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Zusammenfassung Integral Transformationen - Differentialgleichung 1. und 2. Ordnung $7.67
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Zusammenfassung Integral Transformationen - Differentialgleichung 1. und 2. Ordnung

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Der Lernzettel enthält eine Zusammenfassung der Themen - Ableitung/partielle Ableitung, - Differentialgleichung 1. und 2. Ordnung - bestimmtes/unbestimmtes Integral 1 - Mehrfachintegrale -bestimmtes/unbestimmtes Integral 2 - Grenzwertbestimmung Integrale - LaPlace 1. und 2. Ordnung - Fouri...

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  • October 4, 2024
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  • 2023/2024
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Integral Transformation MAT31 Lernzettel für Klausur am On 06
.
.
2024




1 .
Ableitungen
.1.
1
Ableitung 1 .
Ordnung
f(x) =
2 -

7sin(bx) -
7e8x
*
f'(x) =
9 0
2
**
(n(2) -


6 -
7cos((x) -
8 7e
.
*
18 (n(2) 42cs(bx) -
-

562




Hintergrundrechnung :




=> 2 =
innere äußere Ableitung = 9x9
; G
**
>
-
G
**
(n(2) = 9 .
2
**
(n(2) =
*
18 (n(2)

=
-Esin(bx) =
innere äußere Ableitung 6X-6 ; -Esin (6x)t -
Ecos(bx) => 6 -
-


7cos(bx) = -
42cos(bx)

* *
=> -7e8x =
innere äußere Ableitung 8x =
8j-7e
*
--7e
*
= 8 .
-


7e = -

56e




.1 1
1 . . Spezialfälle
Normalform Ableitung COS -
Sin
X
ex e
-


sin -

COS

aX aX
e : 2 COS sin
-




a


"(x 1
/yn
1 -
1
-
1

abx ab (n(a) =
-n -




b : =
X -

nX
-




(m(x) X xn M o
X
n - 1



*
sin COS * =
y
+
2xox = 125


.2
1 . partielle Ableitung
f(x y) ,
= 76x 3y
-




+
EySin(9y
t +
4xy) =


↳ 8 X ableiten

ableiten nach Y
nach X




76x 3"
/2
-




f(x , y) = 6 .
(n(z) + +
4ycos(9y + 4xy) nach x abgeleitet


Hintergrundrechnung Spezialfall :




Ei y -xy* =
4 + X -
2 =(4x xy)
.
*
- I 4+ x .. Ey) =
innere Ableitung

, .3
1 . partielle Ableitung mit
Tangentialebene
f(x y) ,
= z Sin(amX 5ny) ; P(Xo Zo)
+ + , = (3 2 , 9)
,
Ex
f(x y) ,
=
21xy+ 2 cos(anx 5ny) +




Hintergrundrechnung :




75
* 7 (i) äußere Ableitung
7innere
.




-
= 21x3
1

=>



Zahlen von P in die Gleichung einsetzen :

Gacoa Yan(a
21 3 .




= + 52) =
153




f(x y) ,
= z + Sin(amX + 5ny) ; P(Xo - Zo) = (3 2 , 9)
, S
f'(x y) ,
=
-




1x + 5 coS(anx + 5 y)



Hintergrundrechnung :




7 (2 .
·
innere äußere Ableitung =* = - 7. =( )
*
= 7 .


(2) () -
-
·
14xy2



-
2ahlen von P in die Gleichung einsetzen :



y Y ·
14X -
14 3 .




+ 50S(anx 5πy) 5 cos (an 3 + 5n 2) 6 , 615
-



+ =
2. 2 + -
=




Punkt P und den Gleichung
Zahlen aus
Gleichungen in die der
Tangentialebene einsetzen
:



z -


zo =
fx (Xo Xd)(X Xo) fy(xa vd)(y
,
-
·

,
-


y) = z -

zo = (x -

Xo) -
(y yo
-




z - 9 =
(15 , 376 3) (6 , 615 2) - . -




z -
9 = 57 , 1151 + 9
Z = 66 115 ,

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