From 0052d55ab4f28f7db98af8f35bedc933a46cc041 Mon Sep 17 00:00:00 2001 From: fredrikr79 Date: Thu, 30 Oct 2025 17:47:57 +0100 Subject: [PATCH] ex10: problem 2 --- exercise10/2a_even_expansion.png | Bin 0 -> 114537 bytes exercise10/2a_odd_expansion.png | Bin 0 -> 108680 bytes exercise10/2c_even_expansion.png | Bin 0 -> 115315 bytes exercise10/2c_odd_expansion.png | Bin 0 -> 111941 bytes exercise10/main.typ | 2 + exercise10/marine.png | Bin 0 -> 55649 bytes exercise10/problem2.typ | 131 +++++++++++++++++++++++++++++++ 7 files changed, 133 insertions(+) create mode 100644 exercise10/2a_even_expansion.png create mode 100644 exercise10/2a_odd_expansion.png create mode 100644 exercise10/2c_even_expansion.png create mode 100644 exercise10/2c_odd_expansion.png create mode 100644 exercise10/marine.png create mode 100644 exercise10/problem2.typ diff --git a/exercise10/2a_even_expansion.png b/exercise10/2a_even_expansion.png new file mode 100644 index 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zUk?QxtrRNVAm_iD>$TL&)LgN;H_n^17fR)tP7*El`zZ3YsN+1-wa*nfq$%-n>l?yXT+aMQZXmU34qQ-*V|1UDeMoeeEEP2g-+ndIfujgqcZZ{3}TKAse z_1Ao(i&8U+)g2uFDHx1R3DvuRmC;E{`!`%EDM9Ti-94WcF>h`@rab>kL&GoNqa;g;9 zKlfUD`<%bI(1Gin(ff~veV>{6tj0+9)axYMIgelbf9>dSD1BA@E2CN-@XSWKU#!n& z)uZ?86V_O7D(Cyc7U!#Y&3|w5jCCs4-d?yX@WQKH`I+c7=N;1%_RjgbH7~;HC(nY{ za`L8$Z4vL+o^ZI~TRLaXI`Q&F5&fI2t9;u32mUj7`6adaV(j70BZ~#DpNal= 0, -x - x^2)space, quad + h(x) = ccases(x - x^2, x >= 0, -x + x^2) +$ + +we can find the coefficients for these by calculating the integrals from the +definitions of the coefficients + +$ + a_0 (g) & = integral_(-1)^1 g(x) dd(x) \ + & = integral_(-1)^0 (-x - x^2) dd(x) + integral_0^1 (x - x^2) dd(x) \ + & = -1/2 [x^2]_(-1)^0 - 1/3 [x^3]_(-1)^0 + + 1/2 [x^2]_0^1 - 1/3 [x^3]_0^1 \ + & = -1/2 - 1/3 + 1/2 - 1/3 = -2/3 +$ + +$ + a_n (g) & = integral_(-1)^1 g(x) cos(2 pi n x) dd(x) \ + & = integral_(-1)^0 (-x - x^2) cos(2 pi n x) dd(x) + + integral_0^1 (x - x^2) cos(2 pi n x) dd(x) \ + & = 2 integral_0^1 (x - x^2) cos(2 pi n x) dd(x) \ + & = 2 [(x-x^2)/(2 pi n) sin(2 pi n x) + + (1 - 2x)/(4 pi^2 n^2) cos(2 pi n x) + + 2/(8 pi^3 n^3) sin(2 pi n x)]_0^1 \ + & = 2 [-1/(4 pi^2 n^2) - 1/(4 pi^2 n^2)] \ + & = -1/(pi^2 n^2) +$ + +$ + b_n (g) & = integral_(-1)^1 g(x) sin(2 pi n x) dd(x) = 0 +$ + +$ + a_0 (h) & = integral_(-1)^1 h(x) dd(x) \ + & = integral_(-1)^0 (-x + x^2) dd(x) + integral_0^1 (x - x^2) dd(x) \ + & = -1/2 [x^2]_(-1)^0 + 1/3 [x^3]_(-1)^0 + + 1/2 [x^2]_0^1 - 1/3 [x^3]_0^1 \ + & = -1/2 + 1/3 + 1/2 - 1/3 = 0 +$ + +$ + a_n (h) & = integral_(-1)^1 h(x) cos(2 pi n x) dd(x) = 0 +$ + +$ + b_n (h) & = integral_(-1)^1 h(x) sin(2 pi n x) dd(x) \ + & = integral_(-1)^0 (-x + x^2) sin(2 pi n x) dd(x) + + integral_0^1 (x - x^2) sin(2 pi n x) dd(x) \ + & = 2 integral_0^1 (x - x^2) sin(2 pi n x) dd(x) \ + & = 2 [(x^2 - x)/(2 pi n) cos(2 pi n x) + + (1 - 2x)/(4 pi^2 n^2) sin(2 pi n x) + -2/(8 pi^3 n^3) cos(2 pi n x)]_0^1 \ + & = 0 +$ +

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