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13
1)
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1
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3)
,
3
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7
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.
1168
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x
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x ,
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ance
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t
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1
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3)
find
to
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normal
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,
3)
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> (x
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14
Sy
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F
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j K
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Ti
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Si
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S(y 0) + 12 2)
-
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12
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z
2
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.
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25
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j + 2k(x(i
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li
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t
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+
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+
S +32
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-
3 =
1)
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.
-
X + Sy
+ 32
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-
x
I
15 .
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=
X
Parametric
,
,
r=
(1
+ 2+
2
-
3
Standard
=
+ 2+
-
4 +
x
(0 0 0)5
Ti
, ,
v =
,
-
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**
2
=
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kx2i
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v = Y +i
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Ii
point
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3
+
Y
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37
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X
i + 25
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(2
3)
:
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2
!
1
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passed
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.
1
.
+s
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=
alt)
5
.
viCH
=
v(t) = r(t)
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a(t)
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/
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y
-
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,
+
+, 0)(y 227)
+
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25
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+
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3
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fy
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,
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Y
y=
x(
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z = 1
-
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2
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X
subs
simply
or
y =
= 1
z =
1
,
7
.
v
=
a
costitabin + J
v(t) =- Sint :
+ c + k
+ acost]
+
c k-
ala
hic
S
+
X
=
4
=
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a
cost
+
cos2 + )
x+ yz
unimetical
linear
+
=
at
speet.
t
E
Insmake
decrabe
on
over
I
exis
spirals
/
9
.
3coSt :
v =
v(t) = v'(t)
a(t)
=
=
v((t)
-
=
3Sin ti
-
+ Sin + 1
+ 4coS + J
-
3coSti
43in + j + scos + K
-
4cs + 5
-
~
Si + H =
-
speed :
45
m
+ 19134)2
+ 1 - +
"
Shir
t
16 Sin ++ 2sco
+
Shin
+ Twos2
+
+
2
S
=
path :
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x =
3103 t
y=
4(03 +
=
~
S
+
2
y =
=
Ex
Shirt
z
,
=
Saint
15
x4 + y2
,
+ >sin (w +) J
(03 (wt)i
V
=
w
=
2
2
=
-U
Period his
has
derSilt)
+ So it
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i
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nuz
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w
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:
n
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+ (3
a
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ch
32 + 1
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x
)
27
memorize
=
=
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:
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e
H
~
=
can
0
Ven
y
-
crossed
Its
tor
by
c
:.
self
=
LS
=
Rh .
13 .
f(x)
=
4
D = 4x2
.
3
by
X
.
-
Y
=
X
+ y
0
x* + y2 =
analysis
o
everywhere
4.
X
-
y
=
v
y2 xv
1Y = Ex
3
.
4x2 + 942
one
Y)
f(x )
2 =
1
function
variable
E
of
-
3610
4x + 442736
is
except
9
.
f(x
,
y , 2)
0
=
=
X
x2 + y2 + 2
byInspection only
when
x= 0
,
2= 0
II
2
.
f(x y)
=
*
Y
,
X
13
Y=
.
f(x, y)
=
y z
.
0
,
an
15
.
2 = f(x
,
y)
= S
·
↓
L
·
-
2
13
I
.
.
2( )
-
2
1
.
22
4
+
2
=
+
2
0
.
=
DNE
if (xx)
,
+
(0, 0)
h
along
=
x =
x
y
4
f(x , y) =
-
xz y]
=
2
-
22 =
4
-
x2
-
y
x + y2 + 22 = 4
&
+ (x y)z0
,
I
nens o
Sphere but I
..
b)
f(x y)
1
=
X
-
-
,
Let
(1
0
,
Set
0 :
42
X
10 ,
Set + +
0)
,
,
2
0
20
,
:
:
0
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,
,
1)
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Po
⑭dignite
a
eX 1
.
y)
+ (x,
:
lim
(x y) + (2, 1)
f(x , y
,
in
eX1
.
f(x, y)
along
lim
a) lim
(x
,
6)
"in
y) - (0 ) 0)
,
y)
along
5
-
= -
,
f(x , y)
Set
+(x 4)
,
1 h
=
X =
=
f(x,x)
.
(x, y)
(4
X-axis
1 * 10 0)
(X
=
=
>
-
(0, 0)
0)
y *
=
0
0
lig
=
0
=
y =X
-
f(x,
if
M
!
f(x, y)
+h
=
@S(X y)
&
if
(Xo
-
,
,
DNE
Yo
Smooth
Som
(X0 40)
curve
f(x)
(i)
-
any
smoth
A
La
,
along
&
+ (x, Y(2750,
j) +Las
alous
curve
or
has
different
f(x, y)
values
different
along
limit
DNE
craves
than
if
optim
:
Pub
Organic
DNE- need
Thre
is
or
2
-
Yo
,
limit
:
2
Xo
path
diff
check :
x
-
y
lin,
3)
RNE
where
values
Auton :
chem
X(ye(3
show
to
X
-
=
Y
I
=
X +
=
10
i ecoasynt
lim
(x 4) ,
4
4
along X
↳
X
-
=
praxis
1
:
:
=
=
0
0
along
axib
=
100 4
y
x
X :
=
PNE
=
0
8
doesn't
we i
1:
m
(x y) + 0 , 0
4
,
along
·
p axis
along
-
ulos
along
X
y
=
=
y
X
y
2
I
ty
,
(
=
450
axis
=
(x
,
e0
y))
·
X =0
=
=+
y=
F
-
2
alous
↳
His
1%
X4
0
=
axis
lim
&
X00
=
y =0
,
X
(8)
X- 0
*
0
=
along
x
, ,
0
X=
ONE
I im
07
lim
(x ) e10 ,
x
4
Syn+4+ 2
= X
=
4
x2 + x
--2
2
2
=
0
,
= 0
,
4
=
00
0
x=
x
I
=
:
-
DNE
S2+xn + 4
=
52x274
2
+ 2
2
+
=
4
13
I
limits
2
.
Xy +
(iy
.
-
(2
,
-
-
=
=
T
x2
-1
(2)( 1)
2
+
2
alous
.
x
, e0X
+4
2
contrinity
and
1543 10 0y
+
,
3
alons
x, +30,
y
=
x
0
x 4 -(0,
,
=
=
1
Ficasso
I
y
-
X
=
0
400
0
Ex
A
--
,
=
DNE
:
·
14
X =
0
y
along
.
0
=
T
.
2
+
=
xz
6
.
:
T
I
+
>
Iim
doesn't
=
.
↓
exist
.
=
=
I
=
=
y
=
X
y
=
0
+
·
=
IS
=
#
~
f(x , y)
=
.
x y +1
,
=
,
1
=
o
(0)
4)
=
=
E
13 3
chem
.
f(x , y)
first
Partial
divitative
fx
fy
fx
=
fy
=
+x
=
-
14x
=
0
=
-
GXy4
-
22x
y3
tutor :
-
-
x(y4
-
+
sx4y
3x 244 +
20x4
x 4y + 1Sx
35x
0
+ 0
+
32
Rex
A
+
+
Problems :
practic
2
x
=
-
y +2
(3, 2)
-
F
=
3
f
-
-
I
X y4zS
flype
,
-I
,
10, -1 , -1)
3x2442
(x / 2=
,,
-
O
=
fz(X, y , 2)
fg(X
,
Y
,
=
4x3y3z5
=
O
2)
=
=
fun t
-
Sx
0
y424
=
x
=
(1)
=
-
I
7
f,
(x
y)
,
(os(X(y)
=
=
2 cos()
=
f (x , y)
-
=
.
-
(0)(X(y)
:
xy-z
= 23(XSy)
Si
"
S
= cos(-
=_
(os(
-
2)
=
=
4
.
Puz
=
=
Muz-1
2((n2/e
2e
zine-1
A
2/n2
2e
=
x(42)
=
(2ke)
=
=
.
2
In
.
In
2(n2
2
2
e
try
zhe
=
=
=
C
.
e
22
14 !
fix ,
2
.
x
-
+x2(0xC
fz(x, y)
-
:
f(( j 2)
=
t
fangent
2 =
z
=
Plane
5 + Eg(x 1) -, y 2)
-
-
f(X,, y) + f , (x, y)(x x,) f(x y)
-
-
19
=2x
.
f, (x, y)
=
=
fz(x x)
,
·
y
24
=
f(x , x)
=
2))
+
f(() 2) = (n(i (
-
=
(n(1
=
=
(nS
normal
+
&(x
live
-
1)
-
/n(s)
Y(y
enuation
+ 4)
+
2)/
:
·
13
1
.
4
.
=
21 +
=
X=
2/ + 2y2x
= ~
3
.
=
d
=
A
0 +
44x
= 4
3x2y32
=
6 xyz
=
gx y
-
=
=
e-ye
0-ye
de
-
-
Axe
9
&
.
unch
f(x , y) =
fi
fi
partial
devirties
twice
differentiable
continubly
Second
6xy
by
=
=
3x 24
-
=
0
y
fu
=
3x2
fzz
=
0
-
-
to
34
64
fitf
must
64
+ 1
2x2y
f
13 .
=
0
6y) = 0
-
&y
4x4
fz
f
=
could be
exchtle
another
F,
=
=
=
2x2
=
0
=
0
-
-
2yh
44
44
Il
Edevination
albecoul
differentiable
=
Sin(S2) e3X
Aw
Sin (2) e
=
+
·
3
. 4
+ 4
=
Scos(2
3x + 4)
(35 :n (SF) + 43in(S2)
(ySin(S2) +
ScoS(2)))e
4y)
ScoS(2))d
= 0
=
0
&Curre
M
X
+
3x +
Sub
~
w
11
=
13 .
S
I R.
~
& f(t
,
y
,
2)
:
+
g H + f(xhzS(t)ffxy2
,
de
3
.
=
s , (x oh , (v)
+
,
@2(X , Y. XSoh ,
=
i - 2
·
-
13
(1)
7
.
.
al
=
xf(x y, 2)
,
zz
.
.
2)
,
f(x ,"
,
fi
viv
Cut a
=
=
fz
2x23
=
-
3yz
Li +
z=
32
-
=
6x22
- f(x
,
y
-
2)
,
642
=
223 i
-
325 +xxz2
& directional derivative/gradient
restor
1
.
)
al
,
fz
2x
=
f
=
- f(x, y)
-
=
4(x
c
=
,
fz()
-
=
4
Z
245
4i + 25-
-
f (2 , -1)
f (2)
0
24
2xi
=
=
2
-
2)
=
??
+
3
2(y + 1) + 3