From d91ccb84b9a50d8e1fa142163ed493637e92f5e8 Mon Sep 17 00:00:00 2001 From: Jan Meyer Date: Sun, 31 May 2026 17:54:10 +0200 Subject: [PATCH] vault backup: 2026-05-31 17:54:10 --- 11 Homework/Mathe II/Homework 6.pdf | 1528 +++++++++++++++++++++++++++ 11 Homework/Mathe II/Homework 6.typ | 42 + 2 files changed, 1570 insertions(+) create mode 100644 11 Homework/Mathe II/Homework 6.pdf diff --git a/11 Homework/Mathe II/Homework 6.pdf b/11 Homework/Mathe II/Homework 6.pdf new file mode 100644 index 0000000..1181275 --- /dev/null +++ b/11 Homework/Mathe II/Homework 6.pdf @@ -0,0 +1,1528 @@ +%PDF-1.7 +% + +1 0 obj +<< + /Type /Pages + /Count 2 + /Kids [93 0 R 95 0 R] +>> +endobj + +2 0 obj +<< + /Type /Outlines + /First 3 0 R + /Last 6 0 R + /Count 2 +>> +endobj + +3 0 obj +<< + /Parent 2 0 R + /Next 6 0 R + /First 4 0 R + /Last 5 0 R + /Count -2 + /Title (Problem 2) + /Dest 88 0 R +>> +endobj + +4 0 obj +<< + /Parent 3 0 R + /Next 5 0 R + /Title ((a)) + /Dest 86 0 R +>> +endobj + +5 0 obj 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R + /ID [(8fFSuRIZAEIHYQCqPu8a5A==) (8fFSuRIZAEIHYQCqPu8a5A==)] +>> +startxref +45033 +%%EOF \ No newline at end of file diff --git a/11 Homework/Mathe II/Homework 6.typ b/11 Homework/Mathe II/Homework 6.typ index e69de29..531a4dc 100644 --- a/11 Homework/Mathe II/Homework 6.typ +++ b/11 Homework/Mathe II/Homework 6.typ @@ -0,0 +1,42 @@ +#set page( + header: align(right, [ + #set text(size: 9pt) + #set par(leading: .3em) + #pad(y: -.5cm, [ + Mathe II \ + Hausaufgabe 02 \ + Jan Meyer \ + 664237 + ]) + ]) + ) + += Problem 2 + +== (a) +$ f_n (x) = (sin(n x) + n) / (3n + 1) = (sin(n x) / n + 1) / (3 + 1/n) $ +Da $ lim_(n -> oo) sin(n x) / n = 0 $, ist der punktweise Grenzwert: +$ f(x) = lim_(n -> oo) f_n (x) = (0 + 1) / (3 + 0) = 1/3 $ + +== (b) +Zu zeigen: $ f_n (x) = (n x) / (n x^2 + 1) $ konvergiert auf $ [r, 1) $ mit $ r > 0 $ gleichmäßig gegen $ f(x) = 1/x $. +$ |f_n (x) - f(x)| = |(n x) / (n x^2 + 1) - 1/x| = |(n x^2 - (n x^2 + 1)) / (x(n x^2 + 1))| = 1 / (n x^3 + x) $ +Da $ x >= r > 0 $, gilt $ n x^3 + x > n r^3 $. Daraus folgt für das Supremum: +$ sup_(x in [r, 1)) |f_n (x) - f(x)| <= 1 / (n r^3) $ +Wegen $ lim_(n -> oo) 1 / (n r^3) = 0 $ konvergiert die Funktionenfolge gleichmäßig. + += Problem 3 + +Sei $ cal(E) = (bold(m)_0, bold(m)_1) $ die Standardbasis. Die Basiswechselmatrizen von $ cal(B) $ und $ cal(C) $ in $ cal(E) $ sind: +$ M_cal(B) = mat(3, 2; 3, 1), quad M_cal(C) = mat(4, 1; 0, 1) $ +Der Koordinatenvektor von $ bold(p) = bold(m)_0 + bold(m)_1 $ in $ cal(E) $ ist $ bold(p)^cal(E) = vec(1, 1) $. + +== (a) +$ bold(p)^cal(B) = M_cal(B)^(-1) bold(p)^cal(E) = 1/(3-6) mat(1, -2; -3, 3) vec(1, 1) = mat(-1/3, 2/3; 1, -1) vec(1, 1) = vec(1/3, 0) $ + +== (b) +$ T_(cal(C) <- cal(B)) = M_cal(C)^(-1) M_cal(B) = 1/4 mat(1, -1; 0, 4) mat(3, 2; 3, 1) = mat(1/4, -1/4; 0, 1) mat(3, 2; 3, 1) = mat(0, 1/4; 3, 1) $ +$ T_(cal(B) <- cal(C)) = (T_(cal(C) <- cal(B)))^(-1) = 1/(0 - 3/4) mat(1, -1/4; -3, 0) = -4/3 mat(1, -1/4; -3, 0) = mat(-4/3, 1/3; 4, 0) $ + +== (c) +$ bold(p)^cal(C) = T_(cal(C) <- cal(B)) bold(p)^cal(B) = mat(0, 1/4; 3, 1) vec(1/3, 0) = vec(0, 1) $ \ No newline at end of file