calcul diferencial y integral: màxim y mínim

Si ( d_{x}[f(c)] = 0 & f(c)·d_{xx}^{2}[f(c)] < 0 ) ==> ...

... int[ 0 --> f(x) ]-[ x ] d[x] té un màxim en x = c

Si ( d_{x}[f(c)] = 0 & f(c)·d_{xx}^{2}[f(c)] > 0 ) ==> ...

... int[ 0 --> f(x) ]-[ x ] d[x] té un mínim en x = c

Si ( f(c) = 0 & d_{x}[f(c)]^{2} < 0 ) ==> ...

... int[ 0 --> f(x) ]-[ x ] d[x] té un màxim en x = c

Si ( f(c) = 0 & d_{x}[f(c)]^{2} > 0 ) ==> ...

... int[ 0 --> f(x) ]-[ x ] d[x] té un mínim en x = c

Demostració:

d_{x}[ int[ 0 --> f(x) ]-[ x ] d[x] ] = f(x)·d_{x}[f(x)]

d_{xx}^{2}[ int[ 0 --> f(x) ]-[ x ] d[x] ] = d_{x}[f(x)]^{2}+f(x)·d_{xx}^{2}[f(x)]

Exemples:

f(x) = i·( x+(-c) ) <==> d_{x}[f(x)] = i

int[ 0 --> f(x) ]-[ x ] d[x] = (-1)·(1/2)·( x+(-c) )^{2}

f(x) = (-i)·( x+(-c) ) <==> d_{x}[f(x)] = (-i)

int[ 0 --> f(x) ]-[ x ] d[x] = (-1)·(1/2)·( x+(-c) )^{2}

f(x) = ( x+(-c) ) <==> d_{x}[f(x)] = 1

int[ 0 --> f(x) ]-[ x ] d[x] = (1/2)·( x+(-c) )^{2}

f(x) = (-1)·( x+(-c) ) <==> d_{x}[f(x)] = (-1)

int[ 0 --> f(x) ]-[ x ] d[x] = (1/2)·( x+(-c) )^{2}

Si ( d_{x}[f(c)] = 0 & d_{xx}^{2}[f(c)] < 0 ) ==> ...

... int[ 0 --> f(x) ]-[ e^{x} ] d[x] té un màxim en x = c

Si ( d_{x}[f(c)] = 0 & d_{xx}^{2}[f(c)] > 0 ) ==> ...

... int[ 0 --> f(x) ]-[ e^{x} ] d[x] té un mínim en x = c

Demostració:

d_{x}[ int[ 0 --> f(x) ]-[ e^{x} ] d[x] ] = e^{f(x)}·d_{x}[f(x)]

d_{xx}^{2}[ int[ 0 --> f(x) ]-[ e^{x} ] d[x] ] = e^{f(x)}·d_{x}[f(x)]^{2}+e^{f(x)}·d_{xx}^{2}[f(x)]

Exemples:

d_{x}[f(x)] = ( c+(-x) ) <==> d_{xx}^{2}[f(x)] = (-1)

int[ 0 --> f(x) ]-[ e^{x} ] d[x] = e^{(-1)·(1/2)·( c+(-x) )^{2}}+(-1)

Inflexió: ( en x = c+1 || en x = c+(-1) )

d_{x}[f(x)] = ( x+(-c) ) <==> d_{xx}^{2}[f(x)] = 1

int[ 0 --> f(x) ]-[ e^{x} ] d[x] = e^{(1/2)·( x+(-c) )^{2}}+(-1)

Inflexió: ( en x = c+i || en x = c+(-i) )

càlcul diferencial y integral linialitat

d[ f(x)+g(x) ] = ( f(x+h)+g(x+h) )+(-1)·( f(x)+g(x) )

d[ f(x)+g(x) ] = ( f(x+h)+(-1)·f(x) )+( g(x+h)+(-1)·g(x) ) = d[f(x)]+d[g(x)]

d[ s·f(x) ] = ( s·f(x+h) )+(-1)·( s·f(x) ) = s·( f(x+h)+(-1)·f(x) ) = s·d[f(x)]

d_{x}[ int[ f(x)+g(x) ] d[x] ] = f(x)+g(x) = d_{x}[ int[f(x)] d[x] ]+d_{x}[ int[g(x)] d[x] ]

d_{x}[ int[ f(x)+g(x) ] d[x] ] = d_{x}[ int[f(x)] d[x]+int[g(x)] d[x] ]

int[ d_{x}[ int[ f(x)+g(x) ] d[x] ] ] d[x] = int[ d_{x}[ int[f(x)] d[x]+int[g(x)] d[x] ] ] d[x]

int[ f(x)+g(x) ] d[x] = int[f(x)] d[x]+int[g(x)] d[x]

d_{x}[ int[ s·f(x) ] d[x] ] = s·f(x) = s·d_{x}[ int[f(x)] d[x] ]

d_{x}[ int[ s·f(x) ] d[x] ] = d_{x}[ s·int[f(x)] d[x] ]

int[ d_{x}[ int[ s·f(x) ] d[x] ] ] d[x] = int[ d_{x}[ s·int[f(x)] d[x] ] ] d[x]

int[ s·f(x) ] d[x] = s·int[f(x)] d[x]

d[f(x)·g(x)] = (1/2)·( ( 2·f(x+h)·g(x+h) )+(-1)·( 2·f(x)·g(x) ) )

d[f(x)·g(x)] = (1/2)·( ( f(x+h)+(-1)·f(x) )·( g(x+h)+g(x) )+( f(x+h)+f(x) )·( g(x+h)+(-1)·g(x) ) )

d[f(x)·g(x)] = d[f(x)]·g(x)+f(x)·d[g(x)]

f(x)·g(x) = f(x) [o(x)o] int[g(x)] d[x]+int[f(x)] d[x] [o(x)o] g(x)

x^{n} = x^{(n/2)} [o(x)o] ( 4/(n+2) )·x^{((n+2)/2)}

Àlgebra de conjunts

A [ & ] B = 0 <==> A [ || ] B = A [ |o| ] B

[==>]

( x € A || x € B )

( x € A || x € B ) & ¬( x € A & x € B )

( x € A |o| x € B )

[<==]

( x € A & x € B )

( x € A || x € B )

( x € A |o| x € B )

( x € A || x € B ) & ¬( x € A & x € B )

A [ & ] B = 0 <==> A [ \ ] B = A

[==>]

x € A

x € A & ¬( x € A & x € B )

x € A & ( ¬( x € A ) || ¬( x € B ) )

( x € A & ¬( x € A ) ) || ( x € A & ¬( x € B ) )

0 || ( x € A & ¬( x € B ) )

( x € A & ¬( x € B ) )

[<==]

x € A & x € B

( x € A & ¬( x € B ) ) & x € B

tensores

A^{i}_{j}·a^{j} = a^{i}

A^{j}_{i}·a^{i} = a^{j}

(A^{i}_{j}·B^{i}_{j}) = (A·B)^{i}_{j}

(B^{i}_{j}·A^{i}_{j}) = (B·A)^{i}_{j}

A^{i}_{j}·A^{j}_{i} = 1

A^{i}_{j}·A_{i} = A_{j}

A^{i}_{j}·A^{j} = A^{i}

A^{i}_{j}·x+B^{j}_{i} = 0 <==> x = (-1)·(A·B)^{j}_{i}

A^{i}_{j}·x+(-1)·B^{j}_{i} = 0 <==> x = (A·B)^{j}_{i}

A^{i}_{j}·x^{n}+B^{j}_{i} = 0 <==> x = ( (-1)·(A·B)^{j}_{i} )^{(1/n)}

A^{i}_{j}·x^{n}+(-1)·B^{j}_{i} = 0 <==> x = ( (A·B)^{j}_{i} )^{(1/n)}

A^{i}_{j}·x = B^{i}_{j}·y <==> ( x = (A·C)^{j}_{i} & y = (B·C)^{j}_{i} )

A^{j}_{i}·x = B^{j}_{i}·y <==> ( x = (A·C)^{i}_{j} & y = (B·C)^{i}_{j} )

A^{i}_{j}·x^{n} = B^{i}_{j}·y^{m} <==> ...

... ( x = ( (A·C)^{j}_{i} )^{(1/n)} & y = ( (B·C)^{j}_{i} )^{(1/m)} )

A^{j}_{i}·x^{n} = B^{j}_{i}·y^{m} <==> ...

... ( x = ( (A·C)^{i}_{j} )^{(1/n)} & y = ( (B·C)^{i}_{j} )^{(1/m)} )

A^{i}_{j}·( x+B^{i}_{j} ) = C^{j}_{i} <==> x = (A·C)^{j}_{i}+(-1)·B^{i}_{j}

A^{i}_{j}·( x+(-1)·B^{i}_{j} ) = C^{j}_{i} <==> x = (A·C)^{j}_{i}+B^{i}_{j}

A^{i}_{j}·( x^{n}+B^{i}_{j} ) = C^{j}_{i} <==> ...

... x = ( (A·C)^{j}_{i}+(-1)·B^{i}_{j} )^{(1/n)}

A^{i}_{j}·( x^{n}+(-1)·B^{i}_{j} ) = C^{j}_{i} <==> ...

... x = ( (A·C)^{j}_{i}+B^{i}_{j} )^{(1/n)}

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verbos beber y deber o saber y caber

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