Lösung 1.1:1a - Versionsgeschichte

Sekretariat1 um 09:12, 19. Aug. 2009

2009-08-19T09:12:56Z

← Nächstältere Version		Version vom 09:12, 19. Aug. 2009
Zeile 1:		Zeile 1:
	Die Ableitung <math>f^{\,\prime}(-5)</math> entspricht der momentanen Steigung der Funktion im Punkt <math>x=-5</math>, also wie schnell sich der Funktionswert in einer Umgebung von <math>x=-5\,</math> ändert.		Die Ableitung <math>f^{\,\prime}(-5)</math> entspricht der momentanen Steigung der Funktion im Punkt <math>x=-5</math>, also wie schnell sich der Funktionswert in einer Umgebung von <math>x=-5\,</math> ändert.

-	Die Ableitung ist genau dasselbe ~~wir~~ die Steigung der Tangente im Punkt <math>x=-5\,</math>.	+	Die Ableitung ist genau dasselbe wie die Steigung der Tangente im Punkt <math>x=-5\,</math>.

	{\| align="center"		{\| align="center"
Zeile 9:		Zeile 9:
	\|}		\|}

-	~~Nachdem~~ die Tangente eine positive Steigung hat, ist <math>f^{\,\prime}(-5) > 0\,</math>.	+	Da die Tangente eine positive Steigung hat, ist <math>f^{\,\prime}(-5) > 0\,</math>.

-	Im Punkt <math>x=1</math>,hat die Tangente negative Steigung, und daher ist	+	Im Punkt <math>x=1</math> hat die Tangente eine negative Steigung und daher ist
	<math>f^{\,\prime}(1) < 0\,</math>.		<math>f^{\,\prime}(1) < 0\,</math>.

Martin um 09:49, 9. Apr. 2009

2009-04-09T09:49:18Z

← Nächstältere Version		Version vom 09:49, 9. Apr. 2009
Zeile 1:		Zeile 1:
-	~~The derivative~~ <math>f^{\,\prime}(-5)</math> ~~gives the function's instantaneous rate of change at the point~~ <math>x=-5</math>, ~~i.e. it is a measure of how much the function's value changes~~ in ~~the vicinity of~~ <math>x=-5\,</math>.	+	Die Ableitung <math>f^{\,\prime}(-5)</math> entspricht der momentanen Steigung der Funktion im Punkt <math>x=-5</math>, also wie schnell sich der Funktionswert in einer Umgebung von <math>x=-5\,</math> ändert.

-	~~In the graph of the function, this derivative is equal to the slope of the tangent to the graph of the function at the point~~ <math>x=-5\,</math>.	+	Die Ableitung ist genau dasselbe wir die Steigung der Tangente im Punkt <math>x=-5\,</math>.

	{\| align="center"		{\| align="center"
	\|\|{{:1.1 - Figure - Lösung - The graph of f(x) in exercise 1.1:1a with the tangent line at x = -5}}		\|\|{{:1.1 - Figure - Lösung - The graph of f(x) in exercise 1.1:1a with the tangent line at x = -5}}
	\|-		\|-
-	\|\|<small>~~The red tangent line has the equation~~<br>''y'' = ''kx'' + ''m'', ~~where~~ ''k'' = ''f'''(-5).</small>	+	\|\|<small>Die rote Tangente hat die Gleichung <br>''y'' = ''kx'' + ''m'', wo ''k'' = ''f'''(-5).</small>
	\|}		\|}

-	~~Because the tangent is sloping upwards, it has a~~ positive ~~gradient and therefore~~	+	Nachdem die Tangente eine positive Steigung hat, ist <math>f^{\,\prime}(-5) > 0\,</math>.
-	<math>f^{\,\prime}(-5) > 0\,</math>.	+

-	~~At the point~~ <math>x=1</math>, ~~the tangent slopes downwards and this means that~~	+	Im Punkt <math>x=1</math>,hat die Tangente negative Steigung, und daher ist
	<math>f^{\,\prime}(1) < 0\,</math>.		<math>f^{\,\prime}(1) < 0\,</math>.

Zeile 18:		Zeile 17:
	\|\|{{:1.1 - Figure - Lösung - The graph of f(x) in exercise 1.1:1a with the tangent line at x = 1}}		\|\|{{:1.1 - Figure - Lösung - The graph of f(x) in exercise 1.1:1a with the tangent line at x = 1}}
	\|-		\|-
-	\|\|<small>~~The red tangent line has the equation~~<br>''y'' = ''kx'' + ''m'', ~~where~~ ''k'' = ''f'''(1).</small>	+	\|\|<small>Die rote Tangente hat die Gleichung <br>''y'' = ''kx'' + ''m'', wo ''k'' = ''f'''(1).</small>
	\|}		\|}

Tekbot: Solution 1.1:1a moved to Lösung 1.1:1a: Robot: moved page

2009-03-11T09:57:35Z

Solution 1.1:1a moved to Lösung 1.1:1a: Robot: moved page

← Nächstältere Version

Version vom 09:57, 11. Mär. 2009

Tekbot: Robot: Automated text replacement (-Solution +Lösung)

2009-03-10T13:35:04Z

Robot: Automated text replacement (-Solution +Lösung)

← Nächstältere Version		Version vom 13:35, 10. Mär. 2009
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	{\| align="center"		{\| align="center"
-	\|\|{{:1.1 - Figure - ~~Solution~~ - The graph of f(x) in exercise 1.1:1a with the tangent line at x = -5}}	+	\|\|{{:1.1 - Figure - Lösung - The graph of f(x) in exercise 1.1:1a with the tangent line at x = -5}}
	\|-		\|-
	\|\|<small>The red tangent line has the equation<br>''y'' = ''kx'' + ''m'', where ''k'' = ''f'''(-5).</small>		\|\|<small>The red tangent line has the equation<br>''y'' = ''kx'' + ''m'', where ''k'' = ''f'''(-5).</small>
Zeile 16:		Zeile 16:

	{\| align="center"		{\| align="center"
-	\|\|{{:1.1 - Figure - ~~Solution~~ - The graph of f(x) in exercise 1.1:1a with the tangent line at x = 1}}	+	\|\|{{:1.1 - Figure - Lösung - The graph of f(x) in exercise 1.1:1a with the tangent line at x = 1}}
	\|-		\|-
	\|\|<small>The red tangent line has the equation<br>''y'' = ''kx'' + ''m'', where ''k'' = ''f'''(1).</small>		\|\|<small>The red tangent line has the equation<br>''y'' = ''kx'' + ''m'', where ''k'' = ''f'''(1).</small>
	\|}		\|}

Tek um 08:48, 14. Okt. 2008

2008-10-14T08:48:01Z

← Nächstältere Version		Version vom 08:48, 14. Okt. 2008
Zeile 1:		Zeile 1:
-	The derivative	+	The derivative <math>f^{\,\prime}(-5)</math> gives the function's instantaneous rate of change at the point <math>x=-5</math>, i.e. it is a measure of how much the function's value changes in the vicinity of <math>x=-5\,</math>.
-	<math>{f}'\~~left~~( -~~4 \right~~)</math>	+
-	gives the function's instantaneous rate of change at the point	+
-	<math>x=-~~\text{4}~~</math>, i.e. it is a measure of the function's value changes in the vicinity of	+
-	<math>x=-\~~text{4}~~</math>.	+

-	In the graph of the function, this derivative is equal to the slope of the tangent to the ~~curve~~ of function at the point	+	In the graph of the function, this derivative is equal to the slope of the tangent to the graph of the function at the point <math>x=-5\,</math>.
-	<math>x=-\~~text{4}~~</math>.	+

-		+	{\| align="center"
-		+	\|\|{{:1.1 - Figure - Solution - The graph of f(x) in exercise 1.1:1a with the tangent line at x = -5}}
-		+	\|-
-	~~[[Image~~:~~1_1_1_a1~~.~~gif~~\|~~center]]~~	+	\|\|<small>The red tangent line has the equation<br>''y'' = ''kx'' + ''m'', where ''k'' = ''f'''(-5).</small>
-		+	\|}
-	<~~math~~>y=kx+m~~\text{ }</math>~~	+
-	where	+
-	~~<math>~~k={f}'~~\left~~( -~~\text{4} \right~~)</~~math~~>	+
-		+

	Because the tangent is sloping upwards, it has a positive gradient and therefore		Because the tangent is sloping upwards, it has a positive gradient and therefore
-	<math>{f}~~'\left~~( -~~\text{4} \right~~)>0~~</math>.~~	+	<math>f^{\,\prime}(-5) > 0\,</math>.
-		+
-	~~At the point~~	+
-	~~<math>x=~~\~~text{1}</math>~~, ~~the tangent slopes downwards and this means that~~	+
-	~~<math>{f}'\left( \text{1} \right)<0~~</math>.	+

-	~~[[Image:1_1_1_a2~~.~~gif\|center]]~~	+	At the point <math>x=1</math>, the tangent slopes downwards and this means that
		+	<math>f^{\,\prime}(1) < 0\,</math>.

-	<~~math~~>y=kx+m~~\text{ }</math>~~	+	{\| align="center"
-	where	+	\|\|{{:1.1 - Figure - Solution - The graph of f(x) in exercise 1.1:1a with the tangent line at x = 1}}
-	~~<math>~~k={f}'~~\left~~( 1 ~~\right~~)</~~math~~>	+	\|-
		+	\|\|<small>The red tangent line has the equation<br>''y'' = ''kx'' + ''m'', where ''k'' = ''f'''(1).</small>
		+	\|}

Ian um 11:11, 10. Okt. 2008

2008-10-10T11:11:36Z

← Nächstältere Version		Version vom 11:11, 10. Okt. 2008
Zeile 1:		Zeile 1:
-	~~{{NAVCONTENT_START}}~~	+	The derivative
-	The derivative f'(-4)gives the function's instantaneous rate of change at the point x=-4, i.e. it is a measure of the function's value changes in the vicinity of x=-4.	+	<math>{f}'\left( -4 \right)</math>
		+	gives the function's instantaneous rate of change at the point
		+	<math>x=-\text{4}</math>, i.e. it is a measure of the function's value changes in the vicinity of
		+	<math>x=-\text{4}</math>.
		+
		+	In the graph of the function, this derivative is equal to the slope of the tangent to the curve of function at the point
		+	<math>x=-\text{4}</math>.
		+
		+

-	In the graph of the function, this derivative is equal to the slope of the tangent to the curve of function at the point x=-4.

	[[Image:1_1_1_a1.gif\|center]]		[[Image:1_1_1_a1.gif\|center]]

-	Because the tangent is sloping upwards, it has a positive gradient and therefore f'(-4)>0.	+	<math>y=kx+m\text{ }</math>
		+	where
		+	<math>k={f}'\left( -\text{4} \right)</math>
		+
		+
		+	Because the tangent is sloping upwards, it has a positive gradient and therefore
		+	<math>{f}'\left( -\text{4} \right)>0</math>.

-	At the point x=1, the tangent slopes downwards and this means that f'(1)<0.	+	At the point
		+	<math>x=\text{1}</math>, the tangent slopes downwards and this means that
		+	<math>{f}'\left( \text{1} \right)<0</math>.

	[[Image:1_1_1_a2.gif\|center]]		[[Image:1_1_1_a2.gif\|center]]

-	{~~{NAVCONTENT_STOP~~}}	+	<math>y=kx+m\text{ }</math>
		+	where
		+	<math>k={f}'\left( 1 \right)</math>

Tekbot: Lösning 1.1:1a moved to Solution 1.1:1a: Robot: moved page

2008-09-16T14:18:01Z

Lösning 1.1:1a moved to Solution 1.1:1a: Robot: moved page

← Nächstältere Version

Version vom 14:18, 16. Sep. 2008

Oskar um 15:25, 5. Sep. 2008

2008-09-05T15:25:51Z

← Nächstältere Version		Version vom 15:25, 5. Sep. 2008
Zeile 12:		Zeile 12:
	[[Image:1_1_1_a2.gif\|center]]		[[Image:1_1_1_a2.gif\|center]]

-
-	<center> [[Image:1_1_1a-1(2).gif]] </center>
-	{{NAVCONTENT_STOP}}
-	{{NAVCONTENT_START}}
-	<center> [[Image:2_1_1a-2(2).gif]] </center>
	{{NAVCONTENT_STOP}}		{{NAVCONTENT_STOP}}

Oskar um 15:24, 5. Sep. 2008

2008-09-05T15:24:14Z

← Nächstältere Version		Version vom 15:24, 5. Sep. 2008
Zeile 1:		Zeile 1:
	{{NAVCONTENT_START}}		{{NAVCONTENT_START}}
		+	The derivative f'(-4)gives the function's instantaneous rate of change at the point x=-4, i.e. it is a measure of the function's value changes in the vicinity of x=-4.
		+
		+	In the graph of the function, this derivative is equal to the slope of the tangent to the curve of function at the point x=-4.
		+
		+	[[Image:1_1_1_a1.gif\|center]]
		+
		+	Because the tangent is sloping upwards, it has a positive gradient and therefore f'(-4)>0.
		+
		+	At the point x=1, the tangent slopes downwards and this means that f'(1)<0.
		+
		+	[[Image:1_1_1_a2.gif\|center]]
		+
		+
	<center> [[Image:1_1_1a-1(2).gif]] </center>		<center> [[Image:1_1_1a-1(2).gif]] </center>
	{{NAVCONTENT_STOP}}		{{NAVCONTENT_STOP}}
Zeile 5:		Zeile 18:
	<center> [[Image:2_1_1a-2(2).gif]] </center>		<center> [[Image:2_1_1a-2(2).gif]] </center>
	{{NAVCONTENT_STOP}}		{{NAVCONTENT_STOP}}
-
-	[[Image:1_1_1_a1.gif\|center]]
-
-	[[Image:1_1_1_a2.gif\|center]]

Tekbot: Robot: Automated text replacement (-[[Bild: +[[Image:)

2008-08-21T07:48:44Z

Robot: Automated text replacement (-[[Bild: +[[Image:)

← Nächstältere Version		Version vom 07:48, 21. Aug. 2008
Zeile 1:		Zeile 1:
	{{NAVCONTENT_START}}		{{NAVCONTENT_START}}
-	<center> [[~~Bild~~:1_1_1a-1(2).gif]] </center>	+	<center> [[Image:1_1_1a-1(2).gif]] </center>
	{{NAVCONTENT_STOP}}		{{NAVCONTENT_STOP}}
	{{NAVCONTENT_START}}		{{NAVCONTENT_START}}
-	<center> [[~~Bild~~:2_1_1a-2(2).gif]] </center>	+	<center> [[Image:2_1_1a-2(2).gif]] </center>
	{{NAVCONTENT_STOP}}		{{NAVCONTENT_STOP}}

-	[[~~Bild~~:1_1_1_a1.gif\|center]]	+	[[Image:1_1_1_a1.gif\|center]]

-	[[~~Bild~~:1_1_1_a2.gif\|center]]	+	[[Image:1_1_1_a2.gif\|center]]