108 lines
1.4 KiB
Typst
108 lines
1.4 KiB
Typst
|
#import "../template.typ": apply-template
|
||
|
|
#show: apply-template
|
||
|
|
#set page(header: [Mathe am 30.09.2024])
|
||
|
|
|
||
|
|
#outline()
|
||
|
|
|
||
|
|
= Exponentialfunktionen
|
||
|
|
|
||
|
|
== S. 101 Nr. 7
|
||
|
|
|
||
|
|
=== a)
|
||
|
|
|
||
|
|
// $1 dot 0.4^x$
|
||
|
|
|
||
|
|
$
|
||
|
|
f(x)&=1 dot 0.75^(x/1.8)=(0.75^(1/1.8))^x=0.8523^x
|
||
|
|
$
|
||
|
|
|
||
|
|
$1$ → 100% Lichtintensität\
|
||
|
|
$0.75$ → 75% Lichtintensität\
|
||
|
|
$1.8$ → 1,80m Tiefe
|
||
|
|
|
||
|
|
*Aufgabe:* Bis zur nächsten Stunde erklären können wie man zu dieser Formel kommt
|
||
|
|
|
||
|
|
=== b)
|
||
|
|
|
||
|
|
$
|
||
|
|
f(0.6)&=0.8523^0.6 &approx 0.91\
|
||
|
|
f(3.5)&=0.8523^3.5 &approx 0.57\
|
||
|
|
f(9)&=0.8523^9 &approx 0.23
|
||
|
|
$
|
||
|
|
|
||
|
|
=== c)
|
||
|
|
|
||
|
|
$
|
||
|
|
// wo kommt die 0.01 her????
|
||
|
|
log_0.8523 (0.01)&=x\
|
||
|
|
28.82 &approx x
|
||
|
|
$
|
||
|
|
|
||
|
|
Ab einer Tiefe unter ca. $28.82$m ist die Lichtintensität auf unter 1% gesunken.
|
||
|
|
|
||
|
|
== HA
|
||
|
|
|
||
|
|
+ S. 101 Nr. 7a) erklären können
|
||
|
|
+ S. 102 Nr. 8 und Nr. 11
|
||
|
|
|
||
|
|
== Exponentialfunktionen ableiten
|
||
|
|
|
||
|
|
$
|
||
|
|
f(x)&=a^x\
|
||
|
|
f'(x)&=f'(0) dot a^x\
|
||
|
|
"Beispiel:"\
|
||
|
|
f(x)&=2^x & (a>0) \
|
||
|
|
f'(x)&=1 dot 2^x\
|
||
|
|
|
||
|
|
"Nebenrechung:"\
|
||
|
|
f(0)&=2^0=1\
|
||
|
|
f'(0)&=1
|
||
|
|
$
|
||
|
|
|
||
|
|
// #table(
|
||
|
|
// columns: (auto,auto),
|
||
|
|
// table.header(
|
||
|
|
// [$f(x)$], [$f'(0)$],
|
||
|
|
// ),
|
||
|
|
// [$$], [],
|
||
|
|
// [$$], [],
|
||
|
|
// [$$], [],
|
||
|
|
// [$$], [],
|
||
|
|
// [$$], [],
|
||
|
|
// [$$], [],
|
||
|
|
// )
|
||
|
|
|
||
|
|
=== Wiederholung: Ganzrationale Funktionen
|
||
|
|
|
||
|
|
$
|
||
|
|
f(x)&=x^n\
|
||
|
|
f'(x)&=n dot x^(n-1)\
|
||
|
|
\
|
||
|
|
f(x)&=x^n+x^m\
|
||
|
|
f'(x)&=n dot x^(n-1) + m dot x^(m-1)
|
||
|
|
$
|
||
|
|
|
||
|
|
== $e$-Funktionen
|
||
|
|
|
||
|
|
$
|
||
|
|
f(x)=e^x\
|
||
|
|
e approx 2.71828...
|
||
|
|
$
|
||
|
|
|
||
|
|
=== S. 105 Nr. 1
|
||
|
|
*a):*
|
||
|
|
$
|
||
|
|
f(x)&=e^x+1\
|
||
|
|
f'(x)&=e^x
|
||
|
|
$
|
||
|
|
*b):*
|
||
|
|
$
|
||
|
|
f(x)&=e^x+x\
|
||
|
|
f'(x)&=e^x+1
|
||
|
|
$
|
||
|
|
*c):*
|
||
|
|
$
|
||
|
|
f(x)&=e^x+2x²\
|
||
|
|
f'(x)&=e^x+4x
|
||
|
|
$
|