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37 38 39] + /Pg 506 0 R +>> +endobj + +63 0 obj +<< + /Type /StructElem + /S /H4 + /P 53 0 R + /T + /K [29] + /Pg 506 0 R +>> +endobj + +64 0 obj +<< + /Type /StructElem + /S /P + /P 53 0 R + /K [28] + /Pg 506 0 R +>> +endobj + +65 0 obj +<< + /Type /StructElem + /S /P + /P 53 0 R + /K [25 66 0 R 27] + /Pg 506 0 R +>> +endobj + +66 0 obj +<< + /Type /StructElem + /S /Formula + /P 65 0 R + /K [26] + /Pg 506 0 R +>> +endobj + +67 0 obj +<< + /Type /StructElem + /S /H4 + /P 53 0 R + /T + /K [24] + /Pg 506 0 R +>> +endobj + +68 0 obj +<< + /Type /StructElem + /S /P + /P 53 0 R + /K [2 74 0 R 4 73 0 R 8 72 0 R 10 71 0 R 13 70 0 R 15 69 0 R 23] + /Pg 506 0 R +>> +endobj + +69 0 obj +<< + /Type /StructElem + /S /Formula + /P 68 0 R + /K [16 17 18 19 20 21 22] + /Pg 506 0 R +>> +endobj + +70 0 obj +<< + /Type /StructElem + /S /Formula + /P 68 0 R + /K [14] + /Pg 506 0 R +>> +endobj + +71 0 obj +<< + /Type /StructElem + /S /Formula + /P 68 0 R + /K [11 12] + /Pg 506 0 R +>> +endobj + +72 0 obj 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75 76 77 78] + /Pg 504 0 R +>> +endobj + +101 0 obj +<< + /Type /StructElem + /S /Lbl + /P 98 0 R + /K [64] + /Pg 504 0 R +>> +endobj + +102 0 obj +<< + /Type /StructElem + /S /LI + /P 97 0 R + /K [105 0 R 103 0 R] +>> +endobj + +103 0 obj +<< + /Type /StructElem + /S /LBody + /P 102 0 R + /K [104 0 R] +>> +endobj + +104 0 obj +<< + /Type /StructElem + /S /Formula + /P 103 0 R + /K [51 52 53 54 55 56 57 58 59 60 61 62 63] + /Pg 504 0 R +>> +endobj + +105 0 obj +<< + /Type /StructElem + /S /Lbl + /P 102 0 R + /K [50] + /Pg 504 0 R +>> +endobj + +106 0 obj +<< + /Type /StructElem + /S /LI + /P 97 0 R + /K [109 0 R 107 0 R] +>> +endobj + +107 0 obj +<< + /Type /StructElem + /S /LBody + /P 106 0 R + /K [108 0 R] +>> +endobj + +108 0 obj +<< + /Type /StructElem + /S /Formula + /P 107 0 R + /K [38 39 40 41 42 43 44 45 46 47 48 49] + /Pg 504 0 R +>> +endobj + +109 0 obj +<< + /Type /StructElem + /S /Lbl + /P 106 0 R + /K [37] + /Pg 504 0 R +>> +endobj + +110 0 obj +<< + /Type /StructElem + /S 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0 obj +<< + /Type /StructElem + /S /P + /P 53 0 R + /K [19] + /Pg 504 0 R +>> +endobj + +121 0 obj +<< + /Type /StructElem + /S /P + /P 53 0 R + /K [11 123 0 R 14 15 122 0 R 18] + /Pg 504 0 R +>> +endobj + +122 0 obj +<< + /Type /StructElem + /S /Formula + /P 121 0 R + /K [16 17] + /Pg 504 0 R +>> +endobj + +123 0 obj +<< + /Type /StructElem + /S /Formula + /P 121 0 R + /K [12 13] + /Pg 504 0 R +>> +endobj + +124 0 obj +<< + /Type /StructElem + /S /H4 + /P 53 0 R + /T + /K [10] + /Pg 504 0 R +>> +endobj + +125 0 obj +<< + /Type /StructElem + /S /P + /P 53 0 R + /K [9] + /Pg 504 0 R +>> +endobj + +126 0 obj +<< + /Type /StructElem + /S /P + /P 53 0 R + /K [1 128 0 R 3 127 0 R 8] + /Pg 504 0 R +>> +endobj + +127 0 obj +<< + /Type /StructElem + /S /Formula + /P 126 0 R + /K [4 5 6 7] + /Pg 504 0 R +>> +endobj + +128 0 obj +<< + /Type /StructElem + /S /Formula + /P 126 0 R + /K [2] + /Pg 504 0 R +>> +endobj + +129 0 obj +<< + /Type /StructElem + /S /H4 + /P 53 0 R + /T + /K [0] + /Pg 504 0 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179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204] + /Pg 502 0 R +>> +endobj + +144 0 obj +<< + /Type /StructElem + /S /P + /P 53 0 R + /K [142 146 0 R 153 145 0 R 156] + /Pg 502 0 R +>> +endobj + +145 0 obj +<< + /Type /StructElem + /S /Formula + /P 144 0 R + /K [154 155] + /Pg 502 0 R +>> +endobj + +146 0 obj +<< + /Type /StructElem + /S /Formula + /P 144 0 R + /K [143 144 145 146 147 148 149 150 151 152] + /Pg 502 0 R +>> +endobj + +147 0 obj +<< + /Type /StructElem + /S /H3 + /P 53 0 R + /T (3: Forskellige(?) kvadratiske former) + /K [141] + /Pg 502 0 R +>> +endobj + +148 0 obj +<< + /Type /StructElem + /S /P + /P 53 0 R + /K [140] + /Pg 502 0 R +>> +endobj + +149 0 obj +<< + /Type /StructElem + /S /P + /P 53 0 R + /K [81 153 0 R 91 152 0 R 101 151 0 R 124 125 150 0 R 139] + /Pg 502 0 R +>> +endobj + +150 0 obj +<< + /Type /StructElem + /S /Formula + /P 149 0 R + /K [126 127 128 129 130 131 132 133 134 135 136 137 138] + /Pg 502 0 R +>> +endobj + +151 0 obj +<< + /Type /StructElem + /S /Formula + /P 149 0 R + /K [102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123] + /Pg 502 0 R +>> +endobj + +152 0 obj +<< + /Type /StructElem + /S /Formula + /P 149 0 R + /K [92 93 94 95 96 97 98 99 100] + /Pg 502 0 R +>> +endobj + +153 0 obj +<< + /Type /StructElem + /S /Formula + /P 149 0 R + /K [82 83 84 85 86 87 88 89 90] + /Pg 502 0 R +>> +endobj + +154 0 obj +<< + /Type /StructElem + /S /H3 + /P 53 0 R + /T (2: Partielle afledede af simpel skalar-funktion) + /K [80] + /Pg 502 0 R +>> +endobj + +155 0 obj +<< + /Type /StructElem + /S /P + /P 53 0 R + /K [79] + /Pg 502 0 R +>> +endobj + +156 0 obj +<< + /Type /StructElem + /S /P + /P 53 0 R + /K [51 160 0 R 62 159 0 R 65 158 0 R 73 157 0 R 78] + /Pg 502 0 R +>> +endobj + +157 0 obj +<< + /Type /StructElem + /S /Formula + /P 156 0 R + /K [74 75 76 77] + /Pg 502 0 R +>> +endobj + +158 0 obj +<< + /Type /StructElem + /S /Formula + /P 156 0 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author: "Rasmus Rosendahl-Kaa", + course-name: "Matematik 1b", + title: "Funktioner 1 - Opgaver", + date: datetime(year: 2026, month: 02, day: 03), + semester: "Spring 2026" +) + +#set math.mat(delim: "[") +#set math.vec(delim: "[") + += Exercises +=== 1: Funktion eller ej? + +Betragt følger korrespondance mellem $a$ og $b$ værdier: + +#align(center)[ + #table( + columns: 2, + align: center, + [$a$], [$b$], + [1], [0], + [2], [1], + [0], [3], + [1], [2], + ) +] + +Vi betragter funktioner hvis definitionsmængde (domain) er en delmængde af ${0, 1, 2, 3}$ og hvis dispositionsmængde (co-domain) er ${0, 1, 2, 3}$. + +Vi skal bestemme om $f$ og $g$ definerer funktioner, hvis vi lader $f$ følge reglen at første søjle ($a$-værdierne) er input og anden søjle ($b$-værdierne) skal være output af funktionen $f$ og definitionsmængden er ${0, 1, 2}$; og vi lader $g$ følge reglen om at anden søjle er input og første søjle skal være output af funktionen $g$ med definitionsmængde ${0, 1, 2, 3}$. + +Definerer $f$ en funktion? Gør $g$? I bekræftende fald: bestem værdimængden/billedmængden (engelsk: range/image) for funktionen, og afgør om funktionen er injektiv og surjektiv. + +#solution[ + $f$ er ikke en funktion, fordi $a=1$ giver to forskellige værdier. + + $g$ er en funktion. Værdimængden må være $a$'s værdier, så ${0,1,2}$. $G$ er ikke surjektiv da $3$ i dispositionsmængden ikke rammes. Den er heller ikke injektiv, da $1$ rammes 2 gange i dispositionsmængden. +] + +=== 2: Ens funktionsforskrifter? + +Vi betragter funktioner $f_i: RR -> RR$ givet ved: + +$ + f_1(x) &= abs(x) \ + f_2(x) &= cases(x quad & x > 0, -x quad & x <= 0) \ + f_3(x) &= max(x, 0) \ + f_4(x) &= "ReLU"(x) + "ReLU"(-x) +$ + +hvor $x in RR$. + +Nogle af funktionerne er samme funktion. Find dem alle! + +#solution[ +$f_1 (x)$, $f_2 (x)$ og $f_3(x)$ er ens +] + +=== 3: Funktion med ukendt forskrift + +Betragt en funktion $f: RR -> RR$ hvorom der gælder $lim_(x -> 2) f(x) = 5$ og $f(2) = 3$. Hvad kan vi sige om $f$ i punktet $x = 2$? Vælg det korrekte svar: + ++ Funktionen er kontinuert i punktet $x = 2$. ++ Funktionen er differentiabel i punktet $x = 2$. ++ Funktionen er diskontinuert i punktet $x = 2$. ++ Funktionen er ikke veldefineret i punktet $x = 2$. ++ Man kan ikke afgøre ovenstående, da funktionsforskriften ikke er angivet! + +#solution[ +Funktionen er diskontinuert i punktet $x=2$. +] + +=== 4: Ikke-linearitet af ReLU + +Betragt ReLU-funktionen, $"ReLU": RR^n -> RR^n$. Forklar hvorfor funktionen ikke er lineær. + +#solution[ + Funktionen er ikke lineær da når $x<0$ kan man ikke differentiere den. +] + +=== 5: Mulige visualiseringer + +Diskuter om man kan visualisere nedenstående funktioner -- i givet fald plot dem med SymPy/dtumathtools: + ++ En skalarfunktion af to variable $f: RR^2 -> RR, quad f(x_1, x_2) = sqrt(abs(x_1 x_2))$ ++ En skalarfunktion af fire variable $f: RR^4 -> RR, quad f(x_1, x_2, x_3, x_4) = sqrt(abs(x_1 x_2 x_3 x_4))$ ++ En kompleks skalarfunktion af to variable $f: RR^2 -> CC, quad f(x_1, x_2) = sqrt(abs(x_1 x_2)) + i cos(x_1 + x_2)$ ++ Et vektorfelt i 2D $bold(f): RR^2 -> RR^2, quad bold(f)(x_1, x_2) = (-x_2 / 3, x_1 / 3)$ ++ Et vektorfelt i 3D $bold(f): RR^3 -> RR^3, quad bold(f)(x, y, z) = (x^3 + y z^2, y^3 - x z^2, z^3)$ ++ En funktion af formen $bold(r): [0, 10] -> RR^3, quad bold(r)(t) = (cos(t), sin(t), t)$ + +#note-box(title: "Python kommandoer")[ + Følgende kan være nyttige: `dtuplot.plot3d`, `dtuplot.plot_vector`, `dtuplot.plot3d_parametric_line`. +] + +#solution[ + +] + +=== 6: Evaluering af et Neuralt Netværk + +Betragt et simpelt "shallow" neuralt netværk $bold(Phi): RR^2 -> RR$ med ét skjult lag ($L = 2$). Netværket er defineret ved parametrene: + +$ + A_1 = mat(2, 0; -1, 1), quad bold(b)_1 = vec(-1, 0), quad A_2 = mat(-1, 2), quad b_2 = 0 +$ + +hvor aktiveringsfunktionen i det *skjulte lag* er ReLU-funktionen, $bold(sigma)(bold(z)) = "ReLU"(bold(z))$, men aktiveringsfunktionen i det *sidste lag* er identitetsafbildningen $bold(sigma)(bold(z)) = bold(z)$. Netværksfunktionen er altså givet ved: + +$ + bold(Phi)(bold(x)) = A_2 "ReLU"(A_1 bold(x) + bold(b)_1) + b_2 +$ + +==== Spørgsmål a + +Beregn værdien af netværket i punktet $bold(x) = vec(0.5, 1)$. + +#solution[ +$ +mat(-1,2) dot "ReLU"(mat(2,0;-1,1) dot vec(0.5,1)+vec(-1,0)) = \ +mat(-1,2) dot "ReLU"(vec(1,0.5)+vec(-1,0)) = \ +mat(-1,2) dot "ReLU"(vec(0,0.5)) =\ +mat(-1,2) dot vec(0,0.5) = -1 dot 0 + 2 dot 0.5 = 1\ +$ +$Phi(vec(0.5,1))=1$ +] + +==== Spørgsmål b + +Find et punkt $bold(x)$, hvor netværkets output $bold(Phi)(bold(x))$ er negativt. Begrund dit svar. + +#solution[ +Punkt $vec(1,1)$ giver: +$ +mat(2,0;-1,1) dot vec(1,1)+vec(-1,0) =\ +vec(2,0) + vec(-1,0) = vec(1,0)\ +"ReLU"(vec(1,0)) = vec(1,0) =\ +mat(-1,2) dot vec(1,0) = -1 dot 1 + 2 dot 0 = -1 +$ +] + +==== Spørgsmål c + +Hvor mange justerbare parametre (vægte og bias-værdier) har dette netværk totalt? + +#solution[ +Tæller antal værdier i dens parametre: 9 + +] + +==== Spørgsmål d + +Vi erstatter nu aktiveringsfunktionen (ReLU) med en "hard limiter" funktion (en variant af signum-funktionen), som vi kalder $sigma_("step")$. + +Som *skalar*-funktion $sigma_("step"): RR -> RR$ er den defineret ved: + +$ + sigma_("step")(x) = cases(1 quad & "hvis" x >= 0, -1 quad & "hvis" x < 0) +$ + +Som *vektorfunktion* $bold(sigma)_("step"): RR^n -> RR^n$ defineres den ved at anvende skalarfunktionen på hver koordinat: + +$ + bold(sigma)_("step")(bold(z)) = vec(sigma_("step")(z_1), dots.v, sigma_("step")(z_n)) +$ + +Betragt nu netværket med denne nye aktiveringsfunktion: $bold(Phi)(bold(x)) = A_2 bold(sigma)_("step")(A_1 bold(x) + bold(b)_1) + b_2$. + +Angiv den delmængde af definitionsmængden, hvor netværksfunktionen $bold(Phi)(bold(x))$ er *diskontinuert*. + +#solution[ + +] + +=== 7: Visualisering af Netværket + +I denne opgave skal du bruge Python til at visualisere funktionen $bold(Phi)(bold(x))$ fra forrige opgave (med ReLU som aktiveringsfunktion). + +Du skal plotte grafen for netværket over området $x_1, x_2 in [-2, 2]$. + +#note-box[ + Læg mærke til, at grafen består af plane flader, der er "knækket" og sat sammen. Dette skyldes ReLU-funktionen, som er stykkevis lineær. I store netværk der anvendes i praksis sættes millioner af sådanne lineære underrum sammen. +] + +Kan du plotte grafen af det samme neurale netværk hvor $bold(sigma)_("step")$ bruges i stedet for ReLU? Planerne i den nye graf bør ikke "hænge sammen" (hvorfor?). + +#solution[ + +] + +=== 8: Lineær vektorfunktion + +Lad $A in bold(M)_(3 times 5)(RR)$ være givet ved + +$ + A = mat( + 1, 0, 2, 3, 4; + 0, -1, 5, 6, 7; + 0, 0, -3, 8, 9 + ) +$ + +Betragt vektorfunktionen $bold(f): RR^5 -> RR^3$ givet ved $bold(f) = bold(x) |-> A bold(x)$, hvor $bold(x)$ er en søjlevektor i $RR^5$. + +==== Spørgsmål a + +Angiv de 3 koordinatfunktioner for $bold(f)$. + +#solution[ + +] + +==== Spørgsmål b + +Angiv billedmængden $"im"(bold(f))$ for $bold(f)$. + +#solution[ + +] + +==== Spørgsmål c + +Er vektorfunktionen $bold(f)$ surjektiv og/eller injektiv? + +#solution[ + +] + +=== 9: Næste primtal-funktion (frivillig) + +#note-box[Dette er en valgfri ekstraopgave.] + +Lad $f: NN -> NN$ være en funktion, der returnerer det næste primtal (strengt) større end et givet naturligt tal $n$. I denne opgave skal du først vurdere værdien af funktionen for to specifikke input og derefter vise, at funktionen er veldefineret, før du implementerer den i Python. + +==== Spørgsmål a + +Find ved simple overvejelser $f(10)$ og $f(13)$. + +#solution[ + +] + +==== Spørgsmål b + +Argumenter for, at funktionen $f(n)$ er veldefineret. + +#solution[ + +] + +==== Spørgsmål c + +Kan man finde et funktionsudtryk for $f(n)$? Argumenter for, hvorfor det er eller ikke er muligt. + +#solution[ + +] + +==== Spørgsmål d + +Implementer funktionen $f(n)$ i Python, som tager et heltal $n$ som input og returnerer det næste primtal større end $n$. Definer en hjælpefunktion `er_primtal(x)` til at afgøre, om et tal er primtal. + +#solution[ + +] + +==== Spørgsmål e + +Kan du opdatere din Python-funktion fra forrige opgave, så $f$'s definitionsmængde udvides fra $NN$ til $RR$? + +#solution[ + +] + +#pagebreak() + +== Opgaver -- Lille Dag + +=== 1: Størrelse af vektorer + +Betragt følgende tre vektorer i $RR^3$: + +$ + bold(v)_1 = vec(-10, -10, -10), quad + bold(v)_2 = vec(-10, -4, 14), quad + bold(v)_3 = vec(-10, -8, -12) +$ + +Hvilken vektor er længst? Hvilke vektorer er ortogonale på hinanden? Hvilke to vektorer er tættest på hinanden? + +#note-box[ + Vi kan forestille os vektorerne som (geometriske) stedvektorer med begyndelsespunkt i $bold(0) = [0, 0, 0]^T$ og slutpunkt $bold(v)_i$ for hhv $i = 1, 2, 3$. Under tiden skriver man dette som $arrow(bold(0) bold(v)_i)$. +] + +#solution[ + +] + +=== 2: Partielle afledede af simpel skalar-funktion + +Find de partielle afledte $p d v(f, x_1)$ og $p d v(f, x_2)$ for $f(x_1, x_2) = x_1^3 + 3 x_1 x_2 + x_2^3$. Bestem værdien af de partielle afledte i punktet $(x_1, x_2) = (1, 2)$. + +#solution[ + +] + +=== 3: Forskellige(?) kvadratiske former + +Lad $bold(x) = [x_1, x_2]^T$ være en søjlevektor i $RR^2$. Definer: + +$ + A_1 = mat(11, -12; -12, 4), quad + A_2 = mat(11, 0; -24, 4), quad + A_3 = mat(73/5, -36/5; -36/5, 52/5) +$ + +og + +$ + bold(b)_1 = vec(-20, 40), quad + bold(b)_2 = bold(b)_1, quad + bold(b)_3 = vec(-44, 8), quad + c = -60 +$ + +Lad $q_i: RR^2 -> RR$ være givet ved: + +$ + q_i (bold(x)) = bold(x)^T A_i bold(x) + bold(b)_i^T bold(x) + c +$ + +for $i = 1, 2, 3$. Sådanne funktioner kaldes kvadratiske former. + +==== Spørgsmål a + +Gang udtrykket for $q_1(x_1, x_2)$ ud. Først i hånden, så ved hjælp af Python. Gang også udtrykkene for $q_2(x_1, x_2)$ og $q_3(x_1, x_2)$ (i hånden eller Python) ud. + +#solution[ + +] + +==== Spørgsmål b + +Er den kvadratiske matrix $A$ i en kvadratisk form (som fx $bold(x)^T A bold(x)$) entydig givet? + +#solution[ + +] + +==== Spørgsmål c + +Plot grafen af funktionen $q_1$. Plot så nogle niveaukurver. Hvilken geometrisk form har niveaukurverne? Gør det samme for $q_3$. + +#solution[ + +] + +==== Spørgsmål d + +En af funktionerne har et minimum. Hvilken? Hvor ligger det cirka? Hvad kaldes det samme punkt for de funktioner der ikke har et minimum? + +#solution[ + +] + +=== 4: Softmax-funktionen + +I denne opgave ser vi på *softmax*-funktionen. + +==== Spørgsmål a + +Beregn *softmax* af følgende tre vektorer i $RR^3$. Du kan gøre det ved håndkraft (brug lommeregner til eksponentialfunktionen) eller ved hjælp af Python. Angiv svarene med ca. 3 decimaler. + ++ $bold(x)_1 = [1, 2, -5]^T$ ++ $bold(x)_2 = [10, 2, -5]^T$ ++ $bold(x)_3 = [100, 2, -5]^T$ + +#solution[ + +] + +==== Spørgsmål b + +Hvad observerer du, når forskellen mellem den største værdi i inputtet og de andre værdier øges (som i skiftet fra $bold(x)_1$ til $bold(x)_2$ og $bold(x)_3$)? Hvorfor kaldes funktionen mon "soft"-max? + +#solution[ + +] + +==== Spørgsmål c + +Er *softmax*-funktionen kontinuert? + +#solution[ + +] + +==== Spørgsmål d + +Betragt *softmax* som en afbildning fra $RR^n$ til $RR^n$. Er funktionen *injektiv* (en-til-en)? + +Er funktionen *surjektiv* (på) i forhold til dispositionsmængde $RR^n$? + +#solution[ + +] + +=== 5: Kvadratiske former med symmetriske matricer + +Lad $A$ være en vilkårlig $n times n$ matrix, og lad $bold(x)$ være en søjlevektor i $RR^n$. Definer $B$ ved $B = (A + A^T) / 2$. + +==== Spørgsmål a + +Vis at matricen $B$ er symmetrisk. + +#solution[ + +] + +==== Spørgsmål b + +Vis at $bold(x)^T A bold(x) = bold(x)^T B bold(x)$. + +#solution[ + +] + +==== Spørgsmål c + +Konkludér at man altid kan antage at kvadratiske former af formen $q(bold(x)) = bold(x)^T A bold(x) + bold(b)^T bold(x) + c$ er givet ved en *symmetrisk* matrix $A$. + +#solution[ + +] diff --git a/01002 - Matematik 1b/Funktioner 1/Funktioner1.ipynb b/01002 - Matematik 1b/Funktioner 1/Funktioner1.ipynb new file mode 100644 index 0000000..6f4ec79 --- /dev/null +++ b/01002 - Matematik 1b/Funktioner 1/Funktioner1.ipynb @@ -0,0 +1,50 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 11, + "id": "d2493461", + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "2\n", + "2\n" + ] + } + ], + "source": [ + "import sympy\n", + "\n", + "x = sympy.symbols('x')\n", + "h = 2\n", + "relu = sympy.Max(0, h) + sympy.Max(0, -h)\n", + "print(sympy.Max(h,0))\n", + "print(relu)" + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.14.2" + } + }, + "nbformat": 4, + "nbformat_minor": 5 +} diff --git a/01002 - Matematik 1b/Funktioner 1/Funktioner1.pdf b/01002 - Matematik 1b/Funktioner 1/Funktioner1.pdf new file mode 100644 index 0000000..e5c4711 --- /dev/null +++ b/01002 - Matematik 1b/Funktioner 1/Funktioner1.pdf @@ -0,0 +1,4874 @@ +%PDF-1.7 +% + +1 0 obj +<< + /Type /Pages + /Count 6 + /Kids [195 0 R 197 0 R 199 0 R 201 0 R 203 0 R 205 0 R] +>> +endobj + +2 0 obj +<< + /Type /Outlines + /First 3 0 R + /Last 9 0 R + /Count 3 +>> +endobj + +3 0 obj +<< + /Parent 2 0 R + /Next 4 0 R + 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36 0 R 36 0 R 36 0 R 36 0 R 36 0 R 36 0 R 36 0 R 36 0 R 36 0 R 36 0 R 36 0 R 36 0 R 36 0 R 36 0 R 36 0 R 36 0 R 36 0 R 36 0 R 36 0 R 36 0 R 36 0 R 36 0 R 36 0 R 36 0 R 36 0 R 36 0 R 36 0 R 36 0 R 36 0 R 36 0 R 36 0 R 36 0 R 36 0 R 36 0 R 36 0 R 36 0 R 36 0 R 36 0 R 36 0 R 35 0 R 34 0 R 34 0 R 34 0 R 34 0 R 34 0 R 34 0 R 34 0 R 34 0 R 34 0 R 34 0 R 34 0 R 34 0 R 34 0 R 34 0 R 34 0 R 34 0 R 34 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 32 0 R 31 0 R] +endobj + +16 0 obj +[30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 27 0 R 29 0 R 27 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 27 0 R 26 0 R 24 0 R 25 0 R 25 0 R 25 0 R 25 0 R 25 0 R 25 0 R 25 0 R 25 0 R 24 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 21 0 R 22 0 R 22 0 R 22 0 R 22 0 R 22 0 R 22 0 R 22 0 R 22 0 R 22 0 R 20 0 R 20 0 R 20 0 R 20 0 R 20 0 R 20 0 R 20 0 R 20 0 R 20 0 R 20 0 R 20 0 R 20 0 R 20 0 R 20 0 R 20 0 R 20 0 R 20 0 R 20 0 R 20 0 R 20 0 R 20 0 R 20 0 R 20 0 R 20 0 R 20 0 R 20 0 R 20 0 R 20 0 R 18 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R] +endobj + +17 0 obj +<< + /Type /StructElem + /S /Document + /P 10 0 R + /K [159 0 R 158 0 R 151 0 R 150 0 R 149 0 R 148 0 R 147 0 R 146 0 R 142 0 R 138 0 R 136 0 R 133 0 R 130 0 R 129 0 R 127 0 R 126 0 R 124 0 R 123 0 R 121 0 R 119 0 R 117 0 R 116 0 R 115 0 R 113 0 R 111 0 R 110 0 R 109 0 R 107 0 R 105 0 R 103 0 R 101 0 R 100 0 R 98 0 R 97 0 R 95 0 R 94 0 R 93 0 R 92 0 R 91 0 R 89 0 R 87 0 R 86 0 R 84 0 R 82 0 R 81 0 R 78 0 R 76 0 R 75 0 R 73 0 R 72 0 R 71 0 R 70 0 R 69 0 R 65 0 R 61 0 R 60 0 R 57 0 R 56 0 R 54 0 R 53 0 R 51 0 R 50 0 R 49 0 R 46 0 R 45 0 R 44 0 R 43 0 R 39 0 R 37 0 R 36 0 R 35 0 R 34 0 R 33 0 R 32 0 R 31 0 R 30 0 R 27 0 R 26 0 R 24 0 R 23 0 R 21 0 R 20 0 R 18 0 R] +>> +endobj + +18 0 obj +<< + /Type /StructElem + /S /P + /P 17 0 R + /K [154 19 0 R] + /Pg 205 0 R +>> +endobj + +19 0 obj +<< + /Type /StructElem + /S /Formula + /P 18 0 R + /K [155 156 157 158 159 160 161 162 163 164 165] + /Pg 205 0 R +>> +endobj + +20 0 obj +<< + /Type /StructElem + /S /Formula + /P 17 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153] + /Pg 205 0 R +>> +endobj + +21 0 obj +<< + /Type /StructElem + /S /P + /P 17 0 R + /K [116 22 0 R] + /Pg 205 0 R +>> +endobj + +22 0 obj +<< + /Type /StructElem + /S /Formula + /P 21 0 R + /K [117 118 119 120 121 122 123 124 125] + /Pg 205 0 R +>> +endobj + +23 0 obj +<< + /Type /StructElem + /S /Formula + /P 17 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115] + /Pg 205 0 R +>> +endobj + +24 0 obj +<< + /Type /StructElem + /S /P + /P 17 0 R + /K [58 25 0 R 67] + /Pg 205 0 R +>> +endobj + +25 0 obj +<< + /Type /StructElem + /S /Formula + /P 24 0 R + /K [59 60 61 62 63 64 65 66] + /Pg 205 0 R +>> +endobj + +26 0 obj +<< + /Type /StructElem + /S /P + /P 17 0 R + /K [57] + /Pg 205 0 R +>> +endobj + +27 0 obj +<< + /Type /StructElem + /S /P + /P 17 0 R + /K [46 29 0 R 48 28 0 R 56] + /Pg 205 0 R +>> +endobj + +28 0 obj +<< + /Type /StructElem + /S /Formula + /P 27 0 R + /K [49 50 51 52 53 54 55] + /Pg 205 0 R +>> +endobj + +29 0 obj +<< + /Type /StructElem + /S /Formula + /P 27 0 R + /K [47] + /Pg 205 0 R +>> +endobj + +30 0 obj +<< + /Type /StructElem + /S /Formula + /P 17 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45] + /Pg 205 0 R +>> +endobj + +31 0 obj +<< + /Type /StructElem + /S /P + /P 17 0 R + /K [244] + /Pg 203 0 R +>> +endobj + +32 0 obj +<< + /Type /StructElem + /S /P + /P 17 0 R + /K [243] + /Pg 203 0 R +>> +endobj + +33 0 obj +<< + /Type /StructElem + /S /Formula + /P 17 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242] + /Pg 203 0 R +>> +endobj + +34 0 obj +<< + /Type /StructElem + /S /Formula + /P 17 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220] + /Pg 203 0 R +>> +endobj + +35 0 obj +<< + /Type /StructElem + /S /P + /P 17 0 R + /K [203] + /Pg 203 0 R +>> +endobj + +36 0 obj +<< + /Type /StructElem + /S /Formula + /P 17 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202] + /Pg 203 0 R +>> +endobj + +37 0 obj +<< + /Type /StructElem + /S /P + /P 17 0 R + /K [156 38 0 R 160] + /Pg 203 0 R +>> +endobj + +38 0 obj +<< + /Type /StructElem + /S /Formula + /P 37 0 R + /K [157 158 159] + /Pg 203 0 R +>> +endobj + +39 0 obj +<< + /Type /StructElem + /S /P + /P 17 0 R + /K [42 0 R 145 41 0 R 150 40 0 R 155] + /Pg 203 0 R +>> +endobj + +40 0 obj +<< + /Type /StructElem + /S /Formula + /P 39 0 R + /K [151 152 153 154] + /Pg 203 0 R +>> +endobj + +41 0 obj +<< + /Type /StructElem + /S /Formula + /P 39 0 R + /K [146 147 148 149] + /Pg 203 0 R +>> +endobj + +42 0 obj +<< + /Type /StructElem + /S /Formula + /P 39 0 R + /K [144] + /Pg 203 0 R +>> +endobj + +43 0 obj +<< + /Type /StructElem + /S /Formula + /P 17 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143] + /Pg 203 0 R +>> +endobj + +44 0 obj +<< + /Type /StructElem + /S /P + /P 17 0 R + /K [122] + /Pg 203 0 R +>> +endobj + +45 0 obj +<< + /Type /StructElem + /S /Formula + /P 17 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121] + /Pg 203 0 R +>> +endobj + +46 0 obj +<< + /Type /StructElem + /S /P + /P 17 0 R + /K [66 48 0 R 68 47 0 R 75] + /Pg 203 0 R +>> +endobj + +47 0 obj +<< + /Type /StructElem + /S /Formula + /P 46 0 R + /K [69 70 71 72 73 74] + /Pg 203 0 R +>> +endobj + +48 0 obj +<< + /Type /StructElem + /S /Formula + /P 46 0 R + /K [67] + /Pg 203 0 R +>> +endobj + +49 0 obj +<< + /Type /StructElem + /S /Formula + /P 17 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65] + /Pg 203 0 R +>> +endobj + +50 0 obj +<< + /Type /StructElem + /S /P + /P 17 0 R + /K [19] + /Pg 203 0 R +>> +endobj + +51 0 obj +<< + /Type /StructElem + /S /P + /P 17 0 R + /K [16 52 0 R 18] + /Pg 203 0 R +>> +endobj + +52 0 obj +<< + /Type /StructElem + /S /Formula + /P 51 0 R + /K [17] + /Pg 203 0 R +>> +endobj + +53 0 obj +<< + /Type /StructElem + /S /Formula + /P 17 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15] + /Pg 203 0 R +>> +endobj + +54 0 obj +<< + /Type /StructElem + /S /P + /P 17 0 R + /K [326 55 0 R] + /Pg 201 0 R +>> +endobj + +55 0 obj +<< + /Type /StructElem + /S /Formula + /P 54 0 R + /K [327 328 329 330 331] + /Pg 201 0 R +>> +endobj + +56 0 obj +<< + /Type /StructElem + /S /P + /P 17 0 R + /K [325] + /Pg 201 0 R +>> +endobj + +57 0 obj +<< + /Type /StructElem + /S /P + /P 17 0 R + /K [318 59 0 R 320 58 0 R] + /Pg 201 0 R +>> +endobj + +58 0 obj +<< + /Type /StructElem + /S /Formula + /P 57 0 R + /K [321 322 323 324] + /Pg 201 0 R +>> +endobj + +59 0 obj +<< + /Type /StructElem + /S /Formula + /P 57 0 R + /K [319] + /Pg 201 0 R +>> +endobj + +60 0 obj +<< + /Type /StructElem + /S /Formula + /P 17 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317] + /Pg 201 0 R +>> +endobj + +61 0 obj +<< + /Type /StructElem + /S /P + /P 17 0 R + /K [253 64 0 R 257 63 0 R 261 62 0 R] + /Pg 201 0 R +>> +endobj + +62 0 obj +<< + /Type /StructElem + /S /Formula + /P 61 0 R + /K [262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284] + /Pg 201 0 R +>> +endobj + +63 0 obj +<< + /Type /StructElem + /S /Formula + /P 61 0 R + /K [258 259 260] + /Pg 201 0 R +>> +endobj + +64 0 obj +<< + /Type /StructElem + /S /Formula + /P 61 0 R + /K [254 255 256] + /Pg 201 0 R +>> +endobj + +65 0 obj +<< + /Type /StructElem + /S /P + /P 17 0 R + /K [210 68 0 R 226 67 0 R 236 66 0 R 252] + /Pg 201 0 R +>> +endobj + +66 0 obj +<< + /Type /StructElem + /S /Formula + /P 65 0 R + /K [237 238 239 240 241 242 243 244 245 246 247 248 249 250 251] + /Pg 201 0 R +>> +endobj + +67 0 obj +<< + /Type /StructElem + /S /Formula + /P 65 0 R + /K [227 228 229 230 231 232 233 234 235] + /Pg 201 0 R +>> +endobj + +68 0 obj +<< + /Type /StructElem + /S /Formula + /P 65 0 R + /K [211 212 213 214 215 216 217 218 219 220 221 222 223 224 225] + /Pg 201 0 R +>> +endobj + +69 0 obj +<< + /Type /StructElem + /S /P + /P 17 0 R + /K [209] + /Pg 201 0 R +>> +endobj + +70 0 obj +<< + /Type /StructElem + /S /H1 + /P 17 0 R + /T (Kontinuitet) + /K [208] + /Pg 201 0 R +>> +endobj + +71 0 obj +<< + /Type /StructElem + /S /Formula + /P 17 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207] + /Pg 201 0 R +>> +endobj + +72 0 obj +<< + /Type /StructElem + /S /Formula + /P 17 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152] + /Pg 201 0 R +>> +endobj + +73 0 obj +<< + /Type /StructElem + /S /P + /P 17 0 R + /K [74 0 R] +>> +endobj + +74 0 obj +<< + /Type /StructElem + /S /Strong + /P 73 0 R + /K [112] + /Pg 201 0 R +>> +endobj + +75 0 obj +<< + /Type /StructElem + /S /Formula + /P 17 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111] + /Pg 201 0 R +>> +endobj + +76 0 obj +<< + /Type /StructElem + /S /P + /P 17 0 R + /K [77 0 R] +>> +endobj + +77 0 obj +<< + /Type /StructElem + /S /Formula + /P 76 0 R + /K [76 77 78 79 80 81 82 83 84 85] + /Pg 201 0 R +>> +endobj + +78 0 obj +<< + /Type /StructElem + /S /P + /P 17 0 R + /K [80 0 R 60 79 0 R] + /Pg 201 0 R +>> +endobj + +79 0 obj +<< + /Type /StructElem + /S /Formula + /P 78 0 R + /K [61 62 63 64 65 66 67 68 69 70 71 72 73 74 75] + /Pg 201 0 R +>> +endobj + +80 0 obj +<< + /Type /StructElem + /S /Strong + /P 78 0 R + /K [59] + /Pg 201 0 R +>> +endobj + +81 0 obj +<< + /Type /StructElem + /S /H2 + /P 17 0 R + /T + /K [58] + /Pg 201 0 R +>> +endobj + +82 0 obj +<< + /Type /StructElem + /S /P + /P 17 0 R + /K [83 0 R] +>> +endobj + +83 0 obj +<< + /Type /StructElem + /S /Formula + /P 82 0 R + /K [43 44 45 46 47 48 49 50 51 52 53 54 55 56 57] + /Pg 201 0 R +>> +endobj + +84 0 obj +<< + /Type /StructElem + /S /P + /P 17 0 R + /K [35 85 0 R 42] + /Pg 201 0 R +>> +endobj + +85 0 obj +<< + /Type /StructElem + /S /Formula + /P 84 0 R + /K [36 37 38 39 40 41] + /Pg 201 0 R +>> +endobj + +86 0 obj +<< + /Type /StructElem + /S /H3 + /P 17 0 R + /T (Niveaukurver) + /K [34] + /Pg 201 0 R +>> +endobj + +87 0 obj +<< + /Type /StructElem + /S /P + /P 17 0 R + /K [28 88 0 R] + /Pg 201 0 R +>> +endobj + +88 0 obj +<< + /Type /StructElem + /S /Formula + /P 87 0 R + /K [29 30 31 32 33] + /Pg 201 0 R +>> +endobj + +89 0 obj +<< + /Type /StructElem + /S /P + /P 17 0 R + /K [90 0 R] +>> +endobj + +90 0 obj +<< + /Type /StructElem + /S /Formula + /P 89 0 R + /K [1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27] + /Pg 201 0 R +>> +endobj + +91 0 obj +<< + /Type /StructElem + /S /H2 + /P 17 0 R + /T (Visualiseringer af funktioner (1.4 i bogen)) + /K [0] + /Pg 201 0 R +>> +endobj + +92 0 obj +<< + /Type /StructElem + /S /Formula + /P 17 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [251 252 253 254 255 256 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1", + date: datetime(year: 2026, month: 02, day: 03), + semester: "Spring 2026" +) + +#set math.mat(delim: "[") +#set math.vec(delim: "[") + + += Funktioner +For at noget skal være en graf, må én input-værdi kun give én værdi. + += Vektorfunktioner af flere variabler +#definition(title: "3")[ +Funtioner af typen $underline(f): A -> RR^k$ +har $A="dom"(f) in RR^n$ kaldes vektorfunktioner af _n_ variable. + +$A="dom"(f)$ - Definitionsmængden (domæne)\ +$RR^k="codom"(f)$ - Dispositionsmængden (co-domænen)\ +$"im"(f)="Vm"(f)={underline(f)(x)|underline(x) in "dom"(f)}$ - Værdimængden/billedrummet (image/range). Alt hvad funktionen kan ramme\ + +$f=underline(x) |-> f(underline(x))$ + +Input: $underline(x) in RR^n$ - Søjlevektor i $RR^n$ + +Output: $underline(f(x)) in RR^k$ - Søjlevektor i $RR^k$ +] + +#example(title: "1.3.1")[ +Lad $bold(A) in M_(k times n)(RR) = RR^(k times n)$ + +$ +f: RR^n -> RR^k\ +f(underline(x)) = A dot underline(x) (f= underline(x) |-> A underline(x)) +$ +$bold(A)=mat(1,2;2,4)$$ + +bold(A)(underline(x))&=mat(1,2;2,4)vec(x_1,x_2)\ +&=vec(1x_1+2x_2,2x_1+4x_2)\ +&=vec(1,2)x_1 + vec(2,4)x_2 +$ +Ligning: $f(underline(x))=underline(y) quad, underline(y) in "codom"(f)=RR^k$ + +Er den injektiv? Nej, $f(vec(-2,1))=f(vec(0,0))=vec(0,0)$ + +Er den surjektiv? Nej, $"im"(f)="span"_RR (vec(1,2),vec(2,4))="span"_RR (vec(1,2)) = {t vec(1,2)|t in RR} = "col"(bold(A))$ +] +== Koordinatfunktioner +#example()[ +For vektorfunktionen: $bold(A)(underline(x))&=mat(1,2;2,4)vec(x_1,x_2)=vec(f_1 (underline(x)), f_2 (underline(x)))$ + +$f_1 (underline(x)) = 1 x_1 + 2 x_2$ +] + + +$ +underline(f)(underline(x))=vec(f_1 (underline(x)), f_2 (underline(x)), dots.v, f_k (underline(x)))\ +f_i : bold(A) -> RR, i = 1, dots, k\ +bold(A) = "dom"(f) +$ + + +#example(title: "1.2.2")[ +$f: bold(A) -> RR, f(x_1, x_2) = sqrt(16 - x_1^2 - x_2^2)$ + +hvor $bold(A) = "dom"(f) in RR^2$ + +Typisk opgave: Hvad er $bold(A), "codom"(f), "im"(f)$ + +Find størst mulig $bold(A)$ +$ +16 - x_1^2-x_2^2 >= 0\ +4^2 = 16 >= x_1^2 + x_2^2 +$ + +$"codom"(f)=RR$\ + +$ +f(0,0)=sqrt(16-0^2-0^2)=4\ +f(4,0)=sqrt(16-4^2-0^2)=0 +$ +Så $"im"(f)=[0,4]$ +] + +#example(title: "1.3.2 (ReLu)")[ +$ +"ReLu"&: RR -> RR\ +"ReLu"(x) &= "max"(0,x) = cases(x "," x>=0, 0 "," x<0) +$ + +$ +"ReLu": RR^n -> RR^n\ +"ReLu"(underline(x))=vec("ReLu"(x_1),"ReLu"(x_2),dots.v,"ReLu"(x_n)) +$ +] + +== Visualiseringer af funktioner (1.4 i bogen) +$"Graf"(f)={(underline(x), underline(f)(underline(x))) in RR^(n+k) | underline(x) in "dom"(underline(f))}$ + +Se Eks 1.2.2. Ville kræve $n+k<= 3$ + +=== Niveaukurver +Niveaukurver for $c in "codom"(underline(f))$ - c: output + +${underline(x) in "dom"(f) | underline(f)(underline(x))= underline(c)}$ + + +== Neuralt netværk (Def 1.3.2) +*1. lag:* +$underline(x) mapsto sigma_1 (bold(A)_1 underline(x) + underline(b)_1) = underline(z)_1$ + +$bold(A)_1 in RR^(50 times 784)$ + +$ +f_1 : RR^784 -> RR^50\ +sigma_1 : RR^50 -> RR^50\ +sigma_1 = "ReLu" +$ + +*2. lag:* +$ +underline(z)_1 mapsto sigma_2 (bold(A)_2 underline(z)_1 + underline(b)_2) = underline(z)_2 in RR^10\ +bold(A)_2 in RR^(10 times 50), underline(b)_2 in RR^(10 times 1) +$ + +$ +Phi = f_2 compose f_1, quad R^894 arrow RR^10\ +f compose g(x) = f(g(x))\ + +Phi (underline(x)) = sigma_2 (bold(A)_2 sigma_1 (bold(A)_1 underline(x) + underline(b)_1) + underline(b)_2) +$ + += Kontinuitet +#definition(title: "3.2.1")[ +En funktion $f: "dom"(f) arrow RR, quad "dom"(f) in RR$ er kontinuert i $x_0 in A = "dom"(f)$ hvis $x arrow x_0 => f(x) arrow f(x_0)$ (ligning 3.1) +] + +Ligning 3.1 betyder at for ethvert $epsilon > 0$ findes der et $delta > 0$ således at $| x - x_0| < delta => |f(x) - f(x_0)| < epsilon$ + +$ +forall epsilon > 0 exists delta > 0: |x-x_0| < epsilon => |f(x) - f(x_0)| < delta +$ + +Funktionen $f$ er kontinuert hvis den er kontinuert i alle punkter i $"dom"(f)$ + + +#example()[ +Lad $a, b in RR$ + +$ +f(x) = a x + b, quad f: RR arrow RR +$ + +Påstanden: $f$ er kontinuert + +#solution()[ +$ +|f(x) - f(x_0)| &= | a x + b (a x_0 + b)|\ +&= |a x - a x_0| = |a| |x - x_0| +$ + +Lad $epsilon$ være givet. Vi vælger $delta = epsilon/(|a|)$. Så + +$ +|x-x_0| < delta = epsilon/(|a|)\ +=> |f(x) - f(x_0)| = |a||x-x_0|\ +< |a| epsilon/(|a|) = epsilon +$ +] +] + +#example()[ +$ +h: RR arrow RR, quad h(x) = cases(1 quad x=> 0, 0 quad x<0) +$ + +$h$ er diskontinuert i $x_0=0$. Lad $epsilon=1/2$. + +Der findes intet $delta >0$ : +$ +|x - x_0| < delta => |f(x)-f(x_0)| < 1/2\ +|x| < delta => |f(x) - 1| < 1/2 +$ + +Så + +$ +f((-delta)/2) = 0\ +f(delta/2)) = 1 +$ + +$ +|f((-delta)/2)-f(0)|\ += |0-1| = 1 +$ +Så funktionen er ikke kontinuert. +] + +#example()[ +$ +f: RR^2 arrow RR\ +f(x,y) = cases((x^2y)/(x^4 + y^2) quad (x,y) eq.not (0,0), 0 quad (x,y) = (0,0)) +$ + +Er $f$ kontinuert for alle $x,y) in RR^2$? + +#solution()[ +Opstil y som graf af x, så $y = a x, quad a in RR$. Så fås + +$ +f(x, a x) &= (x^2 a x)/(x^4 + a^2x^2)\ +&= (a x)/(x^2 + a^2)\ +&= (a x)/(x^2 + a^2) arrow 0/(0 + a^2) = 0, x arrow 0 +$ +Hvis $y = a x^2, quad a in RR$ +$ +f(x, a x^2) &= (x^2 a x^2)/(x^4 + a^2 x^4)\ +&= a/(1 + a^2) +$ + +Så funktionen er ikke kontinuert $(x,y)=(0,0)$ + +] +] + + + diff --git a/10060 - Physics/Kinematics 1D/Kinematics1D notebook.ipynb b/10060 - Physics/Kinematics 1D/Kinematics1D notebook.ipynb index 0eb1924..107db04 100644 --- a/10060 - Physics/Kinematics 1D/Kinematics1D notebook.ipynb +++ b/10060 - Physics/Kinematics 1D/Kinematics1D notebook.ipynb @@ -397,12 +397,12 @@ }, { "cell_type": "code", - "execution_count": 5, + "execution_count": 15, "metadata": {}, "outputs": [ { "data": { - "image/png": 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", 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", 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" ] @@ -414,14 +414,14 @@ "source": [ "import numpy as np\n", "import matplotlib.pyplot as plt\n", - "t = np.linspace(0,60,1000)\n", + "t = np.linspace(0,15+85/2,1000)\n", "def f(t):\n", - " if t < 20:\n", - " return 0.5 * t\n", - " elif t < 50:\n", + " if t < 10:\n", + " return 1 * t\n", + " elif t < 10+(85/2):\n", " return 10.0\n", " else:\n", - " return 10.0 - 1.0 * (t-50.0)\n", + " return 10.0 - 2.0 * (5-(15+85/2)+t)\n", "v = [f(T) for T in t if True]\n", "plt.figure()\n", "plt.plot(t,v,'b-')\n", @@ -853,7 +853,7 @@ ], "metadata": { "kernelspec": { - "display_name": "Python 3 (ipykernel)", + "display_name": "Python 3", "language": "python", "name": "python3" }, @@ -867,7 +867,7 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.11.4" + "version": "3.14.2" } }, "nbformat": 4,