129 lines
2.1 KiB
Typst
129 lines
2.1 KiB
Typst
#import "@preview/grape-suite:1.0.0": exercise
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#import exercise: project, task, subtask
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#set text(lang: "de")
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#show: project.with(
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title: [Wie werden Funktionen verknüpft],
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seminar: [Mathe Q2],
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// show-outline: true,
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author: "Erik Grobecker",
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date: datetime(day: 9, month: 12, year: 2024),
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)
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= Ganzrationale Funktionen
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Wiederholung:
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$
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f(x) &= underbrace(3x^2, "Pf") #h(0.5em) underbrace(+2x, "Pf") #h(0.5em) underbrace(+1, "Pf")
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$
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Ganzrationale Funktionen bestehen aus einer Summe oder Differenz von Potenzfunktionen(Pf).\
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Man sie ab mittels der Potenz- und Summenregel.
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= Kettenregel
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Beispiel:
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$
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f(x)&= e^(2x+1)\
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g(u)&=e^u\
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t(x)&=2x+1\
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g(t(x)) &= f(x)\
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#line(stroke: (dash: "dashed"))\
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f'(x)&=g'(t(x)) dot t'(x)\
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f'(x) &= e^(2x+1) dot 2
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$
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== Aufgaben
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S. 139 Nr. a) - f)
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$
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a) #h(0.25em) & f'(x) =& 4(x+2)^3\
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b) #h(0.25em) & f'(x) =& 24(8x+2)^2\
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c) #h(0.25em) & f'(x) =& 15(1/2 -5x)^2\
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d) #h(0.25em) & f'(x) =& x(x^2-5)\
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e) #h(0.25em) & f'(x) =& 2e^(2x)\
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f) #h(0.25em) & f'(x) =& -4e^(-4x)
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$
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Nr. 3 a) & b)
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$
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a)\
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f(x) &= 2e^x\
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f'(x) &= 2e^x\
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\
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g(x) &= 0.5(1-3x)^4\
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g'(x) &= -6(1-3x)^3
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\
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b)\
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f(x) &= (5-2x)^4\
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f'(x) &= -8(5-2x)^3\
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\
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g(x)&=4 dot e^(2-x)\
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g'(x)&=-4 dot e^(2-x)
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$
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= Produktregel
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$
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f(x) &= x^2 dot e^(3x)\
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f'(x) &= 2x dot 3e^(3x) #h(1em) ???\
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\
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f(x) &= u dot v\
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f'(x)&= u'(x) dot v(x) + u(x) dot v'(x)\
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u(x)&= x^2\
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u'(x)&=2x\
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v(x)&= e^(3x)\
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v'(x)&=e^(3x) dot 3\
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&=3e^(3x)\
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f'(x)&= 2x dot 3e^(3x) + x^2 dot 3e^(3x)
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$
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$
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g(x)&=e^(3x)\
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t(u)&=e^u\
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j(x)&=3x
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$
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#pagebreak()
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== Übung
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S. 136 Nr. 1a) - d)
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$
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a)\
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f(x)&=2x dot (4x -1)\
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u(x)&=2x\
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u'(x)&=2\
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v(x)&=(4x-1)\
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v'(x)&=4\
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f'(x)&=2 dot (4x-1) + 2x dot 4\
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\
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b)\
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f(x)&=(5x+3) dot (x+2)\
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u(x)&=5x+3\
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u'(x)&=5\
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v(x)&=x+2\
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v'(x)&=1\
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f'(x)&=5 dot (x+2) + (5x+3) dot 1\
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\
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c)\
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f(x)&=(2-5x) dot (x +2)\
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u(x)&=2-5x\
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u'(x)&=5\
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v(x)&=x+2\
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v'(x)&=1\
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f'(x)&=5 dot (x+2) + (2-5x) dot 1\
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\
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d)\
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f(x)&=2x dot e^x\
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u(x)&=2x\
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u'(x)&=2\
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v(x)&=e^x\
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v'(x)&=e^x\
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f'(x)&=2 dot e^x + 2x dot e^x
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$
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