From 899af7fcbcd0b59d35ef8dc1b7b420d3c86c7f11 Mon Sep 17 00:00:00 2001 From: "Christoph J. Scherr" Date: Fri, 19 Jul 2024 15:41:24 +0200 Subject: [PATCH] vorlesung durchgeballert und doofer link weg gemacht --- build/main.pdf | Bin 388769 -> 414779 bytes src/main.typ | 1 - src/vorlesungen/1.typ | 60 +++++++++++++++++++++++++++++++++++++----- 3 files changed, 54 insertions(+), 7 deletions(-) diff --git a/build/main.pdf b/build/main.pdf index d1818b0dcc9f00b7497f4cf82dc415f88ea6c630..53c25d25c4d70f9aa7aa4db7cb43d22658169850 100644 GIT binary patch delta 232250 zcmcG$2Ut^0w=ZsJ(z|r20wTQ=ARtAGR7C}(i6AH~K%|$gRFNiv1Pnz)!9oY=q4yRM zq(}&z&>^%CO1Qk=_kQP`d(XM|p8x-OZt{E9%v$rCot2%L*|TQvN!~Pj!xv5_VRKzQ zNm(gnYT*|bFKTL8I6oVSFnom_#F9c}OK|{SSK*i8ziUU$&o6xsdf)4r6V&+$znsip zMrsWW>L*^#(8q2U^;F;jY$O^f#3&FLF3kyIuY30& zW&f_3`i+*aGJ*X1Wsoeql?TKc*-Kl~OA9)thWqUO?Eu9+!@E{GMsN@ZNRLbYzj8zI zFC%=A8zc;$Vh7Q|v$;uVQ+DY8Rk4x-6arHC%N-q%LJf=)8AZ4N8L1|)adz33wfkK25??^gl2zm@xpVz240Y>^+!U$TiytP0aB54_sjHy=ko z1;L%#1;BTQZVp)`U3KN^9?0yrZnN&i&69a6`F>RSnq_OsS(r02FgXZr0k?rW!Kk{f 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a/src/main.typ +++ b/src/main.typ @@ -38,7 +38,6 @@ #set figure(numbering:"1.1") - // Preabmle /////////////////////////////////////////////////////////////////////////////////////////////////// diff --git a/src/vorlesungen/1.typ b/src/vorlesungen/1.typ index 32773f6..c896b3b 100644 --- a/src/vorlesungen/1.typ +++ b/src/vorlesungen/1.typ @@ -270,9 +270,9 @@ cos(x)^2sin(x^2)x]$, das ist aber falsch, da die Vorzeichen, die beim Ableiten vom $cos$ entstehen, fehlen. -=== Funktionen, Stetigkeit, Ableitungen +=== Grenzwerte und Integration -==== Ex: Grenzwert mit l'Hôpital @Vorlesung[1, P. 59] +==== Ex: Grenzwert mit l'Hôpital @Vorlesung[1, P. 69] #quote()[ Derive the limit @@ -281,8 +281,56 @@ vom $cos$ entstehen, fehlen. ] $ - L &= lim_(x->infinity) (x^2e^x)/(e^x-1)^2 = lim_(x->infinity) f(x) \ - f(x) &= (x^2e^x)/(e^x-1)^2 = g(x)/h(x) \ - g(x) &= x^2e^x => g'(x) = dots \ - h(x) &= (e^x-1)^2 => h'(x) = dots \ + L &= lim_(x->infinity) (x^2e^x)/(e^x-1)^2 = lim_(x->infinity) (g(x))/(h(x)) =^! lim_(x->infinity) (g'(x))/(h'(x)) \ + g(x) &= x^2e^x => g'(x) = (x^2)' dot e^x + x^2 dot e^x = 2 x e^x+x^2e^x = e^x dot (x^2+2x) \ + h(x) &= (e^x-1)^2 => h'(x) = 2(e^x-1) dot e^x = 2e^x dot (e^x-1) \ + => L &= lim_(x->infinity) (e^x dot (x^2+2x))/(2e^x dot (e^x-1)) =lim_(x->infinity) (x^2+2x)/(2e^x -2) =lim_(x->infinity) (x^2+x)/(e^x -1) \ + &= lim_(x->infinity) (x+1)/(e^x-1) = lim_(x->infinity) (1)/(e^x) = 0 checkmark $ + +==== Ex: Integration mit Substitutionsverfahren @Vorlesung[1, P. 76] + +#quote()[ + Calculate the integral + + $ A = integral^(pi/2)_0 cos(x) dot e^(sin(x)) d x $ + + using substitution +] + +$ + A &= integral^(pi/2)_0 cos(x) dot e^(sin(x)) d x = integral^(pi/2)_0 u'(x) dot e(u(x)) d x \ + u(x) &= sin(x) ; u'(x) = cos(x) ; e(x) = e^x ; e'(x) = e^x \ + => A &= integral^(u(pi/2))_(u(0)) e(t) d t \ + &= integral^1_0 e(t) d t = e^1 - e^0 = e-1 checkmark +$ + +#pagebreak() + +==== Ex: Integration mit partieller Integration @Vorlesung[1, P. 76] + +#quote()[ + Calculate the integral + + $ A = integral x sin(x) d x $ + + using partial integration +] + +$ + A &= integral x sin(x) d x = integral u(x) dot v'(x) \ + u(x) &= x ; u'(x) = 1 ; v(x) = -cos(x) ; v'(x) = sin(x) \ + => A &= [u(x) dot v(x)] - integral u'(x) dot v(x) d x \ + &= x dot (-cos(x)) - integral 1 dot (-cos(x)) d x \ + &= x dot (-cos(x)) - (-sin(x)) \ + & = sin(x) - x cos(x) checkmark +$ + +- Es ist sehr verwirrend für mich, wenn die Integrationsgrenzen fehlen 😕 + +An dieser Stelle stimmt die Lösung in @Vorlesung leider wieder nicht. Ich habe +das Ergebnis wie in maschinell überprüfen lassen, und es war +korrekt. Lauf @Vorlesung wäre das Ergebnis: + +$ A = integral x sin(x) d x = x(-cos(x)) - integral sin(x) d x = cos(x)(1-x) +" "#emoji.crossmark $