Zusammenfassung für die Mathe Matura (in English)

Analysis


rates of change
-
absolute change
:
f(b) -


f(a)


----
absolute
change
----
o



a D



-

relative (percentage) change :

f(b)
-



f(a)
f(a)

------
f(b)
-...




f(b) -


f(a)
100
percentage change
>

f(a)
-




f(a)
--
....



a b




difference quotient (average of change/mittlern Änderungsrate)
f(b)-l
:
-

rale




change over an interval !
sequante
G




average change
81

a b




-



differential quotient (instantaneous rak of change/momentane Änderungsrate) :
f(x) =Cimf


change at one specific point !

tangent

instan
,

, Kinematics


displacement s(t)

differentiales velocity v(t) ~
integrate

sacceleration alt)




H) It c
displacement


Sv(t) It < total distance travelled



Rules of differentiation
x" f(x) +"
1
f(x)
=
-
n
-



.
=
a - -
a




k g(x) f(x) k
g'(x)(k constant)
f(x)
-
.
2 = -
= -

...




3 .


f(x) =
g(x) =
h(x) +
f(x)
=

g(x) = h'(x)


Chain Rule : Quotient Rule :




y f(g(x) u(x)
g(x)"
=




y
=

f(x) =
v(x)

n -
1 y' =
g'(x) -


f'(x)
y
=
n
g(x)
-

g(x) f(x) u'(x) v(x) u(x) v'(x)
-

=
-
- .




2
.
et cos(x2 b) -
v(x)
ex .
(Sin(x)5 -
f 3 ex x + 6

(sin(x)"
.




u(r) u'(r) v(x) v'(x)
5 .


(cos(t)) ·


-
sin (x" 3) - .
2x 2x + 1
x2 + 62x2x + 12
f(x) g(x
(2x) (2x 1) (2 + 6)
.




+
2
-
-
.




[2x 1]2 +



Product Rule :




f(x) =
u(x) .

v(x)


f'(x) =
u((x) v(x) .
+ v((x) u(x) -




(x 4)
5
f =
(x 4)af) 5x4 + 2
(x 4)
.
-

= +
.
= -


ex
.

x
-

.




u(r) u'(r) v(x) v'(x)


x55"( 4)22 (x 4) -
.
-