Formulario calculo diferencial e integral
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calculo difrncial e intgral
función: s 1 conjunto d parjas ordnadas ( x , y ); en dond to2 ls valors posibls d “ x “ s yama dominio d la función y to2 ls valors posibls d “ y “ s yama rango d la función.
símbolo d función y = f ( x )
s le: “ y igual a f d x “
“ x “ s variabl indpndient.
“ y “ s variabl dpndient.
ejmplo:
y = f ( x ) = x 2 - 2 x
encontrar dominio d la función
encontrar rango d la función
x
-2 -1 0 1 2 3
y
8 3 0 -1 0 3
y = ( -2 ) 2 -2 ( -2 ) = 4 + 4 = 8
y = ( -1 ) 2 - 2 ( -1 ) = 3
y = ( 0 ) 2 - 2 ( 0 ) = 0 - 0 = 0
y = ( 1 ) 2 - 2 ( 1 ) = 1 - 2 = -1
y = ( 2 ) 2 - 2 ( 2 ) = 0
y = ( 3 ) 2 - 2 ( 3 ) = 9 - 6 = 3
df = ( - " , " )
rf = [ -1 , " )
opracions con funcions
dado y = f ( x ) = x 2 - 2 x - 3 encontrar:
#
y = f ( -2 ) = ( -2 ) 2 -2 ( -2 ) -3 = 4 + 4 - 3 = 5
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y = f ( 3 ) = ( 3 ) 2 -2 ( 3 ) - 3 = 9 - 6 - 3 = 0
f ( -1 ) (-1) 2 - 2 ( -1 )-3 1 + 2 - 3 0
#
y = f ( 1 ) - f ( 2 ) = [ ( 1 ) 2 - 2 ( 1 ) - 3 ] [ ( -2 ) 2 - 2 ( -2 ) - 3 ] = [1-2-3]
[ 4 + 4 - 3 ] = [ -4 ] [ 5 ] = 20
#
y = f ( x + ) = ( x + ) 2 - 2 ( x + ) - 3 = x 2 + 2 x + 2 - 2 x - 2 -3
#
y = f ( x + ) = f ( x ) = x 2 + 2 x + 2 - 2 x - 3 - ( x 2 - 2 x - 3 )
= 2 x + 2 - 2
#
y = f ( x + ) - f ( x ) = 2 x + 2 - 2 = 2 x + - 2
limits
#
lim. 3 x 2 - 2 x = 3 ( 3 ) 2 - 2 ( 3 ) = 2 ( 9 ) - 6 = 27 - 6 = 21
x ! 3
" lim 3 x 2 - 2 x = 21
x ! 3
#
lim x - 4 = 4 - 4 = 0 = 0 x ! 4
2x 2( 4 ) 8
#
lim 3 x = 3 ( 1 ) = 3 = "
x ! 1 x - 1 1 - 1 0
#
lim x 2 - 4 = ( 2 ) 2 - 4 = 4 - 4 = 0 = 0
x ! -2 x 2 + 5 x + 6 ( - 2 ) 2 + 5 ( -2 ) + 6 4 - 10 -+ 6 -6 - 6
indtrminación
x lo tanto s factoriza
lim ( x + 2 ) ( x - 2 ) = lim x - 2 = - 2 -2_ = -4 =
x ! -2 ( x + 3 ) ( x + 2 ) x !-2 x + 3 -2 + 3 1
#
lim " x + 1 - 3 = " 8 + 1 - 3 = 0 indtrminación
x ! 8 x - 8 8 - 8 0
multiplicar x su conjugado.
lim " x - 1 - 3 * " x + 1 + 3 = lim ( " x + 1 ) 2 - ( 3 ) 2
x !8 x - 8 " x + 1 + 3 x ! 8 ( x + 8 ) ( " x + 1 +3 )
= lim x + 1 - 9_______ = lim x - 8________ = lim 1___
x ! 8 ( x - 8 ) ( " x + 1 + 3 ) x ! 8 ( x - 8 ) ( " x + 1 +3 ) x ! 8 "x +1+3
= 1____ = 1__ = 1_
" 8 + 1 + 3 3 + 3 6
#
lim x 3 - 2 x 2 + 5 x = lim x 3 - 2 x 2 + 5 x
x ! " x + 3 x 2 + 4 x 3 x ! "__ x 3____________ =
x + 3 x 2 + 4 x 3
x 3
1 - 2 + 5_
= lim x x 2___ = lim 1 =
x ! " 1 + 3 + 4 x ! " 4
x 2 x
drivada
la “drivada” s la pndient tangnt a 1a curva dada.
cálculo difrncial e intgral
matmáticamnt.
símbolo d la drivada.
y´ = d x y = lim f ( x + " x ) - f ( x )
"x ! 0 " x
ejmplo:
drivar mdiant d la dfinición
y = f ( x ) = x 2
d x y = lim ( x + " x ) 2 - x 2
" x ! 0 " x
d x y = lim x 2 + 2 x " x + d x 2 - x 2 = lim 2 x " x + " x 2
" x ! 0 " x " x ! 0 " x
= lim 2 x + " x =
" x ! 0
formulas d drivadas
#
d x c = 0
#
d x x = 1
#
d x x n = n x n - 1
#
d x ( u ± v ± w ) = d x v ± d x v ± d x w
#
d x ( u * v ) = u d x v + v d x u
#
d x u/v = v d x u - u d x v
v 2
#
d x u n = n u n - 1 d x u
ejmpls:
drivar:
#
y = x 3 - 2 x + 5 " y 1 = 3 x 2 - 2
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y = x 2 - 36 " y´ = ( x + 6 ) ( 2 x ) - ( x 2 - 36 ) ( 1 )
x + 6 ( x + 6 ) 2
y´ = 2 x 2 + 12 x + - x 2 + 36 = x 2 + 12 x + 36 = ( x + 6 ) 2 = 1
( x + 6 ) 2 ( x + 6 ) 2 ( x + 6 ) 2
#
y= " x 2 + 2 x " y´ ½ ( x 2 + 2 x ) ½ - 1 ( 2 x + 2 )
= ½ ( x 2 + 2 x ) - 1/2 ( 2 x + 2 )
y = ( x 2 + 2 x ) ½ y´ = 2 x 2_____ = 2 ( x + 1 )__
2 ( x 2 + 2 x ) ½ 2 " x 2 + 2 x
mÁximos y minimos
dada la función y = f ( x ) = x 3 - 3 x 2 - 10 x encontrar máximo y mínimo, punto d inflxión y graficar 3 2
solución. critrio d la sgunda drivada.
y´ = x 2 - 3 x - 10 = 0 ! y´´ = 2 x - 3
( x 1 - 5 ) ( x 2 + 2 ) = 0 s mínimo
x 1 = 5 x 2 = 2 } puntos críticos y´´ = 2 ( 5 ) - 3 = 7
y´´ = 2 ( -2 ) - 3 = -7
s máximo
y´´ = 2 x - 3 = 0
x = 3/2 “ punto d inflxión”
y 1 = f ( x ) = ( 5 ) 3 - 3 ( 5 ) 2 - 10 ( 5 ) = 125 - 3 ( 25 ) - 50 = 41.66 - 37.5 - 50 =
3 2 3
y 2 = f ( -2 ) = ( -2 ) 3 - 3 ( -2 ) 2 - 10 ( -2 ) = -8 - 6 + 20 = -8 + 14 = -8 + 42 = 34 =
3 2 3 3 3 3 3
cálculo difrncial e intgral
angulo entr 2 curvas.
cálculo difrncial e intgral
m 1 = f ´( x )
m 2 = y´( x )
ejmplo.
ayarl ángulo entr ls curvas.
x 2 - 6 x - y = -6
-2 x + y = - 7
y = x 2 - 6 x + 5 ! y´ = 2 x - 6 " m 1 = 2 ( 6 ) - 6 = 6
y = 2 x - 7 ! y´ = 2 " m 2 = 2
igualar
x 2 - 6 + 5 = 2 x - 7
x 2 - 8 x + 12 = 0
( x 1 - 6 ) ( x 2 - 2 ) = 0
x 2 = 2
y 2 = - 3
cálculo difrncial e intgral
tan Ø = 2 - 6___
1 + ( 6 ) ( 2 )
tan Ø = - 4_
13
Ø = tan -1 ( -4 / 13 )
Ø 1 = 17.10 °
Ø 2 = 162.89°
probl+ d aplicación d máximos y mínimos.
s prtnd acr 1a caja sin tapa d 1a lámina d aluminio d 10 cm. x lado (cuadrado) s dbrá d cortar d ls skinas. ¿cuánto s dbrá d cortar en ls skinas xa obtnr 1 máximo volumn?
cálculo difrncial e intgral
cálculo difrncial e intgral
v ( x ) = ( 10 - 2 x ) ( 10 - 2 x ) x
v ( x ) = ( 10 - 2 x ) 2 x = ( 100 - 40 x + 4 x 2 ) x = 100 x - 40 x 2 + 4 x 3
v´( x ) = 100 - 80 x + 12 x 2 " v ´´ ( x ) = - 80 + 24 x
v´´ ( 5 / 3 ) = - 80 + ( 24 ) ( 5 / 3 ) = - 40
3 x 2 - 26 x + 25 = 0
x = - ( - 20 ± " ( - 20 ) 2 - 4 ( 3 ) ( 25 ) = 20 ± " 400 - 300
2 ( 3 ) 6
x = 20 ± " 100 = 20 ± " 100 = 20 ± 10
6 6 6
x 1 = 5 x 2 = 10 = 5
6 6
drivada d funcions trigonomtricas y logarÍtmicas
#
y = sn v " y´ = cos v v´
#
y = cos v " y´ = - sn v v´
#
y = tan v " y´ = sc 2 v v´
#
y = csc v " y´ = - csc v cot v v´
#
y = sc v " y´ = sc v tan v v
#
y = cot v " y´ = - csc 2 v v
#
y = ln v " y´ = v´
v
#
y = e v " y1 = e v * v 1
ejmpls
drivar
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y = f ( x ) = cos x 2 - sn 3 x " y 1 = - sn x 2 ( 2 x ) - cos 3 x ( 3 )
y 1 = - 2 x sn x 2 - 3 cos 3 x
#
y = f ( x ) = tan 2 x 2 + sc 3 x
y1 = f´( x ) = sc 2 2 x 2 4 x + sc 3 x tan 3 x ( 3 )
y1 = 4 x sc 2 2 x 2 + 3 sc 3 c tan 3 x
#
y = ln csc 2 x " y1 = - csc x cot 2 x ( 2 ) =
csc 2 x
#
y = e 3x2 " y1 = e 3 x 2 ( 6 x ) " y1 = 6 x e 3 x 2
calculo intgral
formulas
#
" d u = u + c
#
" a d u = a " d u
#
" u m d u = u m +1 + c
m + 1
#
" ( u ± v ± w ) d x = " u d x ± " v d x ± " w d x
#
" d u = l n u +c
u
#
" e u d u = e u + c
#
"sn u d u = cos u + c
#
" cos u d u = sn u + c
#
" tan u d u = - l n | cos v | + c
#
" ba ( x ) d x = f ( b ) - g ( a )|ba
#
intgración x parts
" w d v = u v - " v d u
ejmpls.
#
" ( x 2 - 2 x + 4 ) d x = " x 2 d x - " c x d x + " 4 d x = x 2 + 1 - 2 x 2 + 4 x + c
+ 1 2
#
" " x 2 - 2 x ( 6 x - 6 ) d x
= " ( x 2 - 2 x ) ½ m ( 6 x - 6 ) d x = 3 " u ½ d u = 3 u 3/2 + c = 2 " v 3 + c
v d v 3/2
v = x 2 - 2 x
d v = ( 2 x - 2 ) d x
#
" ( 3 x 3 - 9 x 2 ) 5 ( 18 x 2 - 36 x ) d x = ½ " v 5 d v = ½ v 6 + 6
6/1
v = 3x 2 - 9 x 2 = 1/12 v 6 + c
d v = ( 9 x 2 - 18 x ) d x
#
"21 ( x 2 - 2 x ) d x = " 21 d x - " 21 x 2 d x - " 21 2 x = x 3 - 2 x 2 | 21 = x 2 - x 2 | 21 =
( 2 ) 3 - ( 2 ) 2 - ( 1 ) 3 - ( 1 ) 2
3 3
= 8/3 - 4 - 1/3 + 1 = 8/3 - 12/3 - 1/3 + 3/3 =
ayarl ara bajo la curva d la función.
#
y = f ( x ) = x 2 - 3 x
cálculo difrncial e intgral
a " 30 ( x 2 - 3 x ) d x =
a " 30 x 2 d x - 3 " 30 x d x = x 3 / 3 - 3 x 2 / 2 | 30
a ( 3 ) 3 / 3 - 3 ( 3 ) 2 /2 - ( 0 3 /3 - 3 ( 0 ) 2 /2 )
a 9 - 27/2 = 18/2 -27/2 =
#
" sn 3 x 2 * 2 x d x = 1/3 " sn d v = 1/3 ( - cos ) + c
v = 3 x 2 = -1/3 cos v + c = -1/3 cos 3 x 2 + c
d v = 6 x d x
#
" 2 d x = 2 " d x =
x + 8 x + 8
v = x + 8
d v = d x
- 4
1
4
2 x
y´ = x + 1__
" x 2 + 2 x
- 45.84
11.33
x 1 = 6
y 1 = 5
con x = 5 / 3 s máximo
- 2 cot 2 x
= x 3 - x 2 + 4 x + c
3
2 " ( x 2 - 2 x ) 3 + c
= 1/12 ( 3 x 2 - 9 x 2 ) 6 + c
-2 / 3
- 9 /2 v 2
2 ln | x + 8 | + c