Tangenten und Normalen an Funktionen

Berechnung von Tangenten und Normalen an Funktionsgraphen, die durch einen vorgegebenen Punkt verlaufen, der nicht auf dem Graphen der Funktion liegt


Skript, 2010

76 Seiten


Leseprobe


Page 8


&SXJ&X&K'&S++H: >$HX33H +S$H&HXSHXJ&X: !HKPSKB&HX'&P+++&XSJXKKKXKKKXJS&X &&&HHH:X >$HX KH:&H+&HX HJX .$!KB&HX KHX J&HX KK&HX &SH&X 33H +S$HX &SH:&:KH:&H#XOSJX&SHX00H +XSHXJ&X3$!X0<X@XX<@XX>$:&:&'&HXP$XP+&BB+X&PXSHXJ&X F&:&BX &SHX 0$'B&!X UX JS&X ""UB&X JKX U'&X JS&X 00H +G"+&S::H:PG3$!X JS&X B&SSH:X J&X KH:&H+&X KHX J&HX KK&H S!X 00H +X 0X X &!S++&BH#X ===X !S+X J&X .$!KB&HX PSHJX !&SP+X &SH&X 0$'B&!&X >&''HJ&HX P$$&HX JS&X ""UB&X JS&X &&S&H:X ! ! &HH&H# .

2&&+BSSX !&X ""S&S: &S+&HX +&+&HX KKKX &HHX J&X 00H +X HSS+X !&X KKKX J&!X KK&HX J&X 33H +S$HX BS&:+X P$HJ&HX SHX KBB:&!&SH&X K:&X >$:&:&'&HX SP+#X 2&X L'&:KH:

U+X CCCS:X JKKKX JKPPX UX JS&X (PPH:X J&X ===:K'&X &SHX PPBUPPS:&PX 9$HH&&+X !&X &P+&BB+X&J&HX KHH#XPX&J&HXJKHHX>&PPS&J&H&XF&&&H&:&XKH:&:KH:&HXJS&X &HHXU'&KKK+XJKHHXHHXCBBS:X!XSS+S:&HX:&'HSPXXU&H#X

,S&BX J&PX >$BS&:&HJ&HX " SS+PX SP+X &PX J&HX ""UB&HX &SHX KBB:&!&SHX KH&HJ'K&PX $:&&H UX JS&X (PPH:X JS&P&X 0$'B&!P+&BBBH:&HX KHX JS&X IKHJX X :&'&H#X %HP'&P$HJ&&X SJX S&'&SX JKKKKX :&KK+&+X JKPPX &SH&X S!!&X S&J&X &&HJ&X "+ ++XSHXJ&HX===:K'&HX& &HH'KXSJXXHJXJK!S+XJ&!X""UB&X&SH&X&H+PP&&&HJ&X "SS&&S+XSHXJ&XX$B:&SS+S:&HX&K'&S++H:X:&:&'&HXSJ#X2K!S+XSP+X&PXJKHHX!(:BSSX KKKX KHJ&&X 33H +S$H&HX HJX 'SP&X HSS+X '&UU PSS+S:+&X 0$'B&!P+&BBBH:&HX &$B:&SSXKHHH:&&H#

==PXJS&P&X%H+&H+S$HX&:S'+XPSSXJKPPXKKKX+&$&+SPP&XBCC+&H:&HXKP+X>$BBP+CHJS:X >&SS+&+X SJ#X &HH+HSPP&X KBB:&!&SH&X .K++X &J&HX KKPXJ&HX &$'KK++H:&HX KHX &SHH&BH&HX===:K'&HXK':&B&S+&+#

%HX J&HX ===:K'&HX X 'SPX X &J&HX :KHHK+S$HKB&X 33H +S$H&HX :&'$$&HX K+S$HKB&X 33H +S$H&HX P$S&X 33H +S$H&HX !S+X &SH&!X 0KK!&+&X '&KHJ&B+#X 2S&X &K'&S++H:X &$B:+XS!!&XHKKXJ&!XH+&HXJK:&P+&BB+&HX='BKKKPP&!K#

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

=HKB$:X X JS&P&HX '&SJ&HX $:&&HP&SP&HX &J&HX SHX J&HX ===:K'&HX 8X 'SPX 8QX JS&X .$!KB&HX'&&&H&+XSHJ&!XUXJS&X.$!KB&H:B&SSH:&HXJS&X&&S&H:X . ! ! '&UU PSS+S:+XSJ#X

Page 9


image dca983233bdf6f6345b3076101c682a3

RS+XJ&X,KBXJJ:&&&H&+&X===:K'&HXP+&S:&HXJS&XF$$+SH&XHJXJS&X"SS&&S+X!S+X J&XP$BB&X===:K'&HP+&BBBH:&HX'&K'&S+&+X&J&H#X2KHHXSJX&PXKKKX!(:BSSXP&SHX '&SX J&X (PPH:X &SHS:&X 5!!$!!H:PPPS++&X J&&+BSSX X >& U&H#X =HX J&!X :HJPC++BSS&HX$:&&HXCHJ&+XPSSXJKJJX/&J$$XHS+P#X

2S&X S&X >&&HJ&+&HX ===:K'&HX +&+&HX SHX &SH&X !&X $J&X &HS:&X CHBSS&HX 3$!X S!!&X S&J&X SHX >&PPS&J&H&HX &BB&HX KKKX P$JKPPX &SH&X :&HKK&X ,,$JHHH:X J&X ===:K'&X X &SH&X '&P+S!!+&HX &BB&X HSS+X !(:BSSX SP+#X PX KHHX /&J&!X ""UB&X &!!!$B&HX&J&H PSSX&S+&&X===:K'&HXXXXU'&B&:&HXXHJXXX&&H&H#

OS&XSHX===:K'&XX (HH&HXU'&XJS&XSH+&SBBH:XJ&X'&H&XJJXJ&H 33H +S$HP:KK&H SHX&SX&SB&'&H&HX==PPK:&HXU'&XJS&X,KBXJ&XKH:&H+&HXSHX='CH:S: &S+X>$HXJ&X K:&XJ&PX00H +&PX:&!KK+X&J&H#

'&HHKBBPX P&X SH+&&PPKH+X SP+X JS&X :&HKK&X &+KK++H:X >$HX 33H +S$H&HX !S+X &SH&!X 0KK!&+&XS&XSHX===:K'&X#

===X ===:K'&HX X :&'$$&HX K+S$HKB&HX 33H +S$H&HX &:&'&HX &S+&&X SH'BSS &X JKX S&X U'&X JS&X =P !!+$+&HX HJX 0$BP+&BB&HX &SH&X SH+&SBBH:X J&X '&H&X SHX !&&&X >&PPS&J&H&X &SB&'&H&HX &$B:+#X &X HKKX ,:&(S: &S+X J&PX 00H +&PX X &SH&X &SB&'&H&X (HH&HXS&X#X#XSHX===:K'&XXKBB:&!&SH&X==PPK:&HXU'&XJS&X!(:BSS&X =HHKBXJ&XKH:&H+&HX:&+$$$&HX&J&H#

SH&X&S+&&X&+S&&&H:XJ&PX&P+CHJHSPP&PX KHHX&$B:&H SHJ&!XH+&PPP+XSJX$'X PSSX KBB:&!&SH&X &JSH::H:&HX '&SX &SH&X 0KK!&+&H +S$HX P$X $!!BS&&HX BKPP&HX JKPPX&SH&XKH:&H+&XJ&H 33H +S$HP:KK&H &S!KBX'&U+#

image 72f61f8d2289c81f245ac03604f93613

Page 10


Page 11


image 19613450f484579dc6ca3065906559a9
===:K'&X4

===:K'&X84

===:K'&X4

image e7f4f6f8c5ddc557b3584476e448e0e6
===:K'&XQ4

===:K'&X-4 ===:K'&X4 ===:K'&X?4 ===:K'&X4

===:K'&XX4

===:K'&X64

===:K'&X4

===:K'&X84

===:K'&X4

===:K'&XQ4

===:K'&X-4

===:K'&X4

===:K'&X?4 <@

===:K'&X4 <@

===:K'&X4 <@

===:K'&X864 <@ K 8 @ ===:K'&X84

===:K'&X884

===:K'&X84

===:K'&X8Q4 <@

-

Page 12


8 <@ @ #X &&&H&HX "S&X KBB&X KH:&H+&H:B&SSH:&H +<@ &:&'&HX SP+X JS&X 33H +S$HX

KHXJ&H 33H +S$HP:KK&HXJS&XJJXJ&HX00H +X0<GX:&&H P$S&XJS&X:&(S:&HX &UH +&#

(PPH:4

=!X&UH +XX:SB+4

<@ +<@ #

3UXJS&XKH:&H+&X+X:SB+XJS&XB&SSH:4

+<@ !@ ' #8

HJXJK!S+XX$B:+XXUXJ&HX==P&HK'PPHS++X'XJS&XB&SSH:

image af64186f98dfee88a181f8a06e8b9518

2KX K!X &UH +X X J&X 33H +S$HP:KK HJX JS&X KH:&H+&X JS&X :B&SS&X "+&S::H:X K'&HX:SB+4

image 44c607a0ab2c5ddf4533f99ee2f442bf

OSX P&++&HX HHHX JS&X B&SSH:&HX #8X # HJX #Q SHX B&SSH:X # &SHX HJX &KB+&H4

image 271a9977ae1353c27182847ade32e7e6

B&SSH:X #- SJXHKKX@XKKK:&B(P+XHJXSX&KB+&HXJK!S+XJS&X&UH +&#

Page 13


image 74b521dd64754c4ddc7eac9728572199

image 6c4e21204a6ae2e3c3d539f613681e39

2KKKPX&:&'&HXPSSX!S+XJ&X33H +S$HP:B&SSH:XJS&X&UH +&XHJX!S+XB&SSH:X #Q JS&X KH:&H+&HP+&S::H:&HX HJX !S+X B&SSH:X # J&X ==P&HK'PPHS++X J&X KH:&H+&H#

image 224c799d6cd221a9c39741545828a159

Page 14


image 30a8b2e0257e98a0ca4368e8e7cb32f2

&:&'&HXSP+XJS&X 33H +S$HX

+<@X KHX J&HX 33H +S$HP:KK&HX JS&X JJX J&HX 00H +X 0<X :&&H P$S&X JS&X :&(S:&HX&UH +&#

(PPH:4

=!X&UH +XX:SB+4

<@ +<@ 8#

3UXJS&XKH:&H+&X+X:SB+XJS&XB&SSH:4

+<@ !@ ' 8#8

HJXJK!S+XX$B:+XXUXJ&HX==P&HK'PPHS++X'XJS&XB&SSH:

image 86c4dfabf20e8a358c76f89287ce8aa2

2KX K!X &UH +X X J&X 33H +S$HP:KK HJX JS&X KH:&H+&X JS&X :B&SS&X "+&S::H:X K'&HX:SB+4

image e7ecde74128a65f1260833a353ea40d2

OSX P&++&HX HHHX JS&X B&SSH:&HX 8#8X 8# HJX 8#Q SHX B&SSH:X 8# &SHX HJX &KB+&H4

Page 15


image b79f920554d20011e317e490ba37554f

B&SSH:X 8#- SJXHKKX@XKKK:&B(P+XHJXSX&KB+&HXJK!S+XJS&X&UH +&#

image 3ffb3c0657a01972c8b378d50bcb0e31

2KKKPX&:&'&HXPSSX!S+XJ&X33H +S$HP:B&SSH:XJS&X&UH +&XHJX!S+XB&SSH:X 8#Q JS&X KH:&H+&HP+&S::H:&HX HJX !S+X B&SSH:X 8# J&X ==P&HK'PPHS++X J&X KH:&H+&H#

Page 6


image 266ee2d28527fe55cdf1589b8260b16a

image 627fe6e98b9f41e8330a432ac78aab68

&:&'&HXSP+XJS&X 33H +S$HX +<@X KHX J&HX 33H +S$HP:KK&HX JS&X JJX J&HX 00H +X 0<G6--X :&&H P$S&X JS&X :&(S:&HX&UH +&#

(PPH:4

=!X&UH +XX:SB+4

<@ +<@ #

Page 7


3UXJS&XKH:&H+&X+X:SB+XJS&XB&SSH:4

+<@ !@ ' #8

HJXJK!S+XX$B:+XXUXJ&HX==P&HK'PPHS++X'XJS&XB&SSH:

image 7a0cc88e68049a6dce62c846f37e0f75

2KX K!X &UH +X X J&X 33H +S$HP:KK HJX JS&X KH:&H+&X JS&X :B&SS&X "+&S::H:X K'&HX:SB+4

image 930f84dbaf0968ba672733b298dd1b89

OSX P&++&HX HHHX JS&X B&SSH:&HX #8X # HJX #Q SHX B&SSH:X # &SHX HJX &KB+&H4

image 2dfb30e9e6072c6d63963550bf4ceed2

image 52289cb00bd1ab11b17aece66734b494

B&SSH:X #- SJXHKKX@XKKK:&B(P+XHJXSX&KB+&HXJK!S+XJS&X&UH +&#

Page 8


image b2fa013e1b2b5e726b59a6ee11d2d46b

2KX JS&X 2SP S!SHKH+&X B&SH&X KBPX ..BBX SJX :S'+X &PX &SH&X KH:&H+&X KHX <@X JS&X :B&SS&S+S:XJJXJ&HX00H +X0<G6--X>&BCCC+#

8 3UX &SH&X 33H +S$HX &S+&X JHHH:X :S'+X &PX &SX KH:&H+&HX &SH&X

''#X &SH&XKH:&H+&XJS&XJXJ&HX00H +X0<@ 0 0 X>&BKKK&HX&HHX:SB+4

image 34e99a48d31004117ae02ed75130643b

image 2545318e7af63448f950fff98060ae11

image 3fe57de732fb54a04ed67b83f3da0044

Page 9


8 #X &&&H&HX "S&X KBB&X KH:&H+&HG &:&'&HX SP+X JS&X 33H +S$HX

:B&SSH:&HX +<@X KHX J&HX 33H +S$HP:KK&HX JS&X JJX J&HX 00H +X 0<GGX :&&H P$S&XJS&XXX:&(S:&HX&UH +&#

(PPH:4

=!X&UH +XX:SB+4

<@ +<@ Q#

3UXJS&XKH:&H+&X+X:SB+XJS&XB&SSH:4

+<@ !@ ' Q#8

HJXJK!S+XX$B:+XXUXJ&HX==P&HK'PPHS++X'XJS&XB&SSH:

image 3d414e0ff371c36e0d74026b44487060

2KX K!X &UH +X X J&X 33H +S$HP:KK HJX JS&X KH:&H+&X JS&X :B&SS&X "+&S::H:X K'&HX:SB+4

image 9fac169a38d14cf862c8e85f69f4d125

OSX P&++&HX HHHX JS&X B&SSH:&HX Q#8X Q# HJX Q#Q SHX B&SSH:X Q# &SHX HJX &KB+&H4

image b0c8373528a827216fa24c00acbd2445

B&SSH:X Q#- SJXHKKX@XKKK:&B(P+XHJXSX&KB+&HXJK!S+XJS& &UH +&#

Page 10


@ @ - 6

image a7111b770f653db7fc16eac0703bcd88

image 89da754ccc433b65942d42a834d2570a

image 0cfd5a74a8f0c9acde356547fad26144

RS+X JS&P&HX @GO&+&HX UX JS&X &UH +&X &:&'&HX PSSX !S+X J&X 33H +S$HP:B&SSH:X JS&X GO&+&XJ&X&UH +&XHJX!S+XB&SSH:X Q#Q JS&XKH:&H+&HP+&S::H:&HXHJX !S+XB&SSH:X Q# JS& ==P&HK'PPHS++& J&XKH:&H+&H#

image dce90979c5bb78c871dc92e67a44e247

Q

Page 11


image 050d364dd317aad2bcaff37f27e307c0

8 &:&'&HX SP+X JS&X 33H +S$HX #X &&&H&HX "S&X KBB&X KH:&H+&HG

:B&SSH:&HX+<@XKHXJ&HX33H +S$HP:KK&HXJS&XJJXJ&HX00H +X0<8X:&&H P$S&X JS&X:&(S:&HX&UH +&#

(PPH:4

=!X&UH +XX:SB+4

<@ +<@ -#

3UXJS&XKH:&H+&X+X:SB+XJS&XB&SSH:4

+<@ !@ ' -#8

HJXJK!S+XX$B:+XXUXJ&HX==P&HK'PPHS++X'XJS&XB&SSH:

image afe026260c2c6ec85b836f8e73768977

2KX K!X &UH +X X J&X 33H +S$HP:KK HJX JS&X KH:&H+&X JS&X :B&SS&X "+&S::H:X K'&HX:SB+4

<@ !

-

Page 12


8@ !

OSX P&++&HX HHHX JS&X B&SSH:&HX -#8X -# HJX -#Q SHX B&SSH:X -# &SHX HJX &KB+&H4

image a630621f51256f8558e520e895a4eb83

8 8

8 8 @ @ 8@ Q@ Q

8 @ 8@ - 6 -#-B&SSH:X -#- SJXHKKX@XKKK:&B(P+XHJXSX&KB+&HXJK!S+XJS&X&UH +&#

image 18e3683e821ed49863d1f769acf64de3

RS+X JS&P&HX @GO&+&HX UX JS&X &UH +&X &:&'&HX PSSX !S+X J&X 33H +S$HP:B&SSH:X JS&X GO&+&XJ&X&UH +&XHJX!S+XB&SSH:X -#Q JS&XKH:&H+&HP+&S::H:&HXHJX !S+XB&SSH:X -# JS&X==P&HK'PPHS++&XJ&XKH:&H+&H#

Page 13


image 4cb842d58c08ba2cd39ba390a5b6f0c1

8 &:&'&HXSP+XJS&X33H +S$HX #X&&&H&HX"S&XKBB&XKH:&H+&H:B&SSH:&HX

+<@X KHX J&HX 33H +S$HP:KK&HX JS&X JJX J&HX 00H +X 0<886X :&&H P$S&X JS&X :&(S:&HX&UH +&#

(PPH:4

=!X&UH +XX:SB+4

<@ +<@ #

3UXJS&XXKH:&H+&X+X:SB+XJS&XB&SH:4

?

Page 14


+<@ !@ ' #8

HJXJK!S+XX$B:+XXUXJ&HX==P&HK'PPHS++X'XJS&XB&SSH:

image 336fd4cd95d384b3dd1bf71bebfd1ed9

2KX K!X &UH +X X J&X 33H +S$HP:KK HJX JS&X KH:&H+&X JS&X :B&SS&X "+&S::H:X K'&HX:SB+4

<@ !

Q <@ ! #Q

OSX P&++&HX HHHX JS&X B&SSH:&HX #8X # HJX #Q SHX B&SSH:X # &SHX HJX &KB+&H4

image 58854fb38c5d57452442f288799cc1eb

B&SSH:X #- SJXHKKX@XKKK:&B(P+XHJXSX&KB+&HXJK!S+XJS&X&UH +&#

image 883bcaebcbe84113cf979365ec76fbe2

Page 15


RS+X JS&P&HX @GO&+&HX UX JS&X &UH +&X &:&'&HX PSSX !S+X J&X 33H +S$HP:B&SSH:X JS&X GO&+&XJ&X&UH +&XHJX!S+XB&SSH:X #Q JS&XKH:&H+&HP+&S::H:&HXHJX !S+XB&SSH:X # JS&X==P&HK'PPHS++&XJ&XKH:&H+&H#

image ca5b40b310f65972da17ebc033d2d33b

&:&'&HX SP+X JS&X 33H +S$HX <@ @ Q@ #X &&&H&HX "S&X KBB&X KH:&H+&H:B&SSH:&HX

+<@X KHX J&HX 33H +S$HP:KK&HX JS&X JJX J&HX 00H +X 0<GQX :&&H P$S&X JS&X :&(S:&HX&UH +&#

(PPH:4

=!X&UH +XX:SB+4

Page 86


<@ +<@ ?#

3UXJS&XKH:&H+&X+X:SB+XJS&XB&SSH:4

+<@ !@ ' ?#8

HJXJK!S+XX$B:+XXUXJ&HX==P&HK'PPHS++X'XJS&XB&SSH:

image 3b55091285bbc4231636fd1cccf41887

2KX K!X &UH +X X J&X 33H +S$HP:KK HJX JS&X KH:&H+&X JS&X :B&SS&X "+&S::H:X K'&HX:SB+4

<@ !

8 @ Q ! ?#Q

OSX P&++&HX HHHX JS&X B&SSH:&HX ?#8X ?# HJX ?#Q SHX B&SSH:X ?# &SHX HJX &KB+&H4

image eac89cdcdd6f5c721c59d7c15645ba0f

B&SSH:X ?#- SJXHKKX@XKKK:&B(P+XHJXSX&KB+&HXJK!S+XJS&X&UH +&#

8 8@ @ 6

image 8d870186083abe1441f86be13e2fe304

Page 8


RS+X JS&P&HX @GO&+&HX UX JS&X &UH +&X &:&'&HX PSSX !S+X J&X 33H +S$HP:B&SSH:X JS&X GO&+&XJ&X&UH +&XHJX!S+XB&SSH:X ?#Q JS&XKH:&H+&HP+&S::H:&HXHJX !S+XB&SSH:X ?# JS&X==P&HK'PPHS++&XJ&XKH:&H+&H#

Ende der Leseprobe aus 76 Seiten

Details

Titel
Tangenten und Normalen an Funktionen
Untertitel
Berechnung von Tangenten und Normalen an Funktionsgraphen, die durch einen vorgegebenen Punkt verlaufen, der nicht auf dem Graphen der Funktion liegt
Autor
Jahr
2010
Seiten
76
Katalognummer
V145401
ISBN (eBook)
9783640539109
ISBN (Buch)
9783640542635
Dateigröße
792 KB
Sprache
Deutsch
Schlagworte
Tangenten, Normalen, Funktionen, Berechnung, Tangenten, Normalen, Funktionsgraphen, Punkt, Graphen, Funktion
Arbeit zitieren
Dr. Udo Seemann (Autor:in), 2010, Tangenten und Normalen an Funktionen, München, GRIN Verlag, https://www.grin.com/document/145401

Kommentare

  • Noch keine Kommentare.
Blick ins Buch
Titel: Tangenten und Normalen an Funktionen



Ihre Arbeit hochladen

Ihre Hausarbeit / Abschlussarbeit:

- Publikation als eBook und Buch
- Hohes Honorar auf die Verkäufe
- Für Sie komplett kostenlos – mit ISBN
- Es dauert nur 5 Minuten
- Jede Arbeit findet Leser

Kostenlos Autor werden