2. Cálculo Diferencial con “Mathematica”
:f →
2
2
1
; 0
1( )
; 0
2 1
x
x
xf x
ax b
x
x x
+
≤
−=
+
+ +
% + % % (,
(-
% + % ' %. % +
(,
/ % % (, (-
( ,
0 0
lim ( ) lim ( ) (0)
x x
f x f x f+ −
→ →
∃ = =
1
1
1
1
lim)(lim
12
lim)(lim
2
00
200
−=
−=
−
+
=
=
++
+
=
−−
++
→→
→→
b
x
x
xf
b
xx
bax
xf
xx
xx
(- 0 - (, ! (-,
'
2
2
= '(2)= 0
2 1 x
ax b
f
x x =
+
=
+ +
20220)2('
)1(
2
)1(
22
)1(
)1(2)()1(
12
13
334
2'
2
==−+=
+
−−
=
=
+
−−+
=
+
/+/+−+
=
++
+
−=
/
/
aaaf
x
axba
x
baxaax
x
xbaxxa
xx
bax
b
$ (1)* (-
%+ % ' % % + (,
2 + (, % % % )0(')0(' −+
=∃ ff
3. Cálculo Diferencial con “Mathematica”
2
00
2
2 22 0 0
1
(0 ) (0) ( 1)
'(0 ) lim lim
2 1 ( 1) 4
lim lim 4
( 1) ( 1)
hh
a h h
b
ah b
b
f h f h
f
h h
h h h
h h h
+
+
→→
= → →
=
+
−
+ − +
= =
− + + +
= = =
+ +
2
0 0
2
0 0
1
1
(0 ) (0) 1'(0 ) lim lim
1 1 ( 1)
lim lim 1
( 1) ( 1)
h h
h h
h
f h f hf
h h
h h h h
h h h h
− −
−
−
→ →
→ →
+
+
+ − −= =
+ + − +
= = = −
− −
$ (1) (-.! 0,3
≠ 0,1
. % + (,
5. Cálculo Diferencial con “Mathematica”
22( ) '( ) 0 6 6*44 2*144 0 12(3 22 24) 0
max
V t a t t t t t= = − + = − + =
11 7
6
3
11 7 4
3 3
44 4.121 4.3.6.4 22 2 121 72 11 49
6 6 3
t
+
=
−
=
± − ± − ±
= = = =
( )
.min60)1236(12)6(
.max
3
4022
3
4
612
3
4
)226(12)(
VtV
VtV
ttV
=−=
=−=
−=
9@6 ()=) )87 /
= ( 196- /
4
3
3 6 8
t
1000
500
500
s t
V 3 0
V 8 0
a 4 3 0V 0 0
a 6 0
A % %
t (-∞,4/3) 4/3 (4/3,3) 3 (3,6) 6 (6,8) 8 (8,∞)
S(t) S 161 S Smax -1024 Smin S
SgnV + + + 0 - - - 0 +
V(t) V Vmax V V=0 V Vmin V 0 V
Sgn a + 0 - - - 0 + + +
6. Cálculo Diferencial con “Mathematica”
4 % % B % % % CD % % % %
.E ; 4 % .E( 8., ; % + (7
+ ,.7 @%. % ,.7 % (,
% ABO ˆ %
! #$
! % + OBA
x 8 x
v 0.5 t
y 5
A 8,00
B
OBA γ α β= = + *
tan tan
tan tan( )
1 tan *tan
α β
γ α β
α β
+
= + =
−
8
tan ; tan
5 5
x x
α β
−
= =
% ; % % ,.7
0,5 8 0,5
( ) arctg arctg
5 5
t t
tγ
−
= +
( )tγ % % ( ) 0tγ ′ =
7. Cálculo Diferencial con “Mathematica”
2 2
2 2
2 2 2 2
2 2 2 2
2 2 2 2
0,5 0,5
5 5( )
0,25 (8 0,5 )
1 1
5 5
2,5 2,5 10 10
25 0,25 25 (8 0,5 ) 100 100 (16 )
10(100 16 32 100 ) 10(16 32 )
(100 )(100 (16 ) ) (100 )(100 (16 ) )
t
t t
t t t t
t t t t
t t t t
γ ′ = − =
−
+ +
− = − =
+ + − + + −
+ − + − − −
=
+ + − + + −
2
2
2 2
10(16 32 )
( ) 0 0 16 32 0 8
(100 )(100 (16 ) )
t
t t t
t t
γ
−
′ = = − = =
+ + −
máximo.
% % % . (8.% $ (,.7 F 8( 9* 81 (9
8. Cálculo Diferencial con “Mathematica”
; % % -, G + %
E % ; % % % +
! + B % E ' % ;.
' + B E! !; . % % .%
+ % 8, G @' + % -, G @'
! #$
B
20 Kmd
A x
a
C
a - x
% !
E
H % )
20
)(20
20
22
1
xad
t
v
e
t
t
e
V
−+
==
==
H % -
80
2
x
t =
9. Cálculo Diferencial con “Mathematica”
:
8020
)(20 22
xxa
t +
−+
=
2 5
[ ]a0,x,
8020
)(20
)(
22
∈+
+−+
=
xxa
xt
% %$
% % + (, (
% % % 0 ( ,
2 2
4
2 2
1 2( ) 1
'( )
20 802 20 ( )
'( ) 0 80( ) 20 20 ( ) 0
a x
t x
a x
t x a x a x
− −
= +
+ −
= − − + − − =
2 2
2 2 2
2 2
4( ) 20 ( )
16( ) 20 ( )
15( ) 20
a x a x
a x a x
a x
− = + −
− = + −
− =
0
20 20
15 15
a x x a− = = −
∃ 0 ∀ ∈I,. J
% %
( ,.
2 2
1
20
20
a
t
+
=
( . 2 1
80
a
t = +
(
15
20
−a .
2 2
2
3
20 20
20
15 15
20 8
a
t
+ −
= + . %
10. Cálculo Diferencial con “Mathematica”
4% E ( -
H 9 E % B C B ,D ;(1- -
38
; % % % % % %
! #$
% % % % % ( ,.
%% %
B
A
A
( ) ( )
( ) ( )8,080
0,000
→=
→=
B
A
S
S
/ % % + % % %
B %
11. Cálculo Diferencial con “Mathematica”
t=4
t=0t=2
min
V 0
484)2(
02)(
2042)('4)( 2
−=−=
∀=
==−=→−=
A
A
AA
S
mínimottS
tttStttS
E 19., % (-
' + % . %
% + % + ,., % (9
B
( )
00)('
.04)(
00461)('88 2
∀
−=
==//−=→+−=
ttS
máxtS
tttSttS
B
B
BB
; % ,.8 % B
% ,., % (-
% % %
12. Cálculo Diferencial con “Mathematica”
SB
SA
22
)( BA SStd +=
( ) ( )2222
824)( +−+−= ttttd
F 4 % %
( ) ( )
( ) ( ) ( ) ( )
2 22 2
1
2 2
1
2 2
2
( ) 4 2 8
'( ) 2 4 * 2 4 2* 2 8 * 4
4 2 4 8 8 2 8
4 5 6 8
d t t t t
d t t t t t t
t t t t t t
t t t
= − + − +
= − − + − + −
= − − + − − +
= − −
1 2 2
4
5
0
'( ) 0 6 36 160 6 14
5 6 8 0
10 10
t
d t
t t t
−
=
= ± + ±
− − = = = =
A B
t=2 SA(2) = -4
t=2 SB(2) = 0
4 % % E ; 5 % ' % B
13. Cálculo Diferencial con “Mathematica”
' % % % % % 7G
E % E . % ;
% % = G E %
% % + -G @' 9 G @'
! #$
A
6 K m
B
5 K m
P
d
x 6 -x
1 2
2
25 6
;
2 4
d x c x
t t
V V
+ −
= = = =
5 % .
4
6
2
25
)(
2
xx
xt
−
+
+
=
% %$
% % + (, (=
% + %
4
1
252
)('
2
−
+
=
x
x
xt
14. Cálculo Diferencial con “Mathematica”
2 2 2 2
0
25 5
'( ) 0 2 25 0 25 4
3 5
t x x x x x x x= − + = + = = =
% %
( , 4)0( =t
( = 8,7
2
3625
)6( ≈
+
=t
,(
5
5
2
0
55
625
5 5( ) 3,68
2 4
t x
−+
= + = .
15. Cálculo Diferencial con “Mathematica”
K % % + % % % %
% . % %
! #
$
hrrrS ππ 22)( 2
+=
% + % % % $
2
2
r
V
hhrV
π
π ==
2
2
2
3
2
( ) 2 2
2
'( ) 4 0
2
4
2
v
S r r r
r
V
S r r
r
V V
r r
r
π π
π
π
π
= +
= − =
= =
16. Cálculo Diferencial con “Mathematica”
h
y
r
R
L + % % A
! #
!
( ) [ ]
R
8
2
r ==
=−
−
=−
=
/−
/
/+−+//=
∈⋅−=
−+=
⋅=
/
22
2
22
22
2
22
2
222
222
22
2
8
2
2
2
0
2
2
1
24)('
,02)(
2
)(
Rr
r
rR
rR
r
rR
rR
r
rrRrrV
RrrrRrV
rR
H
rhrV
ππ
π
π
!
/ ( )@6 %
[ ]
( )
( )
RrRrRRr
RRrrrRR
RrrRR
rrRrRR
rR
r
rRR
rR
r
rrRrrrV
RrrRRrrV
rRRyRh
3
22
9
8
04129
412494
232
22
2
0
2
2
3
1
3
2
)('
,0
3
1
)(
22222
4222222
2222
22222
22
2
22
22
222
222
22
===+−
+/=−//
−=−
=−+−
−
=−+
=
−
/−
/
/
+−+//
/
=
∈−+=
−+=+=
/
ππ
π
R
r
h
17. Cálculo Diferencial con “Mathematica”
L % % % % % %
. % % %
2 2
2
2 2
2
2
( ) 2
2
'( ) 2 0 0
2
2 0 máximo
2
P r
P r x x
P r
A r r r Pr r
P
A r r P r y x
P
A
π
π
π
π π
π
π
π
π
−
= + =
−
= + = −
= − + = = =
= −
x
r
18. Cálculo Diferencial con “Mathematica”
% + %
9@M +
( )
2
2
2
2
2
2
1
3
4
1
( ) 1
3
'( ) 2 1 2 0
2 4 1 4
0 x a ( ) Vcono
3 9 3 9
Volumen del cono a b
Volumen del cilindro x y
b b x
tg y a x b
a a x a a
x
V x x b
a
x b b x
V x xb x xb
a a a
x V x a b
π
π
ϑ
π
π
π π π
π
=
=
= = = − −
−
= −
/ /
/= − − = − =/
≠ = = ⋅ =
b
y
-a
x x
O
a
