143 lines
1.8 KiB
Typst
143 lines
1.8 KiB
Typst
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#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: [Stunde nach der Klausur],
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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: 5, month: 12, year: 2024),
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)
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#show rect: set align(center)
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= Irgendeine HA
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*S. 113 Nr. 2a)*
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#rect($
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f(x)&=90 dot 0.87^x && |:90 \
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0.5&=0.87^x && | log_(0.87)\
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x &= log_(0.87)(0.5)\
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x &= 4.87
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$)
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*b)*
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$09 dot 0.87^10 = 22.36$\
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Er hat nach 10 Jahren eine Wachstumsgeschwindigkeit von ca. 22,36 cm pro Jahr.
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*c)*
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#rect($
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50 &= 90 dot 0.87^x &&|:90 \
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5/9 &= 0.87^x &&|log \
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x &= log_(0.87)(5/9)
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$)
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*d)*
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#rect($
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f(x)&=90 dot 0.87^x \
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F(x)&=90/ln(0.87) dot 0.87^x
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$)
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*e)*
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#rect($
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integral_0^10 f(x) d x & #sym.tilde.eq 575.71
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$)
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*f)*
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#rect($
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integral_0^20 f(x) d x &= 606\
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606 + 90 &= 696
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$)
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*g)*
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#rect($
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606:20 &= 30.3
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$)
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= Hausaufgabe zur nächsten Stunde
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S. 133 Nr. 1 & 2
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== Nr. 1
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$
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u(x) &= x^2\
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v(x) &= x+2\
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w(x) &= sqrt(x)
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$
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$u+v$:\
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$
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x^2 + (x+2)
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$
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$u dot v$:\
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$
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x^2 dot (x + 2) = x^3 + 2x^2
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$
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$u circle.tiny v$:
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$
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u(v(x))\
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u(x+2)\
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(x+2)^2\
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x^2 + 4
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$
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$w dot v$
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$
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sqrt(x) dot (x+2)\
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sqrt(x) dot x + 2 dot sqrt(x)
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$
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$w circle.tiny v$:
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$
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w(v(x))\
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w(x+2)\
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sqrt(x+2)
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$
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== Nr. 2
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=== a)
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$
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f(x) &= (2x -3)^2\
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"Summe:"\
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f(x)&=(2x -3)^2\
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&=(a+b)^2\
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&=a^2 + 2a b + b^2\
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&= 2x² + (2 dot (2x-3)) + (-3^2)\
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&= 2x² + (4x - 6) + 9\
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\
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"Produkt:"\
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f(x) &= (2x-3)^2\
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&= (2x-3) dot (2x-3)\
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\
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"Kette:"\
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f(x) &= (2x-3)^2\
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g(u) &= u²\
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h(x) &= 2x-3\
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g(u(x)) &= (u(x))²\
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&= (2x-3)^2
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$
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=== b)
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$
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g(x) &= 2 dot e^(3x) -> "ist Produkt"\
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\
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"Kettenregel:"\
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g(u) &= 2 dot e^u\
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u (x) &= 3x\
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g(x) &= g(u(x))\
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&= 2 dot e^(u(x))\
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&= 2 dot e^(3x)
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$
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