Differentiation :
2 1 .
:
The Geometry of Real-valued functions
Functions & Mappings :
vector-valued functions :
IR" ,
m>
scalar-valued functions :
IR
:
312
ex :
IR3 -1 :
f :
(x , Y , 7)
+
(x2 , y2 , 72)
IRO-12 :
9
:
(X, , X2 X3 Xy , Xg
, , , Xo) >
-
(X XzX3XuX5Xo ,
XR
+ Xo'
level Sets ,
curves ,
and surfaces :
cross sections on the X-y axis
eX :
C
ex + y"
=
-x + yz
2 2 Limits
.
:
and Continuity
:
M
open sets for R
Dr(Xo)
&
:
re Dr(Xo) for IR ~
Dr(Xo) for IR2
is
,
⑧ 003
X
Xo-V Xo Notr
&
Dr(Xo)
:
EXtIR :
/IX-Xoll < r3
Theorem :
EXotIR" ,
r >O , Dr(Xo) is an open set
Proof :
d
=
11 % -Yoll
r
d
Xo
11x- Yol
So
S =
r -d =
v-
Prove lly-Yoll > r :
rewrite triangle ineq.
lly Yoll
- :
11 (j () + (X Y o)l) [lly- * 11
- -
+ 11X-Yoll <
S + 11 * -Foll :
r
:. lly-Foll V
eX
:
A =
G(X y) , + 12 1x703
No-X1
:
Mox : No-X
1 %0 y)) + -
< V =
X
Limits :
hm +(*) =
b
X
-
X0
↑ '(Xo) =
him f(x)-f(xo) derivatively ,
definition
X- Xo
X- Xo
, Properties :
B
umf(x) YmfX :-
=
I .
,,
2 f(x) b - Cf(x) a
Lim
m i
=
.
Xe XO
vimg() = g)() =
.
3
limf(x) =, X + Xo
-m(f +
X+ y+2 umx
2 2
e
M, umy + ,
:
+
X+ j lo
= 0 + 1 + 2 =
3
ContinuousFunctions
:
a function is continuous
Limf(x) fix
:
when
:
eX lim 3x2-oxy2 y3 +
3 m X-
61m XIM Y +
As
LY
: :
(X , Y) + (X0 , Yo) * - X0
:
3X02- 6X0Yo + yo3
ex :
f(x y)
,
=
(x y ,
y + x3)
1 + X2
/MX(my) MY) +
I
,
1+ x
No'y ,
Yo + Xo31
1+ Xo2
Composition :
93B" C
A
&
Fog
ex :
Snow f(x Y z)
, ,
=
(x2 + y2 + z23 + sin(z3) is continuous
u-sub
:
u =
X + y2 + z2 ,
v
=
z
30
Lim u + him sin(Vo)
- 40 vevg
=
Up30 + sin(Vol
2 3
.
:
Differentiation
f(x , X Xn)-f(x.
Of (i) um Xjth
:
..., ....
H
=umf(x ni)
+ f -
n
2 1 .
:
The Geometry of Real-valued functions
Functions & Mappings :
vector-valued functions :
IR" ,
m>
scalar-valued functions :
IR
:
312
ex :
IR3 -1 :
f :
(x , Y , 7)
+
(x2 , y2 , 72)
IRO-12 :
9
:
(X, , X2 X3 Xy , Xg
, , , Xo) >
-
(X XzX3XuX5Xo ,
XR
+ Xo'
level Sets ,
curves ,
and surfaces :
cross sections on the X-y axis
eX :
C
ex + y"
=
-x + yz
2 2 Limits
.
:
and Continuity
:
M
open sets for R
Dr(Xo)
&
:
re Dr(Xo) for IR ~
Dr(Xo) for IR2
is
,
⑧ 003
X
Xo-V Xo Notr
&
Dr(Xo)
:
EXtIR :
/IX-Xoll < r3
Theorem :
EXotIR" ,
r >O , Dr(Xo) is an open set
Proof :
d
=
11 % -Yoll
r
d
Xo
11x- Yol
So
S =
r -d =
v-
Prove lly-Yoll > r :
rewrite triangle ineq.
lly Yoll
- :
11 (j () + (X Y o)l) [lly- * 11
- -
+ 11X-Yoll <
S + 11 * -Foll :
r
:. lly-Foll V
eX
:
A =
G(X y) , + 12 1x703
No-X1
:
Mox : No-X
1 %0 y)) + -
< V =
X
Limits :
hm +(*) =
b
X
-
X0
↑ '(Xo) =
him f(x)-f(xo) derivatively ,
definition
X- Xo
X- Xo
, Properties :
B
umf(x) YmfX :-
=
I .
,,
2 f(x) b - Cf(x) a
Lim
m i
=
.
Xe XO
vimg() = g)() =
.
3
limf(x) =, X + Xo
-m(f +
X+ y+2 umx
2 2
e
M, umy + ,
:
+
X+ j lo
= 0 + 1 + 2 =
3
ContinuousFunctions
:
a function is continuous
Limf(x) fix
:
when
:
eX lim 3x2-oxy2 y3 +
3 m X-
61m XIM Y +
As
LY
: :
(X , Y) + (X0 , Yo) * - X0
:
3X02- 6X0Yo + yo3
ex :
f(x y)
,
=
(x y ,
y + x3)
1 + X2
/MX(my) MY) +
I
,
1+ x
No'y ,
Yo + Xo31
1+ Xo2
Composition :
93B" C
A
&
Fog
ex :
Snow f(x Y z)
, ,
=
(x2 + y2 + z23 + sin(z3) is continuous
u-sub
:
u =
X + y2 + z2 ,
v
=
z
30
Lim u + him sin(Vo)
- 40 vevg
=
Up30 + sin(Vol
2 3
.
:
Differentiation
f(x , X Xn)-f(x.
Of (i) um Xjth
:
..., ....
H
=umf(x ni)
+ f -
n
