mathcad - applications of derivatives
DESCRIPTION
MathcadTRANSCRIPT
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5/21/2018 Mathcad - Applications of Derivatives
1/14
VIA UC
Mechanical Engineering Program
Javier I Camacho (164649)
Application of Derivatives
a ) Find the intervals on which f is increasing and decreasing.
b ) Find extreme values,if any.
c ) Which,if any,of the extreme values are global/local.
1 )
f
1
(x)
f1
x( ) x2
3x 3 factor 3 3 x x2
f
1
(x)Domain f1(x) Domain: all x
f
1
(x)(x,0)(Roots) Given f 1
x( ) 0= Find x( )21
2
3
2
21
2
3
2
0.791 3.791( )
f(x)(0,y)(y/axis Interception) f1 0( ) 3
f
1
(x)
f'1
x( )x
f1
x( ) dd
factor 2 x 3
f
1
(x)Domain f'1(x) Domain: all x
f (x)(x,0)(Roots) Given f'1
x( ) 0= Find x( )3
2 1.5
f
1
(x)(0,y)(y/axis Interception) f'1 0( ) 3
f
1
(x)
f''1
x( )2
x
f1
x( )d
d
2
factor 2
f
1
(x)Domain f '' (x) Domain: all x
f (x)(x,0)(Roots) f'' x( ) 0= Undefined
f (x)(0,y)(y/axis Interception) f''
1
0( ) 2
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5/21/2018 Mathcad - Applications of Derivatives
2/14
VIA UC
Mechanical Engineering Program
Javier I Camacho (164649)
10 8 6 4 2 0 2 4 6 8 10
10
8
6
4
2
2
4
6
8
10
f(x)
f1(x) Curve Plot
10
8
6
4
2
0 2 4 6 8 10
10
8
6
4
2
2
4
6
8
10
f '(x)
f1 '(x) Curve Plot
10 8 6 4 2 0 2 4 6 8 10
10
8
6
4
2
2
4
68
10
f ''(x)
f1 ''(x) Curve Plot
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5/21/2018 Mathcad - Applications of Derivatives
3/14
VIA UC
Mechanical Engineering Program
Javier I Camacho (164649)
2 )
f
2
(x)
f2
x( ) 3x3
16x factor x 3 x2
16
f
2
(x)Domain f2(x) Domain: all x
f
2
(x)(x,0)(Roots)
Given f 2 x( ) 0= Find x( ) 04i 3
3
4i 33
f(x)(0,y)(y/axis Interception) f2 0
( ) 0
f
2
(x)
f'2 x( )x
f2 x( ) d
dfactor 9 x
2 16
f
2
(x)Domain f'2(x) Domain: all x
f
2
(x)(x,0)(Roots) Given f'
2
x( ) 0= Find x( )4
3
i 4
3
i
1.333i 1.333i( )
f
2
(x)(0,y)(y/axis Interception) f'2 0
( ) 16
f
2
(x)
f''2 x( ) 2x
f2 x( )d
d
2
factor 18 x
f
2
(x)Domain f '' (x) Domain: all x
f
2
(x)(x,0)(Roots) f''2
x( ) 0=
f
2
(x)(0,y)(y/axis Interception) f''2 0
( ) 0
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5/21/2018 Mathcad - Applications of Derivatives
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VIA UC
Mechanical Engineering Program
Javier I Camacho (164649)
25 20 15 10 5 0 5 10 15 20 25
4 103
3.2 103
2.4 103
1.6 103
800
800
1.6 103
2.4 103
3.2 103
4 103
f(x)
f2(x) Curve Plot
25 20 15 10 5 0 5 10 15 20 25
4 103
3.2 103
2.4 103
1.6 103
800
800
1.6 103
2.4 103
3.2 103
4 103
f '(x)
f2 '(x) Curve Plot
25 20 15 10 5 0 5 10 15 20 25
4 103
3.2 103
2.4 103
1.6 103
800
800
1.6 103
2.4 103
3.2 103
4 103
f ''(x)
f2 ''(x) Curve Plot
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5/21/2018 Mathcad - Applications of Derivatives
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VIA UC
Mechanical Engineering Program
Javier I Camacho (164649)
3)
f
3
(x)
f3
x( )x
23
x 2factor
x2
3
x 2
f
3
(x)Domain
f3(x) Domain: all x except 2
f
3
(x)(x,0)(Roots)
Given
f3 x( ) 0= Find x( ) 3 3( ) 1.732 1.732( )
f
3
(x)(0,y)(y/axis Interception)
f3 0( ) 1.5
f(x)Oblique Asymptote
x
x2
3
x 2
lim
x
x2
3
x 2
lim
ObAsymptoteslope
x
x 1( ) x 3( )
x 2( )2
lim
1
x
f3 x( ) ax b( )lim
0=
x
x2
3
x 2x 2( )
lim
0
x
x2
3
x 2x 2( )
lim
0
fObAsymptote x( ) x 2
f(x)Vertical Asymptote
2x
x2
3
x 2lim
2x
x2
3
x 2lim
fVertAsymptote x= 2=
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5/21/2018 Mathcad - Applications of Derivatives
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VIA UC
Mechanical Engineering Program
Javier I Camacho (164649)
f
3
(x)
f'3
x( )x
f3
x( )d
dfactor
x 1( ) x 3( )
x 2( )2
f
3
(x)Domain
f3(x) Domain: all x except 2
f
3
(x)(x,0)(Roots)
Given
f'2
x( ) 0=
Find x( )4
3i
4
3i
1.333i 1.333i( )
f
3
(x)(0,y)(y/axis Interception)
f'3 0
( ) 0.75
f
3
(x)Vertical Asymptote f'VertAsymptote x= 2=
2x
x 1( ) x 3( )
x 2( )2
lim
2x
x 1( ) x 3( )
x 2( )2
lim
f
3
(x)Horizontal Asymptote
x
x 1( ) x 3( )
x 2( )2
lim
1
x
x 1( ) x 3( )
x 2( )2
lim
1
f'HorizAsymptote
x
x 1( ) x 3( )
x 2( )2
lim
1
x
f' x( ) ax b( )[ ]lim
0=
x
x 1( ) x 3( )
x 2( )2
1( )
lim
0
x
x 1( ) x 3( )
x 2( )2
1( )
lim
0
f'HorizAsymptote
x( ) 1
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5/21/2018 Mathcad - Applications of Derivatives
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VIA UC
Mechanical Engineering Program
Javier I Camacho (164649)
f
3
(x)
f''3 x( ) 2x
f3 x( )d
d
2
factor2
x 2( )3
f
3
(x)Domain
f3'' (x) Domain: all x except 2
f
3
(x)(x,0)(Roots)
f''3 x( ) 0=Undefined
f
3
(x)(0,y)(y/axis Interception)
f''3 0
( ) 0.25
f
3
(x)Vertical Asymptote
2x
2
x 2( )3
lim
2x
2
x 2( )3
lim
f
3
(x)Horizontal Asymptote
x
2
x 2( )3
lim
0
x
2
x 2( )3
lim
0
f''HorizAsymptote
x( ) 0
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5/21/2018 Mathcad - Applications of Derivatives
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VIA UC
Mechanical Engineering Program
Javier I Camacho (164649)
24/07/2014 8 of 14
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5/21/2018 Mathcad - Applications of Derivatives
9/14
VIA UC
Mechanical Engineering Program
Javier I Camacho (164649)
10 8 6 4 2 0 2 4 6 8 10
10
8
6
4
2
2
4
6
8
10
f(x)
f(x) Oblique Asymptote
f3(x) Curve Plot
22
10 8 6 4 2 0 2 4 6 8 10
10
8
6
4
2
2
4
6
8
10
f '(x)
f '(x) Horizontal Asymptote
f3 '(x) Curve Plot
22
10 8 6 4 2 0 2 4 6 8 10
10
8
6
4
2
2
4
6
8
10
f ''(x)
f ''(x) Horizontal Asymptote
f3 ''(x) Curve Plot
22
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5/21/2018 Mathcad - Applications of Derivatives
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VIA UC
Mechanical Engineering Program
Javier I Camacho (164649)
4)
f
4
(x)
f4
x( )x
2x 2
x 3factor
x 1( ) x 2( )
x 3
f (x)Domain
f (x) Domain: all x except 3
f(x)(x,0)(Roots)
Given
f4
x( ) 0= Find x( ) 2 1( ) 2 1( )
f(x)(0,y)(y/axis Interception)
f4
0( ) 0.667
f(x)Oblique Asymptote
x
x 1( ) x 2( )
x 3lim
x
x 1( ) x 2( )
x 3lim
f4ObAsymptoteslope
x x
x 1( ) x 2( )
x 3
d
dlim
1
x
f x( ) ax b( )[ ]lim
0=
x
x 1( ) x 2( )
x 3
x 2( )
lim
0x
x 1( ) x 2( )
x 3
x 2( )
lim
0
f4ObAsymptote x( ) x 2
f(x)Vertical Asymptote
3x
x 1( ) x 2( )
x 3lim
3x
x 1( ) x 2( )
x 3lim
fVertAsymptote x= 3=
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5/21/2018 Mathcad - Applications of Derivatives
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VIA UC
Mechanical Engineering Program
Javier I Camacho (164649)
f
4
(x)
f'4 x( )x
f4 x( )d
dfactor
x 1( ) x 5( )
x 3( )2
f (x)Domain
f (x) Domain: all x except 3
f (x)(x,0)(Roots)
Given
f'4 x( ) 0=
Find x( ) 1 5( ) 1 5( )
f (x)(0,y)(y/axis Interception)
f'4 0( ) 0.556
f (x)Vertical Asymptote f'VertAsymptote x= 3=
3x
x 1( ) x 5( )
x 3( )2
lim
3x
x 1( ) x 5( )
x 3( )2
lim
f (x)Horizontal Asymptote
x
x 1( ) x 5( )
x 3( )2
lim
1
x
x 1( ) x 5( )
x 3( )2
lim
1
f4'HorizAsymptotex
x 1( ) x 5( )
x 3( )2
lim
1
x
f' x( ) ax b( )[ ]lim
0=
x
x 1( ) x 5( )
x 3( )2
1( )
lim
0
x
x 1( ) x 5( )
x 3( )2
1( )
lim
0
f4'HorizAsymptote
x( ) 1
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5/21/2018 Mathcad - Applications of Derivatives
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VIA UC
Mechanical Engineering Program
Javier I Camacho (164649)
f
4
(x)
f''4
x( )2
x
f4
x( ) dd
2
factor8
x 3( )3
f (x)Domain
f '' (x) Domain: all x except 3
f (x)(x,0)(Roots)
f''4
x( ) 0= Undefined
f (x)(0,y)(y/axis Interception)
f''4 0
( ) 0.296
f (x)Vertical Asymptote
3x
8
x 3( )3
lim
3x
8
x 3( )3
lim
f (x)Horizontal Asymptote
x
8
x 3( )3
lim
0
x
8
x 3( )3
lim
0
f4''HorizAsymptote
x( ) 0
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5/21/2018 Mathcad - Applications of Derivatives
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VIA UC
Mechanical Engineering Program
Javier I Camacho (164649)
10 8 6 4 2 0 2 4 6 8 10
50
40
30
20
10
10
20
30
4050
f(x)
f(x) Oblique Asymptote
f4(x) Curve Plot
3
10 8 6 4 2 0 2 4 6 8 10
10
8
6
4
2
2
4
6
8
10
f '(x)
f '(x) Horizontal Asymptote
f4 '(x) Curve Plot
3
10 8 6 4 2 0 2 4 6 8 10
10
8
6
4
2
2
4
6
8
10
f ''(x)f ''(x) Horizontal Asymptote
f4 ''(x) Curve Plot
3
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Mechanical Engineering Program
Javier I Camacho (164649)
Product Design
Youe have been asked to desifn a 1L oil can shaped like right circular cylinder.What dimensions will use the least material?
Volume: Area Circle: Perimeter Circle: Lateral Area:
V r h( ) r2
h= Acircle r( ) r2
= Pcircle r( ) 2 r( )= ALateral a b( ) a b=
h r V V
r2
S.
r h( ) 2 r h 2 r2
S r V 2 V
r2 r
2
S' r V rS r V
d
d4 r
2 V
r2
4 r
2 V
r2 0=
r3
2 V
4 =
1
2
V
=1
2 r
2 h = 1
2r2
h = h 2r=
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