .
!!!!! 4 R-2 $
0 = 0 1 R = 400 !!!!!!
, 200 /
!!!!!!! 8 =
8R
1 R = 200
200
H9!S)+(!(6I2+!6!(90)*;!5%'3+2E!d2!(6I2+!6!032,(2+(!6*C!?)*C+!:2!56*!(902!6(!/.!=%3C+!023!@)*'(2E!\?(23!-!=22I
%?!(:2!5%'3+2<!:2!56*!*%=!(902!6(!OY!=%3C+!6!@)*'(2E!!H:2!@6Q)@'@!(90)*;!+022C!%?!@%+(!:'@6*+!)+!-Z.
=%3C+!6!@)*'(2E!!d)+!(90)*;!+022C!;3%=+!&%;)+()56&&9E
dP
= kP "160 $ P #
6)!43)(2!6!C)??232*()6&!2>'6()%*!(:6(!C2+53)72+!(:)+!+)('6()%*E!!
dt
160
7)!43)(2!(:2!;2*236&!+%&'()%*!(%!(:)+!C)??232*()6&!2>'6()%*E!! P =
1 $ de$160 kt
5) $%&82!(:2!C)??232*()6&!2>'6()%*!?%3!H9E
20 =
160
1 1$ d = 8 1 d = 7
1$ d
+ +160 . .
0 $ 10
-160
160
160
,
/ 0
35
$160 k
$160 k
$160 = .0042
35 =
1 1 $ 7e
=
1e
=
$ 1 1 k = ln$160 k
1 $ 7e
35
35
7
0
0
,
/
C) c+)*;!56&5'&'+!(25:*)>'2+<!?)*C!(:2!36(2!6(!=:)5:!(:2!(90)*;!+022C!)+!)*5326+)*;!=:2*!t!=!Y!=22I+!6*C
t !=!/.!=22I+E
dP
P "5# = 128.824
= .0042"128.824 #" 31.176# = 16.892 wpm/week
dt
dP
P "20# = 159.998
= .0042"159.998#".002# 6 0 wpm/week
dt
2) c+)*;!56&5'&'+!(25:*)>'2+<!?)*C!=:2*!H9D+!(90)*;!;3%=(:!)+!(:2!?6+(2+(E!W*&9!'+2!(:2!56&5'&6(%3
6(!(:2!8239!2*C!=:2*!9%'!56&5'&6(2E
dP
= 160 kP $ kP 2
dt
0 = 160 k $ 2 kP
P = 80
160
1 1 $ 7e$160 kt = 2 1 7e$160 kt = 1
$160 kt
1 $ 7e
+ 1.
1
e$160 kt = 1 t = ln- 0 $160 k = Week 2.892
, 7/
7
80 =
?) H9!C25)C2+!(%!>')(!(:2!5%'3+2!=:2*!:)+!)*5326+2!)*!(90)*;!+022C!)+!&2++!(:6*!/!=%3C+!6!@)*'(2E!c+)*;
;360:)56&!(25:*)>'2+<!=:6(!=22I!=)&&!H9!C25)C2!(%!>')(M
!!!!
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Inverse Trig Functions - 1lasswork
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?'*5()%*E!!1?! f " x# = < x $ C8 ',+&9'f $B " x# = =HEx $ @E
1?!!''f $B!('3*+!%'(!(%!72!6!?'*5()%*!(06++2+!(:2!823()56&!&)*2!(2+()<!(:2!(:2!%3);)*6&!?'*5()%*!f!)+!+6)C!(%!72!invertibleE
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"
#
?'*5()%*!C)C!(%!xE!H:6(!)+! f $B f " x# = x E!1?! f " x# = x < <!(:2*! f $1 " x # = x 6*C! x 2 = x E!U%(2!(:6(!)?!(:2!+6@2
+56&2+!632!'+2C!?%3!(:2!(=%!6Q2+<!(:2*!(:2!;360:+!%?! f 6*C! f $B!632!@)33%3!)@6;2+!=)(:!32+025(!(%!(:2!TY%!&)*2!y!=!xE
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(:2!)*823+2!?'*5()%*!)+!;)82*!79!''x = %*9 y E!4:2*!=2!+%&82!?%3!y<!=2!;2(!''y = %*9 $B x E!!H:2!+9@7%&!arcsin x !)+
E!!d232!632!(:2!;360:+!%?! y = %*9 x 6*C! y = %*9 $B x E
+%@2()@2+!'+2C!(%!:2&0!9%'!C)+()*;')+:!''%*9 $B x ?3%@! B
%*9
x
''
!!
!!!
!
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% ! !(
y = %*9 $B x'*3'.9:'$96('*3' %*9 y = x'.9:'y 7 '$ 8 *
& < <)
y = ;$%$B x'*3'.9:'$96('*3' ;$% y = x'.9:'y 7 &=8 !%
+ ! !.
y = ,.9 $B x'*3'.9:'$96('*3' ,.9 y = x'.9:'y 7 -$ 8 0
, < </
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+ 1.
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!
$
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!!
;$%$B =
7E! !
''<
tan$1 3
5E!! !
3
"
csc$1 $ 2
CE!! !
$
4
#
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+
+
3. 3
2.
6E!! sin-arctan 0 =
7E!! tan-arccos 0 = 1
,
4/ 5
2 /
,
@
<
8
8
A
2
+ $1 1 . 3
5E! sec-sin 0 =
,
3/
8
+ $1 $2 . $2
CE cot-cos
0=
,
5/
21
@
E
B
8
0<
8
PQ6@0&2!O)!P86&'6(2!(:2!?%&&%=)*;E!a6I2!6!0)5('32!(%!C2+53)72!(:2!+)('6()%*E
6E! cos"sin x # = 1 $ x
$1
1$ x2
7E! tan"cos x # =
x
2
B
$1
B
x
8
8
x
!!!!!!!!!!
!!!!!!!!!!!!!!!!!
"
#
$B
5E!! %*9 ;$% < x = B $ A x
<
$1
CE! sin"tan 3 x # =
1
1$ 9x2
B
!!!!!!!!!!!!!!!!!
8
<x
!!!!!!!!!
d
$%!*%=!=2!56*!(6I2!C23)86()82+!%?!)*823+2!(3);!?'*5()%*+E!!G)*C!
%*9 $B x
''dx
$B
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!''y = %*9 x''' 9 ''' %*9 y = x E
"
y = %*9 $B x : %*9 y = x
S-.)'.'#*;,"-&
,+&'.926&'*%'y 8 '$##$%*,&' = 'B8 '+(#$,&9"%& = B
#
B
x
y
'' B$ x
<
''I&7.*9*92'%*:&'*%' B $ x
$)*52! %*9 y = x <!(6I2!(:2!C23)86()82!%?!265:!+)C2
cos y
<
dy
dy
1
= 1 or
=
dx
dx cos y
P6Q@0&2!T)!H6I2!(:2!C23)86()82!%?
y = cos$1 x
6E! dy
$1
=
dx
1$ x2
@x
8
B
or
dy
1
=
dx
1$ x2
y = tan$1 x
7E! dy
1
=
dx 1 $ x 2
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The derivatives of the three inverse trig functions are as
follows9
d
1
sin$1 u# =
"
dx
1 $ u2
d
$1
cos$1 u# =
"
dx
1 $ u2
du
dx
du
dx
1 du
d
tan$1 u# =
"
1 $ u 2 dx
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! dx
PQ6@0&2!Y)!!G)*C!(:2!C23)86()82+!%?
y = sin$1 4 x
6E! dy
4
=
dx
1 $ 16 x 2
+
x.
y = -cos$1 0
,
2/
5E!
y = ,.9 $B x @
7E! dy
@x <
=
C
''dx B $ x
3
+
dy
x.
= 3-cos$1 0
,
2/
dx
2
+ 1.
- 0=
x , 2/
1$
4
$1
2
+
x.
$3-cos$1 0
,
2/
4 $ x2
2
y = x sin$1 x $ 1 $ x 2
CE!! dy
x
$2 x
=
$ sin$1 x $
= sin$1 x
2
2
dx
1$ x
2 1$ x
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!!!!!!!!6*;&2!%?! 8 !36C)6*+!(%!6!02302*C)5'&63!&)*2!(%!(:2!3%6CE
4-";>
x
.H'''M1#-&%%'8 '.%'.9'*95&-%&',-*2'3"9;,*$9H
x
8 = ,.9
B==
''
B=='3,
$B
8
W.,-$6'N.-
d8
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dt
d8
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !=!,/!C2;322+f+25E!d%=!?6+(!)+!(:2!563!;%)*;!)*!?(f+25!6*C!@0:M
dt
d8
dx
B==
=
<
dt B==== $ x dt
!!!!!!!!!!!!!!
dx
$<!
B== dx
=
1
= $Y=HFEF '3,[%&; = CBHPP='7#+
dt
BP= <C==== dt
7E!!G)*C!
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Inverse Trig Functions - >omework
-)!#P86&'6(2!265:!%?!(:2!?%&&%=)*;R
B
.-;;$%
<
6E!
! !
!
!
@
cot $1 3
7E! !
6
sin$1
5E!!
$ 3
2
sec$1 $
CE!!
!
$
3
2 3
3
5!
6
/)!P86&'6(2!(:2!?%&&%=)*;E!a6I2!6!0)5('32!(%!C2+53)72!(:2!+)('6()%*E
+
+
12 . 12
$3 . 4
6E!! cos-arcsin 0 =
7E!! sin-arctan 0 =
,
,
5/ 5
5 / 13
8
E
B<
0@
8
E
"
#
$B
5E! ;%; ;$, A = BF
''
"
2
2
#
$1
CE tan csc 3 =
A
3
8
8
B
B
O)!P86&'6(2!(:2!?%&&%=)*;E!a6I2!6!0)5('32!(%!C2+53)72!(:2!+)('6()%*E
1
B
$1
$B
6E! cos"tan x # =
7E! %&; %*9 x =
2
1$ x
B $ x<
''
"
x
#
B
8
x
8
B
!!!!!!!!!
Ax
$B
5E!! ,.9 %*9 A x =
B $ BC x <
''
"
#
B
8
Ax
"
#
$B
CE! ;$% ,.9 " x $ @# =
''
8
B
B
x < $ C x $ B=
x +@
T)!!G)*C!(:2!C23)86()82+!%?
y = ;$%$B "@ x #
6E! dy
$@
=
B $ Y x<
''dx
y = sin$1 " x 2 $ 1#
2x
2x
7E! dy =
=
2
dx
2x2 $ x4
1 $ " x 2 $ 1#
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y = "tan 2 x #
$1
5E!
y=
5
10"tan 2 x #
4
dy
2
= 5"tan$1 2 x #
=
2
dx
1$ 4 x
1$ 4 x2
$1
4
CE!!
"cos
$1
dy
=
dx 2
dy
=
dx
10 x #
1
"cos
$1
"cos
$1
"
?E!!
2
10 x # 1 $ 100 x
$5
10 x # 1 $ 100 x 2
y = %*9 ;$%$B t
y = .-;,.9 x
B
2E!! dy
=
dx < x "B $ x #
$10
#
dy
= ;$% ;$%$B t
dt
"
#
$B
B$ t<
dy
$t
=
B$ t<
'dt
Y)!!G)*C!6*9!32&6()82!2Q(32@6!%?!''y = .-;%*9 x $ x
dy
1
=
$1 = 0
dx
1$ x2
dy
x = 0, Critical points x = $11
, 1
( 0 "$1, 0# ; "0,1#
dx
y = arcsin x $ x 1
No relative extrema
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!!!!!:63C!(%!326C!7256'+2!%?!(:2!C)+(6*52E!4:2*!(:29!632!5&%+2<!(:2!+);*!)+!:63C!(%!326C!7256'+2!(:2!C3)823!:6+!(%
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!!!!!292!)+!6+!&63;2!6+!0%++)7&2E
.)'L-*,&'8 .%',+&':*33&-&9;&'$3'<'*95&-%&',.92&9,%H
<='3,
E=
@=
8 = ,.9$B $ ,.9 $B
x
x
''
/)'L-*,&'.9'&]".,*$9'3$-' d8
dx
$E=
@=
d8
= <
$ <
''dx x $ <E== x $ Y==
8
'''''M1*,'
B'O*6&'.+&.:
@='3,
N.-
x
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d8
<
<
'='=H'D*9:'x'.9:';$93*-7'"%*92',+&';.6;"6.,$-H'''''E x $ AE== = @ x $ FE== 1 x = @PHF@= ''3,H
'''''''''')+&9'
dx
!!!!
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Inverse Trig Functions Integration - 1lasswork
1?!u!)+!6!C)??232*()67&2!?'*5()%*!%?!x<!6*C!a!>!.!(:2*
du
u
= %*9 $B $ C
<
<
a
a $u
#
#' a du$ u
<
-E!!
#
dx
1$ x
sin x $ C
dx
#
YE
5 $ 2x2
a = 5 , u = x 2 , du = 2 dx
#
x$2
#
4 $ x2
x
4 $ x2
#
2
4 $ x2
x
$ 4 $ x $ 2 sin
$C
2
2
$1
dx
ZE!
5 $ 2x2
1
x 2
1
x 10
sin$1
=
sin$1
$C
5
2
5
2
x$2
dx
2
$4
#x
x
2
# x $ 4 dx $ # x $ 4 dx
ln" x $ 4 #
x
$ tan
$C
2
2
$1
2
2
2
#
ex
dx
9 $ ex
u = ex
du = e x dx
ln 9 $ e x $ C
Ê!
2
dx
# 1$ x
tan$1 x $ C
dx
#
!]E!
dx
dx $
4$x
OE!
2
a = 5 , u = x 2 , du = 2 dx
1
x 2
1
x 10
sin$1
=
sin$1
$C
5
2
5
2
[E!!
B $B u
,.9
$C
a
a
a = 2, u = x, du = dx
x
sin$1 $ C
2
$1
TE
=
dx
#
/E!
2
<
#x
x3
dx
2
$1
+
# -, x $ x
x .
0 dx
$ 1/
2
x 2 ln" x $ 1#
$C
$
2
2
2
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Inverse Trig Functions Integration - >omework
dx
# 1$ 4 x
-E!
a = 1, u = 2 x, du = 2 dx
!!!! 1 $1
tan 2 x $ C
2
# 4$
OE!
2
tan$1
#x
YE!!
dx
2
" x $ 1#
TE!
" x $ 1# $ C
2
x
dx
4
$ 16
8
tan$1
ZE!
!!
x2
$C
4
Ê!
0
2
3 2
= tan$1 3 =
0
sin$1 x
1$ x2
dx
$1
#
1
1$ x2
dx
2
$C
e 2x
dx
9 $ e4 x
Find the area of the region bounded by the curves
-.E!!!
1
, y = 0, x = 0, x = 1
y=
4 $ x2
1
dx
1$ 4 x2
tan$1 "2 x #
dt
a = 3, u = e 2 x , du = 2e 2 x dx
1 $1+ e 2 x .
tan - 0 $ C
6
, 3/
a = 1, u = 2 x, du = 2 dx
!!!! 1
#
"sin x #
]E!!
$ tan$1 "cos x # $ C
#
1$ t4
2
a = 1, u = cos x, du = $ sin xdx
3 2
t
u = sin$1 x du =
sin x
dx !
[E! #
1 $ cos2 x
!!!
#
a = 1, u = t 2 , du = 2 tdt
!!! 1 $1 2
sin t $ C
2
a = 4, u = x 2 , du = 2 xdx
!!! 1
4 $ x2
a = 2, u = x, du = dx
x
sin$1 $ C
2
a = 2, u = x $ 1, du = dx
!!! 1
dx
#
/E!
2
1
!
6
!
x
=
!!!!!! sin
20 6
$1
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Derivatives of Inverse Functions - 1lasswork
g2*236&!S3%7&2@R!G)*C!(:2!C23)86()82!%?!(:2!)*823+2!?'*5()%*!%?! f " x # at x = k .
Method 1R!!$)@0&9!?)*C)*;!(:2!)*823+2!?'*5()%*E!H:)+!=%3I+!=:2*!)(!)+!26+9!(%!;2*236(2!(:2!)*823+2!?'*5()%*E
6)!!G)*C!(:2!)*823+2!?'*5()%*!79!)*(235:6*;)*;!x!6*C!y!6*C!+%&8)*;!?%3!y
7)!!H6I2!(:2!C23)86()82!%?!(:)+!*2=!yE!H:6(!=)&&!72!(:2!C23)86()82!%?!(:2!)*823+2!?'*5()%*E
5)!!S&';!)*!9%'3!;)82*!k!86&'2
Method 29!!!U%(!?)*C)*;!(:2!)*823+2!?'*5()%*!7256'+2!)(!)+!(%%!C)??)5'&(
6)!G)*C!(:2!)*823+2!?'*5()%*!79!)*(235:6*;)*;!x!6*C!y!6*C!+%&8)*;!?%3!y
dy
7)!?)*C! !)@0&)5)(&9
dx
dy
5)!$%&82!?%3! E!1(!=)&&!72!)*!(23@+!%?!yE
dx
C)!i20&652!(:2!86&'2!%?!k ?%3!x!!)*!9%'3!)*823+2!?'*5()%*!?3%@!+(20!6)!67%82!6*C!+%&82!?%3!y
dy
2)!S&';!(:6(!86&'2!%?!y!)*(%!
dx
Txample9!1? f " x # = x 2 , x < 0 <!!?)*C!(:2!C23)86()82!%?! f $1 " x # at x = 4.
MTT>OD 1
6) y = x 2 <!+%!(:2!)*823+2!)+ x = y 2
MTT>OD 2
6)! y = x 2 <!+%!(:2!)*823+2!)+ x = y 2
dy
!!!!!!!!!!!!!!!!!(:232?%32! y = x (?)3+(!>'6C36*()!!!!!!!!!!!!!!!7)!1 = 2 y
dx
dy 1
1
7)! y 4 =
=
5)!!
dx 2 y
2 x
1
1
5)! y 4" 4 # =
=
C)! 4 = y 2 1 y = 2"quad I#
2 4 4
1
1
dy 1
=
=
=
2)!
dx 2 y 2"2# 4
f " x# = x 2
!$',.>&',+&
-&;*#-$;.6'$3
,+&'%6$#&'+&-&^
(/<!T)
$1
''''f'''' " x # = x
_$"').9,',+&
%6$#&'$3',+&
,.92&9,'6*9&'+&-&^
(T<!/)
Note9!!1(!=6+!*252++639!(%!32+(3)5(!(:2!C%@6)*!%?! f " x # !(%! x < 0!+%!(:6(!)(+!)*823+2!)+!6!?'*5()%*R!)E2E!(:6( f " x # !)+
%*2,(%,%*2!(06++2+!(:2!:%3)B%*(6&!&)*2!(2+()E
! www.MasterMathMentor.com!! "#!$%&'()%*+!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!,!-TY!,!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Illegal! (%! 0%+(! %*! 1*(23*2(
TxampleR!!G)*C!(:2!C23)86()82!%?!(:2!)*823+2!?'*5()%*!%?! f " x # = x 3 $ 4 x 2 $ 7 x $ 1 at x = 1.
a2(:%C!-!=)&&!72!(%%!C)??)5'&(E! y = x 3 $ 4 x 2 $ 7 x $ 1!+%!(:2!)*823+2!)+! x = y 3 $ 4 y 2 $ 7 y $ 1
dy
dy
1
1
= 2
!!!6)!!1 = " 3 y 2 $ 8 y $ 7#
dx
dx 3 y $ 8 y $ 7
3
2
!!!7)!!$2(! y $ 4 y $ 7 y $ 1 = 1E!!g360:)56&&9<!9%'!;2(! y = 0.349.
!!!!!!!!!!!!!!!!
!!!!!!!!!!!!!
!!!!5)!
!!!!!!!!!!!!!!!!!!!
dy
1
=
= .219
2
dx " y= .349# 3".349# $ 8".349# $ 7
TxampleR!G)*C!(:2!C23)86()82!%?!(:2!)*823+2!?'*5()%*!%?! f " x # = e x $ ln x at x = 3.
a2(:%C!-!=)&&!72!(%%!C)??)5'&(E! y = e x $ 69 x +%!(:2!)*823+2!)+! x = e y $ ln y
+
dy
y
1 . dy
dy
1
or
6)!1 = -e y $ 0 1
=
=
y
dx
ye $ 1
y / dx
dx e y $ 1
,
y
y
7)!$2(! e $ ln y = 3E!!g360:)56&&9<!9%'!;2(! y = 1.074.
5)!
dy
1
=
= .259
dx " y=1.074 # e1.074 $ 1
1.074
Note:!\?(23!9%'!;360:)56&&9!)*(23+25(<!9%'!56*!26+)&9!;2(!(:2!6*+=23!79!*V23)8(_-<o<o)
2
Txample9 G)*C!(:2!C23)86()82!%?!(:2!)*823+2!?'*5()%*!%?! y = e x , x ( 0.
Inverse Function : x = e y
$%&'()%*R!! ln x = y
2
2
y = ln x = "ln x #
1
2
$1 1
dy 1
1
= "ln x # 2 =
dx 2
x 2 x "ln x # 12
Sample Problems9!!!G)*C!(:2!C23)86()82!%?!(:2!)*823+2!?'*5()%*!%?!!('+2!@2(:%C!/!%*&9!)?!@2(:%C!-!=%*D(!=%3I)
Method I
y = x 3 $ 1 at x = 9
y = x 3 $ 5 x $ 1 at x = 5 Meth. II
y = x $ sin x at x = ! Meth. II
dy
1
3
3
=
!!7)! Inv : x = y $ 5 y $ 1 = 5 1 y = 1 !!5)!! Inv : x = y $ sin y = ! 1 y = !
!6)!! Inv : y = x $ 1 1
dx 3" x $ 1# 2 3
dy
1
1
dy
1
= 2
=
=
= DNE
dy
1
dx " y =1# 3 y $ 5 8
dx " y = !# 1 $ cos y
=
dx " x = 8# 12
! www.MasterMathMentor.com!! "#!$%&'()%*+!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!,!-TZ!,!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Illegal! (%! 0%+(! %*! 1*(23*2(
Derivatives of Inverse Functions - >omework
G%3!(:2!03%7&2@+!72&%=<!?)*C!(:2!C23)86()82!%?! f $1 !?%3!(:2!?'*5()%*! f !6(!(:2!+025)?)2C!86&'2!%?!xE!U%!56&5'&6(%3+E
-E! f " x # = x 3 $ 2 x $ 1 at x = 2
/E! f " x # = 2 x 5 $ x 3 $ 1 at x = 4
Inv : x = y 3 $ 2 y $ 1 = 2 1 y = 1
!!!! dy
1
1
= 2
=
dx " y =1# 3 y $ 2 5
Inv : x = 2 y 5 $ y 3 $ 1 = 4 1 y = 1
!!!!! dy
1
1
=
=
4
2
dx " y =1# 10 y $ 3 y
13
!
!
1
=x=
at x =
2
2
2
$1
Inv : x = sin y 1 y = sin x
1
2
dy
dy
=
1
=
or
2
dx
dx " x = ! 6#
3
1$ x
!
1
!!!!
Inv : x = sin y = 1 y =
2
6
dy
1
2
=
=
+ !.
dx -, y = 0/ cos y
3
6
OE!! f " x # = sin x
YE!! f " x # = x 3 $
$
4
x
x (0
at x = 6
TE!! f " x # = cos2 x
$
!
!
=x=
2
2
at x = 1
cos$1 x
2
dy
dy
$1
1
=
DNE or
2
dx " x = ! 6#
!!! dx " x =1# 2 1 $ x
Inv : x = cos 2 y = 1 1 y = 0
dy
$1
=
= DNE
dx " y = 0# 2 sin 2 y
Inv : x = cos 2 y 1 y =
ZE!!! f " x # = x $ 4
at x = 2
Inv : x = y $ 4 1 y = x 2 $ 4
dy
dy
= 2x 1
4
dx " x = 2#
dx " x = 2#
4
=61 y =2
y
1
1
=
=
4
3 y 2 $ 2 13
y
Inv : x = y 3 $
!!! dy
dx " y = 2#
or
!! Inv : x = y $ 4 = 2 1 y = 8
dy
=
dx " y = 8#
1
=4
1
2 y$4
G%3!(:2!03%7&2@+!72&%=<!?)*C!(:2!C23)86()82!%?! f $1 !?%3!(:2!?'*5()%*!f!6(!(:2!+025)?)2C!86&'2!%?!xE!c+2!56&5'&6(%3+E
[E!! f " x # = x 3 $ 2 x 2 $ 5 x $ 1
at x = 2
Inv : x = y 3 $ 2 y 2 $ 5 y $ 1 = 2 1 y = .737
dy
1
= 2
= .272
dx " y = .737# 3 y $ 4 y $ 5
x
$ sin 2 x at x = 3
2
y
Inv : x = $ sin 2 y = 3 1 y = 4.309
2
!! dy
1
=
= 0.818
1
dx " y = 4.309#
$ 2 sin y cos y
2
Ê!! f " x # =
]E!! f " x # = 3 3 x $ 5
at x = $3
Inv : x = 3 3 y $ 5 = $3 1 y = $7.333
]E!
dy
=
dx " y =$7.333#
1
=9
1
23
"3y $ 5#
-.E! f " x # = xe cos x
at x = 3
Inv : x = ye cos y = 3 1 y = 4.335
1
-.E!! dy
= cos y
= 0.287
dx " y = 4.335# ye "$ sin y # $ e cos y
! www.MasterMathMentor.com!! "#!$%&'()%*+!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!,!-T[!,!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Illegal! (%! 0%+(! %*! 1*(23*2(
1urves Defined by Parametric Tquations - 1lasswork
c*()&!*%=<!=2!:682!722*!32032+2*()*;!;360:+!79!+)*;&2!2>'6()%*+!)*8%&8)*;!863)67&2+!x!6*C!yE!42!=)&&!*%=!+('C9
03%7&2@+!=)(:!=:)5:!O!863)67&2+!632!'+2C!(%!32032+2*(!5'382+E!H:2+2!2>'6()%*+!632!56&&2C!parametric equations.
$'00%+2!6!;%&?23!+(3)I2+!6!;%&?!76&&!(:6(!)+!03%02&&2C!)*(%!(:2!6)3!6(!6*!6*;&2!%?!TY%E!1?!(:2!)*)()6&!82&%5)(9!%?!(:2!76&&
$x 2
)+!ZT!?22(!023!+25%*C<!(:2!%7X25(!?%&&%=+!(:2!06367%&)5!06(:!;)82*!79! y =
$ xE
128
d%=2823<!6&(:%';:!9%'!:682!(:2!06(:!%?!(:2!%7X25(<!9%'!C%!*%(!I*%=!when!(:2!%7X25(!)+!6(!6!;)82*!()@2E!1*!%3C23!(%
C%!(:)+<!=2!)*(3%C'52!6!(:)3C!863)67&2!t<!56&&2C!6!parameterE!"%(:!863)67&2+!x!6*C!y 632!=3)((2*!6+!6!?'*5()%*!%?!t<
6*C!9%'!%7(6)*!(:2!0636@2(3)5!2>'6()%*+R
x = 32 t 2
and
y = $16 t 2 $ 32 t 2
G3%@!(:)+!+2(!%?!2>'6()%*+<!=2!56*!C2(23@)*2!(:6(!6(!(:2!()@2!t!=!.<!(:2!76&&!)+!6(!(:2!0%)*(!(.<!.)E!$)@)&63&9!6(!(:2
()@2!t!=!-<!(:2!76&&!)+!6(!(:2!0%)*(! 32 2 , 32 2 $ 16 E
"
#
Definition of a Plane 1urve
1?!f!6*C!g!632!5%*()*'%'+!?'*5()%*+!%?!t!%*!6*!)*(2386&!I<!(:2*!(:2!2>'6()%*+
!!!!!!!!!!!!!!!!! x = f " t # and y = g" t #
632!56&&2C!parametric equations!6*C!t!)+!56&&2C!(:2!0636@2(23E!H:2!+2(!%?
0%)*(+! " x, y # !%7(6)*2C!6+!t!863)2+!%823!(:2!)*(2386& " t1, t 2 # !)+!56&&2C!(:2!;360:
%?!(:2!0636@2(3)5!2>'6()%*+E!H:2!0636@2(3)5!2>'6()%*+!6*C!(:2!;360:!(6I2*
(%;2(:23!)+!56&&2C!6!0&6*2!5'382E
4:2*!+I2(5:)*;!6!5'382!79!:6*C!32032+2*(2C!79!0636@2(3)5!2>'6()%*+<!9%'
'+2!)*5326+)*;!86&'2+!%?!tE!H:'+!(:2!5'382!=)&&!72!(3652C!%'(!)*!6!+025)?)5
C)325()%*E!H:)+!)+!56&&2C!(:2!orientation!%?!(:2!5'382E!_%'!'+2!633%=+!(%
+:%=!(:2!%3)2*(6()%*E
PQ6@0&2!-)!$I2(5:!(:2!5'382!C2+53)72C!79!(:2!0636@2(3)5!2>'6()%*+R
3t
x = t 2 $ 1 and t = , $ 2 = t = 3
2
t $2 $1 0 1 2 3
!!!!!!! x 3
0
$1 0 3 8
y $3 $1.5 0 1.5 3 4.5
! www.MasterMathMentor.com!! "#!$%&'()%*+!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!,!-T]!,!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Illegal! (%! 0%+(! %*! 1*(23*2(
PQ6@0&2!/)!$I2(5:!(:2!5'382!C2+53)72C!79!(:2!0636@2(3)5!2>'6()%*+R
x = 4 t 2 $ 1 and t = 3t, $ 1 = t = 1.5
t $1 $0.5 0 0.5 1 1.5
0
$1 0 3 8
!!! x 3
y $3 $1.5 0 1.5 3 4.5
!!!!!!!!!!!!!!!!!!!!!!!!!!
U%(2!(:6(!7%(:!2Q6@0&2+!(3652!%'(!(:2!2Q65(!+6@2!;360:E!"'(!(:2!+022C!)+!C)??232*(E!PQ6@0&2!/D+!;360:!)+!(3652C
%'(!@%32!360)C&9E!H:'+!)*!600&)56()%*+<!C)??232*(!0636@2(3)5!2>'6()%*+!56*!72!'+2C!(%!32032+2*(!863)%'+!+022C!6(
=:)5:!%7X25(+!(3682&!6&%*;!06(:+E!!G)*C)*;!6!325(6*;'&63!2>'6()%*!(:6(!32032+2*(+!(:2!;360:!%?!6!+2(!%?!0636@2(3)5
2>'6()%*+!)+!56&&2C!eliminating the parameterE!d232!)+!6!+)@0&2!2Q6@0&2!%?!2&)@)*6()*;!(:2!0636@2(23E
PQ6@0&2!O)!!P&)@)*6(2!(:2!0636@2(23!)*! x = t 2 + 3
!!!!!
e!$%&82!?%3!t!)*!(:2!+25%*C!2>'6()%*+
y
!!! t =
2
and
y = 2t
!!!!!e!$'7+()('(2!)*!(:2!+25%*C!2>'6()%*+!6*C!+)@0&)?9
y2
x
=
$ 3 1 y = ) 4 x $ 12
!!!!!!
4
1
t
and y =
, t ( $4
PQ6@0&2!T)!V2+53)72!(:2!5'382!32032+2*(2C!79!(:2!2>'6()%*+! x =
t$4
t$4
1
1
1
t$4 = 1 t$4= 2 1 t = 2 $4
e!c+2!(:2!x!2>'6()%*<!+%&82!?%3!t
x
x
x
e!$'7+()('(2!t!)*(%!(:2!y!2>'6()%*!6*C!2&)@)*6(2!(:2!5%@0&2Q!?365()%*+E
1
$4
2
x
y=
= 1 $ 4 x 2 ... a parabola opening downward
1
$4$4
x2
e!i26&)B2!(:6(!t >,T<!@26*)*;!(:6(!! lim x " t # = 0
t 23
PQ6@0&2!Y)!$I2(5:!(:2!5'382!32032+2*(2C!79!!!!!!! x = 5 sin 8
and
e!$%&82!?%3! cos8 !!6*C! sin 8 !!)*!7%(:!2>'6()%*+E!!! sin 8 =
0 = 8 < 2!
y = 3 cos8
x
5
cos8 =
y
3
e!c+2!(:2!?65(!(:6(!''%*9 < 8 $ ;$%< 8 = B!(%!?%3@!6*!2>'6()%*!'+)*;!%*&9!x!6*C!y.
x2 y2
$
=1
25 9
e!H:)+!)+!6!;360:!%?!6*!2&&)0+2!52*(232C!6(!(.<!.)!=)(:!823()52+!6(!(Y<.!)!6*C!(,Y<!.)!6*C!@)*%3!6Q)+
!!!!!!!!!!!!!!2*C0%)*(+!6(!(.<!O)<!(,.<!,O)E!U%(2!(:6(!(:2!2&&)0+2!)+!(3652C!counterclockwise!6+ 8 ;%2+!?3%@!.!(%!/"E
e!4:6(!=%'&C!%55'3!)?!(:2!2>'6()%*+!=232! x = 5 sin 8
and
y = 5 cos8 ?
Circle
! www.MasterMathMentor.com!! "#!$%&'()%*+!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!,!-T^!,!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Illegal! (%! 0%+(! %*! 1*(23*2(
c+)*;!(:2!(25:*)>'2!)*!(:2!2Q6@0&2!67%82!9%'!56*!5%*5&'C2!(:6(!(:2!;360:!%?!(:2!0636@2(3)5!2>'6()%*+
!!!!!!!! x = h $ a sin 8
" x $ h#
2
" y $ k#
0 = 8 < 2! )+!(:2!2&&)0+2!((3652C!5&%5I=)+2)!;)82*!79
y = k $ b cos8
and
2
= 1 =:)5:!:6+!52*(23! " h8 k # !6*C!2*C0%)*(+! " a, 0#, "$a, 0#, "0, b# and "0, $b# .
''
a
b2
P&)@)*6()*;!(:2!0636@2(23!)+!6*!6)C!(%!5'382!+I2(5:)*;E!1?!(:2!0636@2(3)5!2>'6()%*+!32032+2*(!(:2!06(:!%?!6!@%8)*;
%7X25(<!(:2!;360:!6&%*2!)+!*%(!+'??)5)2*(!(%!C2+53)72!(:2!%7X25(D+!@%()%*E!_%'!+()&&!*22C!(:2!0636@2(3)5!2>'6()%*+!(%
(2&&!9%'!(:2!position<!direction<!6*C!speed!6(!6!;)82*!()@2E
2
$
PQ6@0&2!Z)!G)*C!6!+2(!%?!0636@2(3)5!2>'6()%*+!(%!32032+2*(!(:2!;360:!%?! y = x $ x 2 '+)*;!265:!%?!(:2!?%&&%=)*;
dy
!!!!!!!!0636@2(23+R!!!!6)! t = x !!!!!!!!!!7E!!(:2!+&%02! m = 6(!(:2!0%)*(! " x, y #
dx
2
!!!!!!6)!H:)+!%*2!)+!26+9E!p'+(!&2(! t = x !)*!(:2!2>'6()%*R!!! x = t, y = t $ t
1$ m
dy
dy
= m = 1$ 2x 1 x =
!!!!!!7)!!$)*52! m = <!C)??232*()6(2!(:2!2>'6()%*!6*C!+%&82!?%3!x
2
dx
dx
2
1 $ m +1 $ m . m $ m 2
$U%=!+'7+()('(2!?%3!x!)*!(:2!%3);)*6&!2>'6()%*E!! y =
0 =
, 2 /
2
4
!!!!!!!U%(2!(:6(!=:2*!;360:2C!)*!0636@2(3)5!@%C2<!(:2!5'382!:6+!6!3);:(!(%!&2?(!%3)2*(6()%*!C2(23@)*2C!79
!!!!!!!!!!!!!!!!!!!(:2!C)325()%*!%?!)*5326+)*;!86&'2+!%?!+&%02!mE!G%3!063(!6)<!(:2!5'382!:6+!(:2!%00%+)(2!%3)2*(6()%*E
PQ6@0&2![)!\(!6*9!()@2!t!=)(:! = = t = B= <!(:2!5%%3C)*6(2+!%?!S!632!;)82*!79!(:2!0636@2(3)5!2>'6()%*+R
x = t $ 2 sin t and
y = 2 $ 2 cos t
!!!!!!!!$I2(5:!(:)+!'+)*;!9%'3!56&5'&6(%3E
t=3
t=9
t=4
t=2
t=10
t=8
t=1
t=5
t=7
t=0
t=6
G'W$*9,%';$--&%#$9:*92',$'*9,&2&-'5.6"&%'$3','.-&'
''%+$)9H'Q,','='B8'W'+.%';$$-:*9.,&%'$3'(0HCP8H'Y<)a'.,'
'',+*%'*9%,.9,8'W'*%'+&.:*92'.67$%,':"&'9$-,+H
G'4+&'3"66';"-5&'*%'9$,',+&'2-.#+'$3'.'3"9;,*$9a'%$7&
'''1'5.6"&%'+.5&'7$-&',+.9'$9&'('5.6"&H
G'4+&'#*;,"-&'%+$)%',+&'10'.9:'(0'.1&%'/",'9$',0.1*%H
G'4+&'/"66&,%'$9',+&'2-.#+'.##&.-'.,'&]".6',*7&'
'''*9,&-5.6%'/",'9$,'.,'&]".6':*%,.9;&%'3-$7'&.;+'$,+&-'
'''/&;."%&'W'%#&&:%'"#'.9:'%6$)%':$)9'.%'*,'7$5&%H'
'''L&')*66'%$$9'%&&'+$)',$';.6;"6.,&',+&'%#&&:'$3'.
'''#.-.7&,-*;';"-5&'.,'.'#$*9,H
PQ6@0&2!])!S636@2(3)5!5'382+!@69!:682!&%%0+<!5'+0+<!823()56&!(6*;2*(+!6*C!%(:23!025'&)63!?26('32+E
and
0=t=5
!!!!!!!g360:!6)!! x = 2 cos t $ 2 cos" 4 t #
y = sin t $ sin" 4 t #
0 = t = 2!
!!!!!!!7)!! x = sin"5 t # and y = sin"6 t #
t=5
!!!!!!!!!!!!!6E!!
t=0
!!!!!!!!!!!!!!!!!!!7E!!!!
!!H:)+!)+!56&&2C!6!Lissajou curveE
! www.MasterMathMentor.com!! "#!$%&'()%*+!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!,!-Y.!,!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Illegal! (%! 0%+(! %*! 1*(23*2(
1urves Defined by Parametric Tquations - >omework
-E!!!#%*+)C23!(:2!0636@2(3)5!2>'6()%*+! x = t
and
y = 2t $ 1
6)!#%@0&2(2!(:2!(67&2
!!!!!!!!!!!!!!!!!
7)!S&%(!(:2!0%)*(+!(x<!y)!)*!(:2!(67&2!6*C!+I2(5:!6!;360:!%?!(:2!0636@2(3)5!2>'6()%*+E!1*C)56(2!(:2
!!!!%3)2*(6()%*!%?!(:2!;360:E
2
2
5)!G)*C!(:2!325(6*;'&63!2>'6()%*!79!2&)@)*6()*;!(:2!0636@2(23E!! x = t 1 y = 2 x $ 1, x < 0
%$! ! (
/E!!!#%*+)C23!(:2!0636@2(3)5!2>'6()%*+! x = 4 cos8 and y = 6 sin 2 8 !%*! ' , *
& 2 2)
6)!#%@0&2(2!(:2!(67&2!!
7)!S&%(!(:2!0%)*(+!(x<!y)!)*!(:2!(67&2!6*C!+I2(5:!6!;360:!%?!(:2!0636@2(3)5!2>'6()%*+E!1*C)56(2!(:2
!!!!%3)2*(6()%*!%?!(:2!;360:E
5)!G)*C!(:2!325(6*;'&63!2>'6()%*!79!2&)@)*6()*;!(:2!0636@2(23E
x
y2
x2
y2
cos2 8 = ,sin 2 8 =
1
$
=1
16
6
256 6
OE!!1*!(:2!?%&&%=)*;!2Q235)+2+<!2&)@)*6(2!(:2!0636@2(23!6*C!5%*?)3@!;360:)56&&9!(:6(!(:2!325(6*;'&63!2>'6()%*+
!!!!!9)2&C!(:2!+6@2!;360:!6+!(:2!0636@2(3)5+E!"2!+'32!9%'!(6I2!C%@6)*!6*C!36*;2!%?!(:2!0636@2(3)5!)*(%!655%'*(E
6E! x = 4 t $ 1 and
y = 2 t $ 3 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!7E!!! x = t $ 3 and
y = t2
+ y $ 3.
y$3
1 x = 40 $1
2
, 2 /
2
!!!!!! t = x $ 3 1 y = " x $ 3#
!!!!!!!
x$7
x = 2y $ 6 $1 1 y =
2
2
3
5E!! x = t and y = 3 $ t !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!CE!!! x = t 2 $ t and y = t 2 $ t
t=
t 2 = x $ t, t 2 = y $ t 1 x $ t = y $ t 1 t =
3
6
!!!!!!!!! x = t 1 y = 3 $ x
!!
y$x
2
2
+ y $ x. y $ x
y =0 $
, 2 /
2
(Cannot solve for y )
! www.MasterMathMentor.com!! "#!$%&'()%*+!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!,!-Y-!,!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Illegal! (%! 0%+(! %*! 1*(23*2(
t
t $2
x$2
t = x$21 y =
x
x = t $ 2 and
2E!!
x = t$3
y=
?E!!
;E!
and
y=t$3
t = y $ 3 1 x = y $ 3$ 3 1 x = y $6
y = x $ 6 or y = 6 $ x, x < 0
x = cos8
x = sec 2 8
and
y = tan 2 8
x = 1 $ tan 2 8 1 x = 1 $ y 1 y = x $ 1
:E!!
and
y = 4 sin 8
x 2 = cos2 8, y 2 = 16 sin 2 8 1
y2
$ x2 =1
16
y = )4 1 $ x 2
TE!!c+2!9%'3!56&5'&6(%3+!(%!;360:!(:2!5'382!32032+2*(2C!79!(:2!0636@2(3)5!2>'6()%*+E!1*C)56(2!(:2!%3)2*(6()%*!%?!(:2
5'382E!1C2*()?9!6*9!0%)*(+!6(!=:)5:!(:2!5'382!)+!*%(!+@%%(:E!!Do not take these problems lightly. Your
task is to come up with an appropriate window to view them. Let your t run from 0 to 2!, 4!, 8!,
etc.
!
!
6E!#95&%)CR!!H:2!5'382!(3652C!79!6!0%)*(!%*!(:2!!!!!
!!!5)35'@?232*52!%?!6!5)35&2!6+!)(!3%&&+!%*!6!+(36);:(!&)*2E
7E!S3%&6(2!#95&%)CR!$6@2!6+!6)!2Q520(!(:2!0%)*(
!!!;%2+!72&%=!(:2!&)*2!(36)&3%6C!(365I)
!!!!!!!! x = 2"8 $ sin 8 #
!!!! x = 28 $ 4 sin 8 and y = 2 $ 4 cos8
and
y = 2"1 $ cos8 #
!!!!!!!!!!!!!!!
5E!d90%595&%)CR! x = 3 cos3 8 and y = 3 sin 3 8 !!!!!!!!!!!!!!!!!!CE!!#'3(6(2!595&%)CR! x = 28 $ sin 8 and y = 2 $ cos8
!!!!!!!!!!!!!!!
2E!4)(5:!%?!\;*2+)R! x = 2 cot 8 and y = 2 sin 2 8
?E!!G%&)'@R! x =
3t
3t 2
and
y
=
1$ t3
1$ t3
!!!!!!!!!!!!!
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Using 1alculus With Parametric Tquations - 1lasswork
1?!6!+@%%(:!5'382!C!)+!;)82*!79!(:2!2>'6()%*+
x = f " t # and y = g" t # <!(:2*!(:2!+&%02!%?!C!6(! " x8 y# !)+
'
dy
dy
= dt , dx dt > 0
dx dx
dt
!!U%(2!(:6(!(:2!+&%02!?%3@'&6!)+!)*!(23@+!%?!t<!*%(!xE!+%!9%'!*22C
(%!?)*C!(:2!86&'2!%?!t!5%332+0%*C)*;!(%!(:2!0%)*(!(x<!y)
PQ6@0&2!-)!G)*C!
dy
?%3!(:2!5'382!;)82*!79! x = cos t and y = $ sin t
dx
dy $ cos t
=
= cot t
dx $ sin t
!!!!!!!!G)*C!(:2!+&%02!%?!(:)+!5'382!6(!(:2!0%)*(!(-<!.)!6*C!(.<!,-)
?
! 3!
B x = 0, t = 0, 2 , 2
? x = 1, t = 0
1 cot t DNE @
1 cot t = 0
@
A y = 0, t = 0
B y = $1, t = !
A
2
dy
"256'+2 )+!6!?'*5()%*!%?!t!<!9%'!56*!'+2!(:2!3'&2!67%82!32026(2C&9!(%!?)*C
dt
higher-order derivatives. !G%3!)*+(6*52R
d %d2y (
d % dy (
'
*
d 3 y d % d 2 y ( dt & dx 2 )
d 2 y d % dy ( dt '& dx *)
=
=
=
'
* = dx
dx
dx 3 dx & dx 2 )
dx 2 dx '& dx *)
dt
dt
2
d y
G)*C!(:2!+25%*C!C23)86()82! 2 !%?!(:2!0636@2(3)5!2>'6()%*!67%82E
dx
d 2 y $ csc 2 t
1
=
= 3
2
dx
$ sin t sin t
1 2
"t $ 2t # <!?)*C!(:2!+&%02!6*C!5%*568)(9!6(!(/<!T)
2
First, you need to find the value of t when
1
t = 2 and y = " t 2 $ 2 t # = 4. You know that from the x - equation, t = 4.
2
And t = 4 satisfies the y - equation as well.
PQ6@0&2!/)!?%3!(:2!5'382!;)82*!79! x = t and y =
+
1 .
2- t1 2 $ 1 2 0
dy
t $1
d y
,
2t /
=
= 2" t 3 2 $ t1 2 #, 2 =
= 4t $ 4
1
1
dx
dx
$1 2
$1 2
t
t
2
2
2
slope = 12,concavity = 12
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M1.7#6&'@)''4+&'#.-.7&,-*;';"-5&'*%'2*5&9?
x = t $ 2 sin t and y = 2 $ 2 cos t
dy
dx
dy
$2 sin t
= $2 sin t, = 1 $ 2 cos t, =
dt
dt
dx 1 $ 2 cos t
6)!?)*C!(:2!+&%02!6(!t!=!-E
dy
$2 sin1
=
= 20.879
!!!!!!!!!!!!!!!! dx 1 $ 2 cos1
!!!!!7)!4:232!)+!(:2!5'382!:%3)B%*(6&M
5)!=:232!)+!(:2!5'382!823()56&M
2 sin t = 0
1 $ 2 cos t = 0 1 cos t = 1 2
t = 0, !, 2!,...
t = ! 3, 5! 3, 7! 3,...
42!I*%=!=2!56*!?)*C!(:2!635!&2*;(:!%?!6!5'382!C!;)82*!79! y = h " x # !%823!(:2!)*(2386&! & x1, x 2 % !79!!
x2
s=
#
2
x2
2
1 $ & h 4" x #% dx =
+ dy .
1 $ - 0 dx
, dx /
#
x1
x1
1?!C#)+!32032+2*(2C!79!(:2!0636@2(3)5!2>'6()%*+! x = f " t #
x2
!s=
#
x2
2
1 $ & h 4" x #% dx =
x1
#
x1
x2
Arc Length
#
s =
x1
2
+ dy .
1 $ - 0 dx <!9%'!56*!=3)(2
, dx /
2
+ dy .
1 $ - 0 dx =
, dx /
2
b
=
"dx dt # $ "dy
#
2
"dx dt #
a
b
=
and y = g" t #, a = t = b
#
a
2
2
2
x2
+ dy dt .
1$ 0 dx
, dx dt /
#
x1
2
dt # + dx .
- 0dt =
, dt /
+ dx . + dy .
- 0 $ - 0 dt =
, dt / , dt /
b
b
#
a
2
"dx dt #
2
2
$ " dy dt # + dx .
- 0dt
+ dx .
, dt /
- 0
, dt /
# & f 4"t #% $ &g4"t #%
2
dt
a
Note that this formula only works when the curve does not intersect itself
on the interval a = t = b and the curve must be smooth.
PQ@0&2!T)!\!5)35&2!%?!36C)'+!r!:6+!325(6*;'&63!2>'6()%*! x 2 $ y 2 = r 2 !6*C!0636@2(3)5!2>'6()%*+!
!!!!!!!!!!!!!!!!!! x = r cos8 y = r sin 8 E!$:%=!(:6(!)(+!635!&2*;(:!(5)35'@?232*52)!)+!/"!'+)*;
!!!!!!6)!i25(6*;'&63!635!&2*;(:!?%3@'&6 !!!!!!!!!!!!!!!!!!!!!!!!!!7)!S636@2(3)5!635!&2*;(:!?%3@'&6
dy
$x
y = r2 $ x 2 1
=
dx
1$ x2
dx
dy
r
r
= $r sin 8, = r cos8
x2
r2
dt
dt
A = 4 # 1$ 2
dx = 4 #
dx
2
2
2
r
$
x
r
$
x
!
!2
2
0
0
2
2
2
2
4
A = 4 # r sin 8 $ r cos 8 d8 = # r d8
r
!!!!!!!!!!!!!!!!!!!!!!
% $1 x (
1
0
0
A = 4r #
dx
=
r
4
sin
'&
r2 $ x 2
r *)0
0
+!.
!2
A = 4 r8 % 0 = 4- 0 = 2!
+
.
!
,2/
A = 4 r sin$1 1 $ 4 r sin$1 0 = 4 r- 0 = 2!
,2/
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PQ6@0&2!Y)!!\!5)35&2!%?!36C)'+!-!3%&&+!63%'*C!(:2!5)35'@?232*52!%?!6!&63;23!5)35&2!%?!36C)'+!YE!H:2!20)595&%)C
!!!!!!!!!!!!!!!!!!!!!(3652C!79!6!0%)*(!%*!(:2!5)35'@?232*52!%?!(:2!+@6&&23!5)35&2!)+!;)82*!79
x = 5 sin t $ sin 5 t
and
y = 5 cos t $ cos 5 t
.)'V-.#+',+&'#.-.7&,-*;'&]".,*$9%'3$-'
&$6, 6%, &$6, 6%
t = 0 to 2!
b*&)'*,'c$$7'%]".-&H
!!!!!!!
7)!U%(2!(:2!5'382!:6+!5'+0!0%)*(+!=:2*! t = 0 and t = ! 2 E!"2(=22*!(:2+2!0%)*(+<! dx dt and dy dt 632
!!!!*%(!+)@'&(6*2%'+&9!B23%!(5'+0+)E!$%!(:2!0%3()%*!%?!(:2!5'382!;2*236(2C!?3%@! t = 0 and t = ! 2 !)+
!!!!+@%%(:E!$%<!(%!?)*C!(:2!(%(6&!C)+(6*52!(3682&2C!79!(:2!0%)*(!=2!56*!?)*C!(:2!635!&2*;(:!)*!(:2!-+(
!!!!>'6C36*(!6*C!@'&()0&9!(:2!32+'&(!79!TE!4:)&2!)(!@69!72!0%++)7&2!X'+(!(%!)*(2;36(2!?3%@!.!(%!/"!(6*C!)(!)+!)*
!!!!!!!!!!!!!!!!!(:)+!56+2<!)(!56*!72!C6*;23%'+E!1(!)+!26+)23!(%!)*(2;36(2!%*!+25()%*+!9%'!632!+'32!)(!)+!C)??232*()67&2E
!2
L=4#
"5 cos t $ 5 cos 5t #
2
2
$ "$5 sin t $ 5 sin 5 t # dt
0
!2
L = 4 # 25 cos2 t $ 50 cos t cos 5 t $ 25 cos2 5 t $ 25 sin 2 t $ 50 sin t sin 5 t $ 25 sin 2 5 tdt
0
!2
!2
!2
L = 4 # 50 $ 50 sin t sin 5 t $ 50 cos t cos 5 tdt = 20 # 2 $ 2 sin t sin 5 t $ 2 cos t cos 5 t dt = 20 # 2 $ 2 cos 4 tdt
0
0
!2
0
!2
!2
L = 20 # 2 $ 2 cos 2"2 t # dt = 20 # 2 $ 2"1 $ 2 sin 2 2 t # dt = 20 # 2 sin 2 t dt = 40
0
0
0
Area of a surface of revolution
1?!6!+@%%(:!5'382!C!)+!;)82*!79! x = f " t # and y = g" t #, a = t = b <!6*C!C C%2+!*%(!)*(23+25(!)(+2&?< !(:2*!(:2!6326
%?!(:2!+'3?652!%?!328%&'()%*!67%'(!(:2!5%%3C)*6(2!6Q2+!)+!;)82*!79R
t= b
S = 2!
# g"t #
t= a
t= b
S = 2!
# f "t #
t= a
2
2
2
2
+ dx . + dy .
- 0 $ - 0 dt - Revolution about x - axis : g" t # < 0
, dt / , dt /
+ dx . + dy .
- 0 $ - 0 dt - Revolution about y - axis : f " t # < 0
, dt / , dt /
PQ6@0&2!Z)!G)*C!(:2!+'3?652!6326!=:2*! x = 4 sin t and y = 4 cos t
!3
S = 2! # 3 sin t
"$3 sin t #
2
0=t=
!
!)+!3%(6(2C!67%'(!(:2!x!6Q)+E
6
2
$ " 3 cos t # dt
0
!3
!3
S = 2! # 3 sin t " 3# dt = 18! # sin t dt
0
0
+1 .
!3
S = 18!&$ cos t % 0 = $18!- $ 10 = 9!
,2 /
! www.MasterMathMentor.com!! "#!$%&'()%*+!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!,!-YY!,!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Illegal! (%! 0%+(! %*! 1*(23*2(
Using 1alculus With Parametric Tquations - >omework
2
1*!(:2!?%&&%=)*;!2Q235)+2+<!?)*C! dy dx and d y
dx 2
6*C!286&'6(2!265:!6(!(:2!)*C)56(2C!86&'2!%?!(:2!0636@2(23EE
x= t
x = 3t
and
y = 4t $ 1
t =2
-E! dx
dy
dy 4 d 2 y
=3, =41
= ,
=0
dt
dt
dx 3 dx 2
and
y = 4t $1
t=4
2
dx
1 dy
dy
d 2 y 8"1 2 t #
=
, =41
= 8 t, 2 =
=8
!!!!!!!!!!!!!!!!!!!!!!!/E!
dt 2 t dt
dx
dx
1 2t 2
dy
d2y
= 16, 2
=8
dx & t = 4 %
dx & t = 4 %
5!
x = 2 t $ 2 and y = t $ 4 t
t =1
4
2
$1
dx
dy
dy 2 t $ 4
d y 1
dx
dy
dy
d2y
,
=
$
2
sin
=
2
cos
1
=
$
cot
,
=
2
,
=
2
$
4
1
=
=
$
2
,
=
t
t
t
t
t 2=
OE!
!!!TE
2
2 sin 3 t
2
2
dt
dt
dx
dx
dt
dt
dx
dx
dy
d2y
1
dy
d2y
= $1, 2
= 2
= $1, 2
=
dx & t =1%
dx & t =1% 2
dx & t = 5 ! 4 %
dx & t = 5 ! 4 %
x = 2 cos t
2
x= t
and
y = t$2
and
y = 2 sin t
t=
t =1
dx
1 dy
1
dy
=
, =
1
=
dt 2 t dt 2 t $ 2
dx
1
t
$1
"
2
d 2 y 2 t $ 2 " t $ 2#
YE!! 2 =
1
dx
2 t
t
t$2
x = sin 3 t
!!!!!!!
dy
1 d2y
$1 1
=
, 2
=
dx & t =1%
3 dx & t =1% 9 3
and
y = cos3 t
t=
3!
4
dx
dy
= 3 sin 2 t cos t , = $3 cos2 t sin t
dt
dt
!ZE!!
2
csc 2 t
dy
d y
1
= $ cot t, 2 =
=
2
4
dx
dx
3 sin t cos t 3 sin t cos t
2
$8
dy
d y
= 1, 2
=
dx & t = 3 ! 4 %
dx & t = 3 ! 4 % 3 2
1*!(:2!?%&&%=)*;!/!2Q235)+2+<!?)*C!(:2!2>'6()%*!%?!(:2!(6*;2*(!&)*2!6(!(:2!)*C)56(2C!0%)*(!%*!(:2!5'382E
2
and y = 2 sin 2 8 at "2,1#
tan 8
dx
$2 dy
= 2 , = 4 sin 8 cos8
dt sin 8 dt
x=
[E!
dy
= $2 sin 3 8 cos8,
dx
y $1 =
+ 2 . + 2 . $1 !!
dy
!
!
x = 2,8 = , y = 1,8 = 1
= $2- 0 - 0 =
dx & t =! 4 %
4
4
, 2 /, 2 / 2
3
$1
" x $ 2#
2
! www.MasterMathMentor.com!! "#!$%&'()%*+!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!,!-YZ!,!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Illegal! (%! 0%+(! %*! 1*(23*2(
(
x = !4 cos$ and y = 3 + 2 sin $ at !2,3 + 3
)
dx
dy
= 4 sin $ , = 2 cos$
dt
dt
8. dy 1
dy
"
"
3
x = !2,$ = , y = 3 + 3,$ = #
= cot $,
=
dx 2
dx [ t =" 3 ]
3
3
6
(
)
y ! 3+ 3 =
3
( x + 2)
6
In the following exercises, find all points of horizontal and vertical tangency to the curve.
x = 2 ! t and
y = t2
dx
dy
= !1, = 2 t
9. dt
dt
Horiz : t = 0 Point (2,0)
Vert : None
x = 8 cos2 $ and
y = !4 sin $
dx
dy
= 16 cos$ sin $, = !4 cos$
dt
11. dt
Horiz : None
Vert : t = 0, ", 2",…
Point : (8, 0)
x = t 2 + t ! 4 and
y = 2t 3 ! 6t
dx
dy
= 2 t + 1, = 6 t 2 ! 6
dt
dt
13.
Horiz : t = ±1 Points : (!2, !4 ), (!4, 4 )
!1
Vert : t = ,
2
% !17 11(
Point : '
, *
& 4 4)
x = t + 4 and
y = 2t 2 + 6t + 1
dx
dy
= 1, = 4 t + 6
dt
dt
10.
% 5 !7 (
!3
Horiz : t =
Point : ' , *
&2 2 )
2
Vert : None
x = 2$ and
y = 2(1 ! cos$ )
dx
dy
= 2, = 2 sin $
12. dt
dt
Horiz : t = 0, ", 2",… Points : (0, 0), (2", 4 ), ( 4 ", 0), (6", 4 ),...
Vert : None
x = 2 cos$ and
y = !2 sin 2$
dx
dy
= !2 sin $, = !4 cos 2$
dt
dt
14.
" 3" 5"
Horiz : t = , , ... Pts : 2 , !2 , ! 2 , 2 ,
4 4 4
Vert : t = 0, ", 2",… Point : (2, 0), (!2, 0)
(
- 157 -
)(
)(
)(
)
2 , !2 , ! 2 , 2
G)*C!(:2!635!&2*;(:!%?!(:2!;)82*!5'382!%*!(:2!)*C)56(2C!)*(2386&E!#6&5'&6(%3+!023@)((2C!%*!-Y!6*C!-ZE
x = t 2 $ t and
-YE!
y = 4t3 + 2
,%
&$11
x= t
dx
dy
= 2 t $ 1, = 12 t 2
dt
dt
-ZE!
1
L=
4 t 2 $ 4 t $ 1 $ 144 t 4 dt = 8.842
#
$1
x = 2e$ t sin t and
y = 2e$ t cos t
!
L=
# &
2
% &
2e$ t "cos t $ sin t # $
0
!
L=
and
y=3 t
&1, 2%
dx
1 dy
1
= 12, = 23
dt 2 t dt 3t
2
1
1
L= #
$ 4 3 dt = 0.489
4 t 9t
1
&0, !%
%
2e$ t "$ sin t $ cos t #
2
dt
2e$2 t "cos2 t $ 2 sin t cos t $ sin 2 t # $ 2e$2 t "sin 2 t $ 2 sin t cos t $ cos2 t # dt
#
0
!
L=
-[E!!
4 e$2 t dt
#
Expand - cancellation and sin 2 t $ cos2 t = 1
0
!
L = 2 # e$ t dt = &$2e$ t %
0
!
0
+1 . +
1.
L = $2- ! $ 10 = 2-1 $ ! 0
,e
/ , e /
t5
1
x = $ 3 and
y=t
10 6 t
dx t 4
1 dy
= $ 4,
=1
dt 2 2 t
dt
3
L=
#
1
3
L=
-]E!!!
#
1
t8 1
1
$ $ 8 +1 dt =
4 2 4t
2
+ t4
1 .
- $ 4 0 dt =
, 2 2t /
&1, 3%
3
#
1
t8 1
1
$ $ 8 dt
4 2 4t
3
+ t4
1 .
# -, 2 $ 2t 4 0/ dt
1
3
% t5
1 ( 243 1
1 1
L = ' $ 3* =
$
$ $ = 24.361
&10 6 t )1 10 162 10 6
The graph of this parametric is a horizontal line whose length is x " 3# $ x "1#.
! www.MasterMathMentor.com!! "#!$%&'()%*+!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!,!-Y]!,!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Illegal! (%! 0%+(! %*! 1*(23*2(
-Ê!!H:2!06(:!%?!6!+%5523!76&&!)+!@%C2&2C!79!(:2!!2>'6()%*+! x = "100 cos 30 o # t and y = "100 sin 30 o # t $ 16 t 2
!!!!!!!!!=:232!x!6*C!y!632!@26+'32C!)*!?22(E!!!g360:!(:2!06(:!%?!(:2!03%X25()&2!6*C!'+2!(:2!)*(2;36()%*!56067)&)()2+
!!!!!!!!%?!(:2!56&5'&6(%3!(%!6003%Q)@6(2!(:2!635!&2*;(:!%?!(:2!06(:!6*C!(:2!C)??232*52!72(=22*!(:2!635!&2*;(:
!!!!!!!!6*C!(:2!36*;2!%?!(:2!+%5523!76&&E
Range : 100 sin 30 o t $ 16 t 2 = 0
3.125
L=
t "50 $ 16 t = 0# 1 t = 3.125 sec
!!!!!!
x = 100 cos 30 o " 3.125# = 270.633 ft
o 2
# "100 cos 30 # $ "50 $ 32t #
2
dt
0
!! L = 284.994 ft C Difference = 14.361 ft
3t
3t 2
and
y
=
, t < 0 <!+I2(5:!)(!%*!9%'3!56&5'&6(%3E!!H:)+!;360:
1$ t3
1$ t3
!!!!!!!)+!56&&2C!6!?%&)'@E!1(!&%%I+!&)I2!6!&26?!(?%&)6;2)E
/.E!!g)82*!(:2!0636@2(3)5!2>'6()%*+! x =
!!!!!!!!!6)!V2(23@)*2!=:6(!86&'2+!%?!t!;)82!6!:%3)B%*(6&!(6*;2*(E
3
3
2
2
dy "1 $ t #"6 t # $ 3t " 3t # 6 t $ 6 t 4 $ 9 t 4 3t "2 $ t #
=
=
=
2
2
2
!!! dt
"1 $ t 3 #
"1 $ t 3 #
"1 $ t 3 #
Horizontal tangents at t = 0, t = 3 2
!!!!!!!!!7)!\003%Q)@6(2!(:2!635!&2*;(:!%?!(:2!5&%+2C!&%%0E!$2(!'0!(:2!)*(2;36&!6*C!'+2!9%'3!56&5'&6(%3!6003%03)6(2&9
3
L=
#
0
2
2
% d + 3t .( % d + 3t 2 .(
$' dt = 4.915
' 3 0*
3 0*
& dt ,1 $ t /) & dt ,1 $ t /)
/-E!!G)*C!(:2!6326!%?!(:2!+'3?652!;2*236(2C!79!328%&8)*;!(:2!5'382!67%'(!(:2!;)82*!6Q2+E!!!c+2!56&5'&6(%3!%*!6)E
x = 6 cos8 and y = 6 sin 8,
x = t and y = 8 $ 4 t,
i) x $ axis
6)!
2
A = 2! # "8 $ 4 t # 17 dt
0
A = 207.25
&0, 2%
i) x $ axis
ii) y $ axis
2
A = 2! # t 17 dt
0
A = 51.812
!2
7E!! A = 2! # 6 sin t 36 dt
0
!2
&0,! 2%
ii) y $ axis
!2
A = 2! # 6 cos t 36 dt
0
!2
A = 72! # sin t dt
A = 72! # cos t dt
A = 72!
A = 72!
0
0
! www.MasterMathMentor.com!! "#!$%&'()%*+!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!,!-Y^!,!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Illegal! (%! 0%+(! %*! 1*(23*2(
Polar Tquations - 1lasswork
42!:682!6&=69+!'+2C!(:2!#63(2+)6*!(*6@2C!6?(23!i2*22!V2+563(2+)!5%%3C)*6(2!+9+(2@E!H:)+!+92(2@!)+!76+2C!%*!6
;3)C!%?!02302*C)56&63!&)*2+E!a6*9!(%=*+!6*C!5)()2+!632!@%C2&2C!(:)+!=69E!H:2!polar coordinate system!)+!76+2C
%*!6!5)35'&63!@%C2&E!H:2!5)(9!%?!S63)+!)+!@%C2&2C!=)(:!0%&63!5%%3C)*6(2+!=)(:!(:2!\35!C2!H3)%@0:2!6+!(:2!52*(23E
H%!?%3@!(:2!0%&63!5%%3C)*6(2!+9+(2@<!=2!+(63(!=)(:!6!0%)*(!H 56&&2C!(:2!0%&2!%3!(:2!%3);)*E!P65:!0%)*(!P!)*!(:2
0&6*2!)+!6++);*2C!0%&63!5%%3C)*6(2+! " r,8 # !6+!?%&&%=+R!!r!)+!(:2!C)325(2C!C)+(6*52!?3%@!H!(%!P!6*C! 8 !)+!(:2!C)325(2C
6*;&2C<!5%'*(235&%5I=)+2!?3%@!0%&63!6Q)+!(%!+2;@2*(! OP E!!H:2!C)6;36@!72&%=!+:%=+!(:322!0%)*(+!%*!(:2!0%&63
5%%3C)*6(2!+9+(2@E!1(!)+!5%*82*)2*(!(%!&%56(2!0%)*(+!=)(:!32+025(!(%!6!;3)C!%?!5%*52*(3)5!5)35&2+!)*(23+25(2C!79!36C)6&
&)*2+!(:3%';:!(:2!0%&2E!!H:2!&)*2! 8 = 0)+!56&&2C!(:2!polar axis.
![<
(A8<![@)
(@8![C)
=
!
(A8'0![A)'$-'(0A8@![A)
@![<
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
H%!5%*823(!(%!6*C!?3%@!(:2!0%&63!+9+(2@!(%!(:2!5%%3C)*6(2!+9+(2@<!9%'!@'+(!I*%=!(:2!?%&&%=)*;!32&6()%*+:)0+E
" x .y # or "r,8 #
r
y
x
<
<
<
''y = r %*9 8''''''''''''''''r = x $ y
x = r ;$%8''''''''''''''' ,.9 8 =
y
8
PQ6@0&2!-)!#%*823(!(:2!?%&&%=)*;!0%&63!0%)*(+!(%!325(6*;'&63!5%%3C)*6(2+E
+
!.
7E!! -2 3, 0
6E! "5,!#
,
6/
"
#
!!!!!! 3, 3
!!!!!!! "$5, 0#
PQ6@0&2!/)!#%*823(!(:2!?%&&%=)*;!325(6*;'&63!0%)*(+!(%!0%&63!5%%3C)*6(2+E
6E!!(,Y<!,Y)
7E!(.<!,/)
+ 5! .
+ 3! .
!!!!!!!! -5, 0
!!!!!!!! -2, 0
, 4/
, 2/
PQ6@0&2!O)!#%*823(!(:2!?%&&%=)*;!0%&63!2>'6()%*+!(%!325(6*;'&63!2>'6()%*+E
6)!!r!=!/
7)!! 8 = 2! 3
2
2
2
2
!!!!! x $ y = 2 1 x $ y = 4
!!!! tan 8 = $ 3 1
PQ6@0&2!T)!#%*823(!(:2!?%&&%=)*;!0%&63!2>'6()%*+!(%!325(6*;'&63!2>'6()%*+E
3
r=
1 $ sin 8
r = $2 csc 8
6)!
$2
r=
1 r sin 8 = $2 1 y = $2
sin 8
y
= $ 3 1 y = $x 3
x
2
2
2
2
7)! r $ y = 3 1 x $ y $ y = 3 1 x $ y = 3 $ y
9 $ x2
x $ y = 9 $ 6y $ y 1 y =
6
2
2
2
! www.MasterMathMentor.com!! "#!$%&'()%*+!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!,!-Z.!,!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Illegal! (%! 0%+(! %*! 1*(23*2(
To find the slope of a tangent line to a polar graph, suppose we have a differentiable function given by r = f (! ) .
To convert to polar form, we know that x = r cos! and y = r sin ! . So, it follows that:
x = r cos! = f (! ) cos! and y = r sin ! = f (! ) sin !
dy
given in the previous section, the slope of a differentiable function of ! =
dx
f (! ) cos! + f "(! ) sin !
dy
dy d!
dx
$ 0 at ( r,! )
=
=
provided that
dx
dx d!
d!
# f (! ) sin ! + f "(! ) cos!
Using the parametric form of
From the above theorem, we can make the following observations:
dy
dx
= 0 provided that
$0
• to find horizontal tangents to polar equations, set
d!
d!
dx
dy
• to find vertical tangents to polar equations, set
= 0 provided that
$0
d!
d!
dy
dx
• if
and
are simultaneously zero, we make no conclusion about tangent lines.
d!
d!
Example 5) Find horizontal and vertical tangent lines to
r = 2(1 # sin ! )
x = 2(1 # sin ! )(cos! ) = 2(cos! # sin ! cos! )
r = 2 sin !, 0 % ! % &
x = 2 sin ! cos!
dx
= 2[sin ! (# sin ! ) + cos! (cos! )]
d!
& 3&
sin 2 ! = cos2 ! ' ! = , (Vertical)
4 4
a.
2
y = 2 sin !
dy
&
= 4 sin ! cos! ' ! = 0, , &(Horiz)
d!
2
dx
= 2 # sin ! # (# sin 2 ! + cos2 ! )
dt
dx
= 2[# sin ! + 2 sin 2 ! # 1] = 2(2 sin ! + 1)(sin ! # 1)
dt
7& 11& &/
!= ,
, (Vertical)
6 6 2/
[
b.
]
y = 2(1 # sin ! )(sin ! ) = 2(sin ! # sin 2 ! )
dy
= 2(cos! # 2 sin ! cos! ) = 2 cos! (1 # 2 sin ! )
dt
& 5& &/ 3&
! = , , , (Horiz)
6 6 2/ 2
If r = f (! ) = 0 at ! = ( and f "(( ) $ 0, then the line ! = ( is tangent to the pole.
Example 6) Find the tangent lines at the pole to r = 4 cos 3! . Confirm graphically.
& 3& 5& 7& 9& 11&
cos 3! = 0 ' 3! = , , , , ,
2 2 2 2 2 2
& & 5& 7& 3& 11&
!= , , , , ,
6 2 6 6 2 6
- 161 -
Polar Tquations - >omework
G%3!265:!%?!(:2!?%&&%=)*;!0%&63!5%%3C)*6(2+<!?)*C!(:2!5%332+0%*C)*;!325(6*;'&63!5%%3C)*6(2+E
+ !.
-E!! -6, 0 = "0, 6#
, 2/
+ 7! . + $ 2 $ 2 .
,
/E! -$1, 0 = 0
,
4/ , 2
2 /
+ $! .
OE! -$4, 0 = $2, 2 3
,
3/
"
#
G%3!265:!%?!(:2!?%&&%=)*;!325(6*;'&63!5%%3C)*6(2+<!?)*C!two!5%332+0%*C)*;!0%&63!5%%3C)*6(2+E
+ 1 $ 3 . + $! . + 5! .
+
+ 3! . + $! .
3! . +
7! .
YE!! - ,
TE!! "$3, 3# = - 3 2 , 0,- 3 2 , 0
0 = -1, 0,-1, 0 !!!!!!!!!!!!!!!!!ZE! "0, $4 # = - 4, 0,- 4, 0
,
, 2 /, 2 /
4 /,
4/
,2 2 / , 3 / , 3 /
G%3!265:!%?!(:2!?%&&%=)*;!325(6*;'&63!2>'6()%*+<!5:6*;2!)(!(%!0%&63!?%3@!6*C!5%*?)3@!%*!9%'3!56&5'&6(%3E
xy = 12
5x $ y = 7
[E!! 5 r cos8 $ r sin 8 = 7
7
r=
5 cos8 $ sin 8
" x $ 1#
2
"r cos8 #"r sin8 # = 12
]E!!! r 2 sin 8 cos8 = 12
r=)
$ y2 =1
x2 $ y2 $ 4x = 0
x2 $ 2x $ 1$ y2 = 1
2
2
2
2
Ê! E! r cos 8 $ 2 r cos8 $ r sin 8 = 0
12
sin 8 cos8
-.E!!
2
r $ 2 r cos8 = 0
r 2 $ 4 r cos8 = 0 1 r 2 $ 4 r cos8 $ 4 cos2 8 = 4 cos2 8
"r $ 2 cos8 #
2
= 4 cos2 8
r $ 2 cos8 = )2 cos8 1 r = $4 cos8
r" r $ 2 cos8 # 1 r = 2 cos8
G%3!265:!%?!(:2!?%&&%=)*;!0%&63!2>'6()%*+<!5:6*;2!)(!(%!325(6*;'&63!?%3@!6*C!5%*?)3@!%*!9%'3!56&5'&6(%3E
tan 2 8 = 9
r=4
--E!
x 2 $ y 2 = 16
-/E!
y = ) 16 $ x 2
r = 4 sin 8
+ y.
r = 4- 0 1 x 2 $ y 2 = 4 y
,r/
2
2
-OE!! y $ 4 y $ 4 = 4 $ x
" y $ 2#
2
= 4 $ x2
y = 2 ) 4 $ x2
y2
=9
x2
y = )3 x
1
1 $ cos8
r $ x =1
r=
-TE!!
x2 $ y2 = x $1
x2 $ y2 = x2 $ 2x $ 1
y = ) 2x $ 1
! www.MasterMathMentor.com!! "#!$%&'()%*+!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!,!-Z/!,!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Illegal! (%! 0%+(! %*! 1*(23*2(
G%3! r = 2 $ 3sin 8 <!?)*C!
dy
6*C!(:2!+&%02!%?!(:2!(6*;2*(!&)*2!6(!(:2!?%&&%=)*;!0%&63!0%)*(+E
dx
y = sin 8 "2 $ 3 sin 8 # = 2 sin 8 $ 3 sin 2 8
x = cos8 "2 $ 3 sin 8 # = 2 cos8 $ 3 sin 8 cos8
dx
dy
= $2 sin 8 $ 3 cos2 8 $ 3 sin 2 8
= 2 cos8 $ 6 sin 8 cos8
d8
d8
dy
2 cos8 $ 6 sin 8 cos8
=
dx $2 sin 8 $ 3 cos2 8 $ 3 sin 2 8
"
#
"
-YE!! 5, 5! 2 1 m = 0
#
-ZE! $1, 3! 2 1 m = 0
-[E! "2,!# 1 m =
$2
3
G%3!265:!%?!(:2!?%&&%=)*;<!?)*C!(:2!0%)*(+!%?!:%3)B%*(6&!6*C!823()56&!(6*;2*59!()?!6*9)
r = 3 $ sin 8
x = cos8 " 3 $ sin 8 # = 3 cos8 $ sin 8 cos8
dx
= $3 sin 8 $ cos2 8 $ sin 2 8
d8
-]E!!
$3 sin 8 $ 1 $ 2 sin 2 8 = 0
y = sin 8 " 3 $ sin 8 # = 3 sin 8 $ sin 2 8
dy
= 3 cos8 $ 2 sin 8 cos8
d8
cos8 " 3 $ 2 sin 8 # = 0
! 3!
Horiz : 8 = ,
2 2
2 sin 2 8 $ 3 sin 8 $ 1 = 0
Vert : 8 = .284, 2.857
r = sin 8 cos2 8
-Ê!!
0 =8 '!
x = sin 8 cos3 8
dx
= cos4 8 $ 3 sin 2 8 cos2 8
d8
cos2 8 "cos2 8 $ 3 sin 2 8 # = 0
1
2
!+ ! 5!
Vert : 8 = , ,
2+ 6 6
cos8 = 0,sin 8 = )
y = sin 2 8 cos2 8
dy
= 2 sin 2 8 cos3 8 $ 2 sin 3 8 cos2 8
d8
2sin8 2 cos8 2 "cos8 $ sin 8 # = 0
!+ ! 3!
Horiz : 8 = 0, , ,
2+ 4 4
G)*C!(:2!&)*2+!(6*;2*(+!6(!(:2!0%&2!(%
r = 4 "1 $ cos8 #
!!/.E!! 1 $ cos8 = 0
8 =0
r = 3 sin 28
sin 28
/-E!! 28 = 0, !, 2!, 4 !
! 3!
8 = 0, , !, , 2!
2
2
! www.MasterMathMentor.com!! "#!$%&'()%*+!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!,!-ZO!,!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Illegal! (%! 0%+(! %*! 1*(23*2(
Area, Arc Length a Surface Area in Polar Tquations - 1lasswork
4+&'.-&.'$3'.'%&;,$-'$3'.';*-;6&')*,+'-.:*"%'-'.9:';&9,-.6'.926&''8
1
*%'2*5&9'/(' A = 8 r 2 #-$5*:&:',+.,' 8 *%'7&.%"-&:'*9'-.:*.9%H
2
r
8
r
H%!?)*C!(:2!6326!%?! r = f "8 # !72(=22*!(=%!86&'2+!%?!(:2!6*;&2<!*6@2&9! D and E <!063()()%*!(:2!)*(2386&! &D , E % !)*(%
n!2>'6&!+'7)*(2386&+R!! D ' 81 ' 8 2 ' 8 3 ' ... ' 8 n $1 ' 8 n ' E E
E $D
= 58
i26&)B2!(:6(!(:2!36C)'+!%?!(:2!i(:!+25(%3!=! f "8 i # !6*C!(:6(!(:2!52*(36&!6*;&2!%?!(:2!i(:!+25(%3!)+!
n
n
n
+ 1.
+ 1.
2
2
$%<! A 6 F- 0 58 & f "8 i #% !6*C!(:'+! A = lim F- 0 58 & f "8 i #% !!=:)5:!&26C+!(%!(:2!?%&&%=)*;R
n 23
, 2/
, 2/
i= 1
i= 1
1?! f !)+!5%*()*'%'+!6*C!*%*,*2;6()82!%*!(:2!)*(2386&! &81,8 2 % <!(:2*!(:2!6326!%?!(:2!32;)%*!7%'*C2C!79!(:2
;360:!%?! r = f "8 # !72(=22*!(:2!36C)6&!&)*2+! 8 = 81 and 8 = 8 2 !)+!;)82*!79
8
8
2
1 2
1 2 2
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! A = # & f "8 #% d8 = A = # r d8
2 81
2 81
PQ6@0&2!-)!G)*C!(:2!6326!%?!%*2!02(6&!%?!(:2!3%+2!5'382!;)82*!79! r = 4 cos 38 E
K$,&',+.,'-*2+,'#&,.6'*%',-.;&:'.% 8
*%',-.;&:'3--$7'0![C',$'![CH'!$')&'
;.9'"%&',+&'#-&;&:*92'3$-7"6.',$'
3*9:',+&'.-&.H'
!!!!!!!!G)3+(<!&2(D+!;360:!)(E!!!!!!!!!!!!!!!
!6
+ 1 .! 6
2
2
!!!!!!!!$2(!'0!(:2!)*(2;36&R!!! A = 2- 0 # 16 cos 38 d8 = 16 # cos 38 d8
, 2/ 0
0
!!!!!!!!1(!C%2+!*%(!600263!(:6(!(:)+!56*!72!)*(2;36(2CE!"'(!'+2!(:2!0%=23!32C'5)*;!?%3@'&6!
1 $ cos 28
cos2 8 =
2
!6
!6
+ 1.
A = 2- 0 # 16 cos2 38 d8 = 16 # cos2 38 d8
, 2/ 0
0
!6
!6
!6
+1 $ cos 68 .
%4
(
!6
16 # 0 d8 = 8 # "1 $ cos 68 # d8 = &88 % 0 $ ' sin 68* = 4.189
,
/
&3
)0
2
0
0
!!!!!!buestionR!H%!?)*C!(:2!(%(6&!6326!&9)*;!)*+)C2!6&&!O!02(6&+<!=:9!56*D(!9%'!)*(2;36(2!?3%@!.!(%!/"M
r switches from positive to negative in this region.
! www.MasterMathMentor.com!! "#!$%&'()%*+!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!,!-ZT!,!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Illegal! (%! 0%+(! %*! 1*(23*2(
PQ6@0&2!/)!G)*C!(:2!6326!%?!(:2!32;)%*!&9)*;!72(=22*!(:2!)**23!6*C!%'(23!&%%0+!%?!(:2!&)@6q%*! r = 2 sin 8 $ 1E
8
_$"')*66'9&&:',$'3*9:',+&'5.6"&%'$3''''''')+&-&',+&'
*99&-'6$$#'%,.-,%'.9:'%,$#%'.9:')+&-&',+&'$",&-'
6$$#'%,.-,%'.9:'%,$#%H''L+.,'.-&',+&(d
!!!!!!!!!
2 sin 8 $ 1 = 0 1 sin 8 =
!!!!!!!!$2(!'0!(:2!)*(2;36&!?%3!(:2!)**23!&%%0E
1
! 5!
18 = ,
2
6 6
!!!!!!!!!!!!!!!!!!!!!!$2(!'0!(:2!)*(2;36&!?%3!(:2!%'(23!&%%0E
+ 1 .! 2
2
2- 0 # "2 sin 8 $ 1# d8 = .544
, 2 /! 6
+ 1 .3 ! 2
2
2- 0 # "2 sin 8 $ 1# d8 = 8.881
, 2 /5 ! 6
$%!(:2!(%(6&!6326!)+M!! 8.337
PQ6@0&2!O)!G)*C!(:2!5%@@%*!)*(23)%3!%?! r = 1 $ cos8 and r = 1!E
!+.:&'*9',+&';$77$9'*9,&-*$-'$3',+&',)$'2-.#+%H
''''''
!!!!!!!!!
!$%!(:2!(%(6&!6326!)+!;)82*!79!(:2!2Q032++)%*R!!
! + 1. !
2
$ 2- 0 # "1 $ cos8 # d8 = 1.927
2 , 2 /! 2
Points of intersection of a polar graph
"256'+2!6!0%)*(!@69!72!32032+2*(2C!)*!@%32!(:6*!=69!'+)*;!0%&63!5%%3C)*6(2+<!9%'!@'+(!72!5632?'&!)*!?)*C)*;
)*(23+25()%*!0%)*(+!%?!(=%!0%&63!;360:+E!_%'3!56&5'&6(%3!C%2+!*%(!:682!6*!1UHPi$P#H1WU!?'*5()%*!)*!0%&63
@%C2E!$%<!9%'!:682!(%!C%!)(!6&;2736)56&&9<!7'(!=)(:!6!I*%=&2C;2!%?!(:2!;360:E
PQ6@0&2!T)!!G)*C!(:2!)*(23+25()%*!%?!(:2!;360:+! r = 2 $ 4 cos8 and r = 2 E!!L2(D+!C%!)(!79!+'7+()('()%*E
!!
! 3!
2 $ 4 cos8 = 2 1 2 cos8 = 0 1 8 = ,
2 2
! @!
_$"'%&&',+.,'($"'2&,'8 = 8 H
< <
e",',+&-&'.-&';6&.-6(',+-&&'#$*9,%'$3'*9,&-%&;,*$9H'4+&'-&.%$9',+.,',+&',+*-:'#$*9,'
).%'9$,'3$"9:'*%',+.,'*,':*:'9$,'$;;"-')*,+',+&'%.7&';$$-:*9.,&%'.%',+&',)$'2-.#+%H'
H:2!0%)*(+!%?!)*(23+25()%*!632! "2, ! 2# and "2, 3 ! 2# E!H:2!(:)3C!0%)*(!)+!2)(:23! "2, !# or "$2, 0# E
_%'!56*!5%@0632!(:2!03%7&2@!%?!?)*C)*;!)*(23+25()%*+!%?!(=%!0%&63!;360:+!=)(:!(:2!03%7&2@!%?!?)*C)*;!5%&&)+)%*
0%)*(+!%?!3652!563+!;%)*;!63%'*C!6!(365I!%3!+6(2&&)(2+!5)35&2)*;!(:2!263(:E!H:2!563+!%3!+6(2&&)(2+!@69!%55'09!(:2!+6@2
0%)*(!)*!+0652!7'(!=)&&!*%(!5%&&)C2!6+!&%*;!6+!(:2!3265:!(:2!0%)*(+!%?!)*(23+25()%*!6(!C)??232*(!()@2+!( 8 !86&'2+)E!\
5%&&)+)%*!%55'3+!%*&9!6(!(:2!0%)*(+!%?!)*(23+25()%*!(:6(!632!3265:2C!6(!(:2!+6@2!()@2!( 8 !86&'2)E
! www.MasterMathMentor.com!! "#!$%&'()%*+!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!,!-ZY!,!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Illegal! (%! 0%+(! %*! 1*(23*2(
Arc Length in Polar Form
L2(! f !!72!6!?'*5()%*!=:%+2!C23)86()82!)+!5%*()*'%'+!%*! &81,8 2 % E!H:2!&2*;(:!%?!(:2!;360:!?3%@! 8 = 81 to 8 = 8 2 !)+
82
!!!!!!!!!!!!!!!!!!!!!!!!!!!!! s =
# & f "8 #%
2
2
$ & f 4"8 #% d8 =
81
82
#
81
2
+ dr .
r $ - 0 d8
, d8 /
2
PQ6@0&2!Y)!!G)*C!(:2!&2*;(:!%?!(:2!635!?3%@! 8 !=!.!(%! 8 !=!/"!?%3!(:2!563C)%)C! f "8 # = r = 2 $ 2cos8 !6*C!?)*C
!!!!!!!!!!!!!!!!!!!!(:2!C)??232*52!72(=22*!(:6(!6*C!(:2!5)35&2!=)(:!36C)'+!Yf/E!$2(!'0!6*C!(:2*!'+2!(:2!56&5'&6(%3E
!!!!!!!!!!!!!!!!!!!!!!!!!!!!563C)%)C
!!!!!!!!!!!5)35&2
r = 2 $ 2 cos8
r=
5
2
!!!!! r4 = $2 sin 8
!!!!!!!!!!!!!!!!!!!!
!
L=2#
"2 $ 2 cos8 #
2
!
+ 5.
C = 2!- 0 = 15.708
, 2/
$ 4 sin 2 8 d8
0
!
L = 2 # 4 $ 8 cos8 $ 4 cos2 8 $ 4 sin 2 8 d8
0
!
L = 2 # 8 $ 8 cos8 d8 = 16
0
Surface Area in Polar Form
L2(! f !72!6!?'*5()%*!=:%+2!C23)86()82!)+!5%*()*'%'+!%*! &81,8 2 % E!H:2!6326!%?!(:2!+'3?652!?%3@2C!79!328%&8)*;!(:2
;360:!%?! r = f "8 # !?3%@! 8 = 81 to 8 = 8 2 67%'(!(:2!)*C)56(2C!&)*2!)+R
82
S .A. = 2! # f "8 # sin 8
81
2
82
+ dr .
2
2
4
f
8
$
f
8
d
8
=
2
!
& " #% & " #%
# r sin8 r 2 $ -, d8 0/ d8 !!!!!!!!!!!!!,!67%'(!(:2!&)*2! 8 = 0
81
82
S .A. = 2! # f "8 # cos8
& f "8 #%
81
2
2
82
+ dr .
2
!
4
$ & f "8 #% d8 = 2! # r cos8 r 2 $ - 0 d8 !!!!!!!!!!!!,!67%'(!(:2!&)*2! 8 =
, d8 /
2
81
!
PQ6@0&2!Z)!G)*C!(:2!+'3?652!6326!?%3@2C!79!328%&8)*;!(:2!5)35&2! f "8 # = r = $cos8 !67%'(!(:2!&)*2! 8 = $ E
2
4+*%'*%'.';-$%%'%&;,*$9'$3',+&'$/f&;,
;-&.,&:'/(':$*92',+&'-$,.,*$9H'L+.,
3.7*6*.-'$/f&;,':$&%',+*%';-&.,&'*9'@Sd
''
bagel, doughnut, torus
!
% $! 2
(
S = 2 '2! # $ cos8 cos8 cos2 8 $ sin 2 8 d8*
& 0
)
$! 2
S = 4!
2
# $ cos 8 d8 = 9.870
0
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Area, Arc Length a Surface Area in Polar Tquations - >omework
1*!(:2!?%&&%=)*;<!;360:!%*!9%'3!56&5'&6(%3!6*C!?)*C!(:2!6326!%?!(:2!32;)%*E!$2(!'0!6*C!'+2!(:2!56&5'&6(%3E
r = 6 cos 38
+ 1 .! 6
2
-E!!W*2!02(6&!%?!! A = 2- 0 # 36 cos 38 d8
, 2/ 0
r = 2 sin 28
1
/E!W*2!02(6&!%?! A =
2
A = 9.425
2
# 4 sin 28 d8
0
A = 1.571
r = sin 58
1
OE!!W*2!02(6&!%?! A =
2
!2
r = 2 $ sin 8 (above polar axis)
!5
2
# sin 58 d8
0
1
TE!!1*(23)%3!%?! A =
2
!
# "2 - sin8 #
2
d8
0
A = 3.069
A = 0.157
Between the loops of r = 1 $ 2 cos8
r = 1 $ 2 cos8
+ 1 .! 3
2
YE!1**23!&%%0!%?! A = 2- 0 # "1 $ 2 cos8 # d8
, 2/ 0
A = .544
ZE
+ 1. !
2
Outer : A = 2- 0 # "1 $ 2 cos8 # d8 = 8.881
, 2 /! 3
+ 1 . 2!
2
Inner : A = 2- 0 # "1 $ 2 cos8 # d8 = .544
, 2 /5 ! 3
Difference = 8.337
G)*C!(:2!0%&63!0%)*(+!%?!)*(23+25()%*!%?!(:2!;360:+!%?
r = 2 $ cos8 and r = 2 $ cos8
2 $ cos8 = 2 $ cos8
[E!! 2 cos8 = 0
! 3! + ! . + 3! .
8 = , 1 -2, 0,-2, 0
, 2/ , 2 /
2 2
r = 1 $ 2 cos8 and r = 4 cos8
1 $ 2 cos8 = 4 cos8
Ê!! 2 cos8 = 1 1 cos8 =
1
2
! 5! + ! . + 5! .
8 = , 1 -2, 0,-2, 0
, 3/ , 3 /
3 3
r = 1 $ cos8 and r = 1 $ sin 8
1 $ cos8 = 1 $ sin 8
]E!! $ cos8 = sin 8 1 tan 8 = $1
3! 7! +
2 3! . +
2 7! .
8= ,
1 -1 $
, 0,-1 $
, 0
4 4
2 4 /,
2 4/
,
r = 10 sin 28 and r = 5
10 sin 28 = 5
1
! 5! 13! 17!
,
-.E!! sin 28 = 1 28 = , ,
2
6 6 6 6
! 5! + ! . + 5! . + 13! . + 17! .
8 = , 1 -5, 0,-5, 0,-5,
0,-5,
0
, 12 / , 12 / , 12 / , 12 /
3 3
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c+2!9%'3!56&5'&6(%3+!(%!;360:!(:2!0%&63!2>'6()%*+!6*C!?)*C!(:2!6326!%?!(:2!)*C)56(2C!32;)%*E
Common interior of r = 4 "1 $ sin 8 # and
Outside r = 2 sin 8 and inside r = 2
r = 4 "1 $ sin 8 #
+ 1 .! 2
2
!
2
A
4
2
=
!
$
--E!!
-/E!
- 0 # 4 sin 8 d8
+ 1.
2
,
/
2 0
A = 4- 0 # & 4 "1 $ sin 8 #% d8
, 2/ 0
A = 4 ! $ ! = 3!
A = 11.398
Inside the lemniscate of r = 25 cos 28
!4
+ 1.
-OE! A = 4- 0 # 25 cos 28 d8
, 2/ 0
The sonar signal from a submarine is modeled by
r = a cos2 8 . Find the area of the geographical
!!!!!!!!-TE! region between the two curves for a = 4 and a = 6
+ 1 .! 2
A = 4- 0 # " 36 cos4 8 $ 16 cos4 8 # d8
, 2/ 0
A = 25
A = 23.562
c+2!9%'3!56&5'&6(%3+!(%!;360:!(:2!0%&63!2>'6()%*+!6*C!?)*C!(:2!&2*;(:!%?!(:2!5'382!655'36(2!(%!O!C25)@6&!0&652+E
!
r = 38
0 =8 =
r = 2 $ sin 8
0 = 8 = 2!
2
2!
-YE!! A =
# "2 $ sin8 #
2
!2
2
-ZE!! A =
$ cos 8 d8
A = 6.238
A = 13.365
r = sec 8
0 =8 =
!4
-[E!!
#
!
(No calculator)
4
2
2
sec 8 $ "sec 8 tan 8 # d8
!4
#
0
r = e8
!4
sec 2 8 "1 $ tan 2 8 # d8 =
!4
A = tan 8 % 0 = 1
# sec
0 = 8 = 2! (No calculator)
2!
-]E!! A =
0
A=
98 2 $ 9 d8
0
0
A=
#
2
#
e 28 $ e 28 d8
0
8 d8
0
A = 2e8
%
2!
0
= 2 "e 2 ! $ 1#
"graphs a straight line#
c+2!9%'3!56&5'&6(%3+!(%!;360:!(:2!0%&63!2>'6()%*+!%823!(:2!;)82*!)*(2386&!6*C!?)*C!(:2!+'3?652!6326!=:2*!3%(6(2C
67%'(!(:2!;)82*!&)*2E
!
!
!
r = 2 sin 8 0 = 8 =
8=
"No calculator#
r = e$8
0 =8 =
8 =0
2
2
2
!2
-Ê!!
A = 2! # 2 sin 8 cos8 4 sin 2 8 $ 4 cos2 8 d8
0
!2
A = 8! # sin 8 cos8 d8
0
1
A = 4!% 0 = 4!
!2
/.E!!
A = 2! # e$8 sin8 e$28 $ e$28 d8
0
!2
A = 2! # 2 sin 8 e$28 d8
0
"surface area of sphere#
A = 1.624
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x dx

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