{"id":39933,"date":"2025-11-04T15:07:19","date_gmt":"2025-11-04T13:07:19","guid":{"rendered":"https:\/\/mahifi.se\/?page_id=39933"},"modified":"2026-07-30T19:13:49","modified_gmt":"2026-07-30T17:13:49","slug":"39933-2","status":"publish","type":"page","link":"https:\/\/mahifi.se\/?page_id=39933","title":{"rendered":"Ma4 – Uppgiftsbibliotek"},"content":{"rendered":"\t\t
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Uppgiftsbibliotek - Ma4 (Under konstruktion) <\/a><\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t
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H\u00e4r kan du s\u00f6ka p\u00e5 uppgifter fr\u00e5n Mahifis uppgiftsbibliotek. V\u00e4lj vilket omr\u00e5de du vill jobba med samt vilken sv\u00e5righetsgrad. D\u00e4refter f\u00e5r du ett antal slumpm\u00e4ssigt utvalda uppgifter. Alla uppgifter har facit och vissa har \u00e4ven l\u00f6sningsf\u00f6rslag. Du kan byta uppgift genom att trycka p\u00e5 uppgiftsnumret till v\u00e4nster n\u00e4r du s\u00f6ker p\u00e5 uppgifter. Du kan \u00e4ven skapa en diagnos d\u00e4r det genereras uppgifter som man enbart kan se facit p\u00e5 efter att man tryckt p\u00e5 knappen Visa facit\/Klar l\u00e4ngst ner.Om uppgiften har f\u00f6ljande symbol intill sig\u00a0\"\ud83e\uddee\"<\/span> betyder det att du f\u00e5r anv\u00e4nda digitala hj\u00e4lpmedel (t.ex. minir\u00e4knare eller geogebra) p\u00e5 uppgiften.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t

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\n\t\t\t\t\t\r\n\r\n\r\n\r\nSlumpgenerator Ma1c<\/title>\r\n<script src=\"https:\/\/cdn.jsdelivr.net\/npm\/mathjax@3\/es5\/tex-mml-chtml.js\"><\/script>\r\n<style>\r\n body { font-family: Arial, sans-serif; margin: 0px; }\r\n button {\r\n padding: 8px 12px;\r\n border: none;\r\n border-radius: 6px;\r\n background-color: #4747D7;\r\n color: white;\r\n cursor: pointer;\r\n font-size: 14px;\r\n margin: 4px 4px 4px 0;\r\n }\r\n button:hover { background-color: #3636b0; }\r\n button:focus, button:active { background-color: #4747D7 !important; color: white !important; outline: none; }\r\n label { margin-right: 15px; }\r\n .uppgift { margin-bottom: 30px; font-size: 16px; line-height: 1.4; text-align: left; }\r\n .facit, .losningsforslag { display: none; margin-top: 5px; font-style: italic; color: #333; font-size: 16px; line-height: 1.4; }\r\n .poang { color: black; font-weight: normal; margin-left: 5px; font-size: 16px; display: inline-block; }\r\n img { max-width: 400px; height: auto; display: block; margin: 10px 0; }\r\n .controls { margin-bottom: 12px; }\r\n #uppgifterContainer { margin-top: 16px; }\r\n #diagnosKlarContainer { margin-top: 12px; }\r\n .meta { margin-top: 6px; }\r\n .losningsforslag { color: #555; font-size: 15px; }\r\n .losningsforslag, .facit { margin-bottom: 8px; }\r\n .byt-uppgift { cursor: pointer; color: #4747D7; }\r\n .byt-uppgift:hover { text-decoration: underline; }\r\n\r\n #diagnosPopup, #utokadPopup {\r\n display: none;\r\n position: fixed;\r\n top: 10%;\r\n left: 50%;\r\n transform: translateX(-50%);\r\n background-color: white;\r\n padding: 24px;\r\n border: 2px solid #4747D7;\r\n border-radius: 10px;\r\n z-index: 1000;\r\n width: 600px;\r\n max-height: 80vh;\r\n overflow-y: auto;\r\n box-shadow: 0 4px 12px rgba(0,0,0,0.2);\r\n}\r\n\r\n\/* ==== ADAPTIV DIAGNOS DESIGN ==== *\/\r\n.adaptiv-header {\r\n background: linear-gradient(90deg, #4747D7, #3a3ac5);\r\n color: white;\r\n padding: 12px 16px;\r\n border-radius: 10px 10px 0 0;\r\n font-weight: bold;\r\n font-size: 18px;\r\n text-align: center;\r\n}\r\n\r\n.adaptiv-info {\r\n background-color: #f2f3ff;\r\n border: 1px solid #d8dbff;\r\n border-radius: 8px;\r\n padding: 10px 14px;\r\n font-size: 14px;\r\n margin-bottom: 12px;\r\n color: #333;\r\n}\r\n\r\n.adaptiv-uppgift {\r\n background: white;\r\n border: 1px solid #ddd;\r\n border-radius: 10px;\r\n padding: 18px;\r\n margin-top: 12px;\r\n box-shadow: 0 2px 8px rgba(0,0,0,0.1);\r\n}\r\n\r\n.adaptiv-uppgift b {\r\n font-size: 17px;\r\n color: #222;\r\n}\r\n\r\n.adaptiv-progress {\r\n margin-top: 6px;\r\n font-size: 14px;\r\n color: #555;\r\n}\r\n\r\n.adaptiv-btn {\r\n background-color: #4747D7;\r\n color: white;\r\n border: none;\r\n padding: 8px 14px;\r\n border-radius: 6px;\r\n cursor: pointer;\r\n font-size: 14px;\r\n margin: 6px 6px 0 0;\r\n transition: background-color 0.2s;\r\n}\r\n.adaptiv-btn:hover {\r\n background-color: #3a3ac5;\r\n}\r\n\r\n.adaptiv-facit {\r\n margin-top: 12px;\r\n background-color: #f7f8ff;\r\n border-left: 4px solid #4747D7;\r\n padding: 10px 14px;\r\n border-radius: 6px;\r\n color: #333;\r\n}\r\n\r\n.adaptiv-l\u00f6sning {\r\n margin-top: 6px;\r\n font-size: 14px;\r\n color: #444;\r\n}\r\n\r\n#uppgifterContainer {\r\n margin-top: 20px;\r\n}\r\n\r\n\/* Popup f\u00f6r adaptiv diagnos *\/\r\n#adaptivPopup {\r\n animation: fadeIn 0.25s ease;\r\n border: 2px solid #4747D7;\r\n border-radius: 12px;\r\n box-shadow: 0 4px 14px rgba(0,0,0,0.2);\r\n\r\n}\r\n\r\n\r\n#adaptivPopup h3 {\r\n color: black;\r\n text-align: center;\r\n margin-bottom: 16px;\r\n}\r\n\r\n@keyframes fadeIn {\r\n from { opacity: 0; transform: translateY(-10px); }\r\n to { opacity: 1; transform: translateY(0); }\r\n}\r\n\r\n\r\n\/* \u2705 g\u00f6r flera kolumner automatiskt *\/\r\n#utokadPopup .kategori-gruppar {\r\n display: grid;\r\n grid-template-columns: repeat(auto-fit, minmax(200px, 1fr));\r\n gap: 16px 24px;\r\n align-items: start;\r\n}\r\n\r\n #diagnosPopup input[type=\"number\"], #utokadPopup input[type=\"number\"] { width: 62px; margin-left: 5px; margin-right: 15px; }\r\n #totalPoang { margin-top: 10px; font-weight: bold; }\r\n\r\n .digital-icon {\r\n margin-left: 6px;\r\n font-size: 16px;\r\n vertical-align: middle;\r\n }\r\n<\/style>\r\n<\/head>\r\n<body>\r\n\r\n<div class=\"controls\">\r\n <div>\r\n <label><input type=\"checkbox\" id=\"trig\"> Trigonometriska formler\/ekvationer<\/label>\r\n <label><input type=\"checkbox\" id=\"funktioner\"> Trigonometriska funktioner<\/label>\r\n <label><input type=\"checkbox\" id=\"derivata\"> Derivata\/analys av grafer<\/label>\r\n <label><input type=\"checkbox\" id=\"integraler\"> Integraler<\/label>\r\n <label><input type=\"checkbox\" id=\"tillampintegral\"> Till\u00e4mpningar integraler<\/label>\r\n <\/div>\r\n <div>\r\n <label><input type=\"checkbox\" id=\"komplex\"> Komplexa tal\/polynomekvationer<\/label>\r\n <label><input type=\"checkbox\" id=\"geogebra\"> Geogebra<\/label><button id=\"utokadBtn\">Avancerad s\u00f6kning<\/button>\r\n <\/div>\r\n\r\n <div style=\"margin-top:8px;\">\r\n <label><input type=\"checkbox\" id=\"E\"> E-niv\u00e5<\/label>\r\n <label><input type=\"checkbox\" id=\"C\"> C-niv\u00e5<\/label>\r\n <label><input type=\"checkbox\" id=\"A\"> A-niv\u00e5<\/label>\r\n <\/div>\r\n\r\n <button id=\"genereraBtn\">S\u00f6k uppgifter<\/button>\r\n <label>Antal uppgifter:\r\n <input type=\"number\" id=\"antal\" min=\"1\" max=\"50\" value=\"5\">\r\n <\/label> \r\n <div><\/div>\r\n <button id=\"genereraAllaBtn\">Visa alla uppgifter<\/button>\r\n <div><\/div>\r\n <button id=\"skapaDiagnosBtn\">Skapa diagnos<\/button>\r\n<div><\/div>\r\n<button id=\"startAdaptivBtn\">Starta adaptiv diagnos<\/button>\r\n<div><\/div>\r\n\r\n <button id=\"laddaKodBtn\">Ladda uppgifter<\/button>\r\n <label>Skriv kod: \r\n <input type=\"text\" id=\"kodInput\" placeholder=\"t.ex. f1a2f3\" style=\"width:150px;\">\r\n <\/label>\r\n\r\n<div id=\"uppgifterContainer\"><\/div>\r\n<div id=\"diagnosKlarContainer\"><\/div>\r\n<div id=\"diagnosKodVisning\" style=\"margin-top:10px; font-size:14px; color:#333;\"><\/div>\r\n\r\n\r\n<!-- Popup f\u00f6r diagnos -->\r\n<div id=\"diagnosPopup\">\r\n <h3>Skapa diagnos<\/h3>\r\n <p style=\"font-size:14px; color:#333; margin-bottom:10px;\">\r\n H\u00e4r kan du skapa en diagnos utifr\u00e5n dina valda moment och sv\u00e5righetsgrader. Notera att du m\u00e5ste kryssa i dessa innan du skapar diagnosen.\r\n <\/p>\r\n <label>E-po\u00e4ng: <input type=\"number\" id=\"epoang\" min=\"0\" value=\"0\"><\/label>\r\n <label>C-po\u00e4ng: <input type=\"number\" id=\"cpoang\" min=\"0\" value=\"0\"><\/label>\r\n <label>A-po\u00e4ng: <input type=\"number\" id=\"apoang\" min=\"0\" value=\"0\"><\/label><br><br>\r\n <label><input type=\"checkbox\" id=\"slumpmasigt\"> Slumpm\u00e4ssigt<\/label>\r\n <div id=\"totalPoang\">Totalt valt: 0<\/div><br>\r\n <hr style=\"margin:10px 0;\">\r\n<label for=\"diagnosKod\">Diagnoskod:<\/label>\r\n<input type=\"text\" id=\"diagnosKod\" placeholder=\"t.ex. f1f3f7a2\" style=\"width:180px; margin-left:8px;\">\r\n<button id=\"laddaDiagnosKodBtn\">Ladda diagnoskod<\/button>\r\n <button id=\"startDiagnosBtn\">Starta diagnos<\/button>\r\n <button id=\"stangPopupBtn\">Avbryt<\/button>\r\n<\/div>\r\n\r\n<!-- Popup f\u00f6r ut\u00f6kad s\u00f6kning -->\r\n<div id=\"utokadPopup\">\r\n <h3>Avancerad s\u00f6kning<\/h3>\r\n\r\n <div class=\"kategori-gruppar\">\r\n <div>\r\n <b>Trigonometriska formler\/ekvationer<\/b><br>\r\n <label><input type=\"checkbox\" class=\"underkategori\" value=\"formler\"> Trigonometriska formler<\/label><br>\r\n <label><input type=\"checkbox\" class=\"underkategori\" value=\"ekvationer\"> Trignometriska ekvationer<\/label><br>\r\n <\/div>\r\n \r\n <div>\r\n <b>Trigonometriska funktioner<\/b><br>\r\n <label><input type=\"checkbox\" class=\"underkategori\" value=\"funktioner\"> Funktioner<\/label><br>\r\n <label><input type=\"checkbox\" class=\"underkategori\" value=\"grafer\"> Grafer<\/label><br>\r\n <label><input type=\"checkbox\" class=\"underkategori\" value=\"problemfunk\"> Probleml\u00f6sning funktioner<\/label><br>\r\n <\/div>\r\n <div>\r\n <b>Derivata\/analys av grafer<\/b><br>\r\n <label><input type=\"checkbox\" class=\"underkategori\" value=\"deriveringsregler\"> Deriveringsregler<\/label><br>\r\n <label><input type=\"checkbox\" class=\"underkategori\" value=\"derivatagraf\"> Grafer och derivata<\/label><br>\r\n <label><input type=\"checkbox\" class=\"underkategori\" value=\"analysgraf\"> Analys av grafer\/asymptoter <\/label><br>\r\n <label><input type=\"checkbox\" class=\"underkategori\" value=\"problemderivata\"> Probleml\u00f6sning derivata<\/label><br>\r\n <\/div>\r\n\r\n <div>\r\n <b>Integraler<\/b><br>\r\n <label><input type=\"checkbox\" class=\"underkategori\" value=\"primitiv\"> Primitiva funktioner<\/label><br>\r\n <label><input type=\"checkbox\" class=\"underkategori\" value=\"primintegral\"> Integraler med primitiv funktion<\/label><br>\r\n <label><input type=\"checkbox\" class=\"underkategori\" value=\"grafintegral\"> Grafer och integraler<\/label><br>\r\n <\/div>\r\n <div>\r\n <b>Probleml\u00f6sning integraler<\/b><br>\r\n <label><input type=\"checkbox\" class=\"underkategori\" value=\"integralproblem\"> Probleml\u00f6sing integraler<\/label><br>\r\n <label><input type=\"checkbox\" class=\"underkategori\" value=\"normal\"> Normalf\u00f6rdelning<\/label><br>\r\n <label><input type=\"checkbox\" class=\"underkategori\" value=\"rotation\"> Rotationsvolymer<\/label><br>\r\n <\/div>\r\n <div>\r\n <b>Komplexa tal\/Polynomekvationer <\/b><br>\r\n <label><input type=\"checkbox\" class=\"underkategori\" value=\"komplexa\"> Komplexa tal<\/label><br>\r\n <label><input type=\"checkbox\" class=\"underkategori\" value=\"polynom\"> Polynomekvationer<\/label><br>\r\n <\/div>\r\n <div>\r\n <b>Geogebra<\/b><br>\r\n <label><input type=\"checkbox\" class=\"underkategori\" value=\"geogebraderivata\"> Derivata<\/label><br>\r\n <label><input type=\"checkbox\" class=\"underkategori\" value=\"geogebraekvation\"> Ekvationer<\/label><br>\r\n <\/div>\r\n<\/div>\r\n\r\n \r\n\r\n <br>\r\n <label><input type=\"checkbox\" id=\"utokadE\"> E-niv\u00e5<\/label>\r\n <label><input type=\"checkbox\" id=\"utokadC\"> C-niv\u00e5<\/label>\r\n <label><input type=\"checkbox\" id=\"utokadA\"> A-niv\u00e5<\/label><br> <label><input type=\"checkbox\" id=\"visaKoderUtokad\"> Visa uppgiftskoder<\/label><br><br>\r\n\r\n <label>Antal uppgifter: <input type=\"number\" id=\"utokadAntal\" min=\"1\" max=\"50\" value=\"5\"><\/label> <button id=\"startUtokadBtn\">Starta s\u00f6kning<\/button>\r\n <button id=\"visaAllaUtokadBtn\">Visa alla uppgifter<\/button>\r\n <button id=\"stangUtokadBtn\">St\u00e4ng<\/button> <br><br>\r\n\r\n<\/div>\r\n\r\n<div id=\"adaptivPopup\" style=\"display:none; position: fixed; top:10%; left:50%; transform:translateX(-50%);\r\n background-color:white; padding:24px; border:2px solid #4747D7; border-radius:10px; z-index:1000; width:600px; max-height:80vh; overflow-y:auto; box-shadow:0 4px 12px rgba(0,0,0,0.2);\">\r\n <h3>V\u00e4lj omr\u00e5den f\u00f6r adaptiv diagnos<\/h3>\r\n <p style=\"font-size:14px; color:#333; margin-bottom:10px;\">\r\n En adaptiv diagnos \u00e4r en diagnos d\u00e4r du g\u00f6r uppgifter p\u00e5 en s\u00e4rskild niv\u00e5. Om du f\u00e5r 3 r\u00e4tt i rad g\u00e5r du upp en niv\u00e5, om du f\u00e5r 3 fel i rad g\u00e5r du ner en niv\u00e5. Det \u00e4r ett s\u00e4tt att repetera uppgifter utifr\u00e5n sin egna niv\u00e5. \r\n <div class=\"kategori-gruppar\">\r\n <!-- Kopiera samma struktur som i ut\u00f6kadPopup -->\r\n <div>\r\n <b>Trigonometriska formler\/ekvationer<\/b><br>\r\n <label><input type=\"checkbox\" class=\"underkategori\" value=\"formler\"> Trigonometriska formler<\/label><br>\r\n <label><input type=\"checkbox\" class=\"underkategori\" value=\"ekvationer\"> Trignometriska ekvationer<\/label><br>\r\n <\/div>\r\n \r\n <div>\r\n <b>Trigonometriska funktioner<\/b><br>\r\n <label><input type=\"checkbox\" class=\"underkategori\" value=\"funktioner\"> Funktioner<\/label><br>\r\n <label><input type=\"checkbox\" class=\"underkategori\" value=\"grafer\"> Grafer<\/label><br>\r\n <label><input type=\"checkbox\" class=\"underkategori\" value=\"problemfunk\"> Probleml\u00f6sning funktioner<\/label><br>\r\n <\/div>\r\n <div>\r\n <b>Derivata<\/b><br>\r\n <label><input type=\"checkbox\" class=\"underkategori\" value=\"deriveringsregler\"> Deriveringsregler<\/label><br>\r\n <label><input type=\"checkbox\" class=\"underkategori\" value=\"derivatagraf\"> Grafer och derivata<\/label><br>\r\n <label><input type=\"checkbox\" class=\"underkategori\" value=\"problemderivata\"> Probleml\u00f6sning derivata<\/label><br>\r\n <\/div>\r\n\r\n <div>\r\n <b>Integraler<\/b><br>\r\n <label><input type=\"checkbox\" class=\"underkategori\" value=\"primitiv\"> Primitiva funktioner<\/label><br>\r\n <label><input type=\"checkbox\" class=\"underkategori\" value=\"primintegral\"> Integraler med primitiv funktion<\/label><br>\r\n <label><input type=\"checkbox\" class=\"underkategori\" value=\"grafintegral\"> Grafer och integraler<\/label><br>\r\n <\/div>\r\n <div>\r\n <b>Probleml\u00f6sning integraler<\/b><br>\r\n <label><input type=\"checkbox\" class=\"underkategori\" value=\"integralproblem\"> Probleml\u00f6sing integraler<\/label><br>\r\n <label><input type=\"checkbox\" class=\"underkategori\" value=\"normal\"> Normalf\u00f6rdelning<\/label><br>\r\n <label><input type=\"checkbox\" class=\"underkategori\" value=\"rotation\"> Rotationsvolymer<\/label><br>\r\n <\/div>\r\n <div>\r\n <b>Komplexa tal\/Polynomekvationer <\/b><br>\r\n <label><input type=\"checkbox\" class=\"underkategori\" value=\"komplexa\"> Komplexa tal<\/label><br>\r\n <label><input type=\"checkbox\" class=\"underkategori\" value=\"polynom\"> Polynomekvationer<\/label><br>\r\n <\/div>\r\n <div>\r\n <b>Geogebra<\/b><br>\r\n <label><input type=\"checkbox\" class=\"underkategori\" value=\"geogebraderivata\"> Derivata<\/label><br>\r\n <label><input type=\"checkbox\" class=\"underkategori\" value=\"geogebraekvation\"> Ekvationer<\/label><br>\r\n <\/div>\r\n<\/div>\r\n <br>\r\n <button id=\"startAdaptivUppgifterBtn\">Starta<\/button>\r\n <button id=\"stangAdaptivBtn\">St\u00e4ng<\/button>\r\n<\/div>\r\n\r\n<script>\r\n \r\n \/\/ ---- Fr\u00e5gebank ----\r\n const fragor = {\r\n trig: [\r\n{\r\n fraga: `L\u00f6s ekvationen \\\\(sinx=\\\\frac{1}{2}\\\\). Svara i grader.`,\r\n svar: `\\\\(x=30^{\\\\circ}+360^{\\\\circ}\\\\cdot n\\\\) och \\\\(x=150^{\\\\circ}+360^{\\\\circ}\\\\cdot n\\\\)`,\r\n niva: \"E\",\r\n poang: [2,0,0],\r\n underkategori: [\"ekvationer\"],\r\n kod: \"a1\"\r\n},\r\n{\r\n fraga: `L\u00f6s ekvationen \\\\(2cosx=1\\\\) .`,\r\n svar: `\\\\(x=\\\\pm 60^{\\\\circ}+360^{\\\\circ}\\\\cdot n\\\\)`,\r\n niva: \"E\",\r\n poang: [2,0,0],\r\n underkategori: [\"ekvationer\"],\r\n kod: \"a2\"\r\n},\r\n{\r\n fraga: `L\u00f6s ekvationen \\\\(cos2x=\\\\frac{\\\\sqrt{3}}{2}\\\\). Svara i grader.`,\r\n svar: `\\\\(x=\\\\pm 15^{\\\\circ}+180^{\\\\circ}\\\\cdot n\\\\)`,\r\n niva: \"E\",\r\n poang: [3,0,0],\r\n underkategori: [\"ekvationer\"],\r\n kod: \"a3\"\r\n},\r\n{\r\n fraga: `L\u00f6s ekvationen \\\\(2sin\\\\frac{x}{2}=\\\\sqrt{3}\\\\) .`,\r\n svar: `\\\\(x=120^{\\\\circ}+720^{\\\\circ}\\\\cdot n\\\\) och \\\\(x=240^{\\\\circ}+720^{\\\\circ}\\\\cdot n\\\\)`,\r\n niva: \"C\",\r\n poang: [2,1,0],\r\n underkategori: [\"ekvationer\"],\r\n kod: \"a4\"\r\n},\r\n{\r\n fraga: `L\u00f6s ekvationen \\\\(2sinxcosx=1\\\\) .`,\r\n svar: `\\\\(x=45^{\\\\circ}+180^{\\\\circ}\\\\cdot n\\\\) `,\r\n niva: \"C\",\r\n poang: [2,1,0],\r\n underkategori: [\"ekvationer\"],\r\n kod: \"a5\"\r\n},\r\n{\r\n fraga: `L\u00f6s ekvationen \\\\((cosx-sinx)(cosx+sinx)=\\\\frac{1}{\\\\sqrt{2}}\\\\). Svara i radianer .`,\r\n svar: `\\\\(x=\\\\pm \\\\frac{\\\\pi}{8}+ \\\\pi \\\\cdot n\\\\) `,\r\n niva: \"C\",\r\n poang: [0,2,0],\r\n underkategori: [\"ekvationer\"],\r\n kod: \"a6\"\r\n},\r\n{\r\n fraga: `L\u00f6s ekvationen \\\\(sinx=-1\\\\). Svara i radianer.`,\r\n svar: `\\\\(x= \\\\pi + 2\\\\pi \\\\cdot n\\\\) `,\r\n niva: \"E\",\r\n poang: [2,0,0],\r\n underkategori: [\"ekvationer\"],\r\n kod: \"a7\"\r\n},\r\n{\r\n fraga: `L\u00f6s f\u00f6ljande ekvation \\\\(sinx+cosx=1\\\\). Svara i radianer`,\r\n svar: `\\\\(x=0 + 2\\\\pi \\\\cdot n\\\\) eller \\\\(x=\\\\frac{\\\\pi}{2} + 2\\\\pi \\\\cdot n\\\\)`,\r\n niva: \"C\",\r\n poang: [0,2,0],\r\n underkategori: [\"ekvationer\"],\r\n kod: \"a8\"\r\n},\r\n{\r\n fraga: `Best\u00e4m ett exakt v\u00e4rde p\u00e5 uttrycket \\\\(sin(75^{\\\\circ})cos(285^{\\\\circ})\\\\)`,\r\n svar: `\\\\(\\\\frac{1}{4}\\\\)`,\r\n niva: \"A\",\r\n poang: [0,0,2],\r\n underkategori: [\"formler\"],\r\n kod: \"a9\"\r\n},\r\n{\r\n fraga: `Best\u00e4m ett exakt v\u00e4rde f\u00f6r f\u00f6ljande uttryck \\\\(sin(\\\\pi)+cos(\\\\pi)\\\\)`,\r\n svar: `-1`,\r\n niva: \"E\",\r\n poang: [1,0,0],\r\n underkategori: [\"formler\"],\r\n kod: \"a10\"\r\n},\r\n{\r\n fraga: `Best\u00e4m ett exakt v\u00e4rde f\u00f6r f\u00f6ljande uttryck \\\\(2sin(15^{\\\\circ})cos(15^{\\\\circ})\\\\)`,\r\n svar: `\\\\(\\\\frac{1}{2}\\\\)`,\r\n niva: \"C\",\r\n poang: [0,1,0],\r\n underkategori: [\"formler\"],\r\n kod: \"a11\"\r\n},\r\n{\r\n fraga: `Best\u00e4m ett exakt v\u00e4rde f\u00f6r uttrycket \\\\(sin^2(\\\\frac{\\\\pi}{5})+cos^2(\\\\frac{\\\\pi}{5})\\\\)`,\r\n svar: `1 Tips: det \u00e4r trig.ettan`,\r\n niva: \"C\",\r\n poang: [0,1,0],\r\n underkategori: [\"formler\"],\r\n kod: \"a12\"\r\n},\r\n{\r\n fraga: `Best\u00e4m ett exakt v\u00e4rde f\u00f6r f\u00f6ljande uttryck \\\\(sin^2x+cos^2x+sin(30^{\\\\circ})\\\\)`,\r\n svar: `\\\\(\\\\frac{3}{2}\\\\)`,\r\n niva: \"E\",\r\n poang: [1,0,0],\r\n underkategori: [\"formler\"],\r\n kod: \"a13\"\r\n},\r\n{\r\n fraga: `Best\u00e4m ett exakt v\u00e4rde f\u00f6r f\u00f6ljande uttryck \\\\(cos(51\\\\pi)\\\\)`,\r\n svar: `-1`,\r\n niva: \"C\",\r\n poang: [0,1,0],\r\n underkategori: [\"formler\"],\r\n kod: \"a14\"\r\n},\r\n{\r\n fraga: `Joakim menar att uttrycket \\\\(sin(\\\\frac{40 \\\\pi}{79})\\\\) \u00e4r v\u00e4ldigt n\u00e4ra ett v\u00e4rde men \u00e4r inte exakt. Vilket v\u00e4rde t\u00e4nker han p\u00e5.`,\r\n svar: `V\u00e4ldigt n\u00e4ra 1 eftersom \\\\(\\\\frac{40\\\\pi}{79}\\\\) \u00e4r v\u00e4ldigt n\u00e4ra \\\\(\\\\frac{\\\\pi}{2}\\\\)`,\r\n niva: \"C\",\r\n poang: [0,1,0],\r\n underkategori: [\"formler\"],\r\n kod: \"a15\"\r\n},\r\n{\r\n fraga: `Visa att f\u00f6ljande likhet st\u00e4mmer \\\\(\\\\frac{sin2x}{2cosx}=sinx\\\\)`,\r\n svar: `Skriv om v\u00e4nsterledet med hj\u00e4lp av sinus f\u00f6r dubbla vinkeln.`,\r\n niva: \"E\",\r\n poang: [2,0,0],\r\n underkategori: [\"formler\"],\r\n kod: \"a16\"\r\n},\r\n{\r\n fraga: `Visa att f\u00f6ljande likhet st\u00e4mmer \\\\(1-\\\\frac{cos^2x}{1+sinx}=sinx\\\\)`,\r\n svar: `Skriv v\u00e4nsterledet p\u00e5 gemensam n\u00e4mnare och f\u00f6renkla sedan med de trigonometriska identiterna f\u00f6r att .`,\r\n niva: \"C\",\r\n poang: [0,2,0],\r\n underkategori: [\"formler\"],\r\n kod: \"a17\"\r\n},\r\n{\r\n fraga: `Visa att f\u00f6ljande likhet st\u00e4mmer \\\\(\\\\sqrt{\\\\frac{1}{cos^2x}-1}=tanx\\\\)`,\r\n svar: `Skriv om v\u00e4nsterledet till gemensam n\u00e4mnare`,\r\n niva: \"A\",\r\n poang: [0,1,1],\r\n underkategori: [\"formler\"],\r\n kod: \"a18\"\r\n},\r\n{\r\n fraga: `Visa att f\u00f6ljande likhet st\u00e4mmer \\\\(sin^2x+cos^2x+sin(2\\\\pi)=1\\\\)`,\r\n svar: `Skriv om v\u00e4nsterledet med trigettan och v\u00e4rdet f\u00f6r \\\\(sin(2\\\\pi)=0\\\\)`,\r\n niva: \"E\",\r\n poang: [1,0,0],\r\n underkategori: [\"formler\"],\r\n kod: \"a20\"\r\n},\r\n{\r\n fraga: `Visa att f\u00f6ljande likhet st\u00e4mmer \\\\(cos^4x-sin^4x=cos2x\\\\)`,\r\n svar: `Anv\u00e4nd konjugatregeln f\u00f6r v\u00e4nsterledet och forts\u00e4tt f\u00f6renkla d\u00e4refter.`,\r\n niva: \"A\",\r\n poang: [0,1,1],\r\n underkategori: [\"formler\"],\r\n kod: \"a21\"\r\n},\r\n{\r\n fraga: `Visa att f\u00f6ljande likhet g\u00e4ller \\\\((sinx+cosx)^2-1=sin2x\\\\)`,\r\n svar: `Utveckla v\u00e4nsterledet med kvadreringsregeln och f\u00f6renkla sedan vidare.`,\r\n niva: \"C\",\r\n poang: [1,2,0],\r\n underkategori: [\"formler\"],\r\n kod: \"a22\"\r\n},\r\n{\r\n fraga: `Vi anger vinklar i radianer. Best\u00e4m om sin10 \u00e4r ett positivt tal eller negativt tal med en motivering. `,\r\n svar: `Det \u00e4r ett negativt tal eftersom \\\\(3\\\\pi\\\\) \u00e4r lite mindre \u00e4n 10 och \u00e4r lika med noll kommer det som g\u00e5r \u00f6ver \\\\(3\\\\pi\\\\) till \\\\(4\\\\pi\\\\) att vara negativt.`,\r\n niva: \"C\",\r\n poang: [0,1,0],\r\n underkategori: [\"formler\"],\r\n kod: \"a23\"\r\n},\r\n{\r\n fraga: `Best\u00e4m ett exakt v\u00e4rde f\u00f6r uttrycket \\\\(cos^2(75^{\\\\circ})-sin^2(255^{\\\\circ})\\\\)`,\r\n svar: `\\\\(\\\\frac{-\\\\sqrt3}{2}\\\\)`,\r\n niva: \"A\",\r\n poang: [0,1,1],\r\n underkategori: [\"formler\"],\r\n kod: \"a24\"\r\n},\r\n{\r\n fraga: `Best\u00e4m ett exakt v\u00e4rde p\u00e5 \\\\(tan15^{\\\\circ}\\\\)`,\r\n svar: `\\\\(\\\\frac{\\\\sqrt3-1}{sqrt3+1\\\\)`,\r\n niva: \"A\",\r\n poang: [0,0,2],\r\n underkategori: [\"formler\"],\r\n kod: \"a25\"\r\n},\r\n{\r\n fraga: `L\u00f6s ekvationen \\\\(cos^2x-\\\\frac{sin2x}{2sinx}=0\\\\)`,\r\n svar: `\\\\(x=\\\\pm \\\\frac{\\\\pi}{2}+2\\\\pi \\\\cdot n\\\\) eller \\\\(x= 2\\\\pi \\\\cdot n\\\\)`,\r\n niva: \"A\",\r\n poang: [0,1,1],\r\n underkategori: [\"ekvationer\"],\r\n kod: \"a26\"\r\n},\r\n{\r\n fraga: `Best\u00e4m ett exakt v\u00e4rde f\u00f6r f\u00f6ljande uttryck \\\\(sin(\\\\frac{\\\\pi}{10})+sin(-\\\\frac{\\\\pi}{10})+sin^2(0,21\\\\pi)+cos^2(-0,21\\\\pi)\\\\)`,\r\n svar: `1`,\r\n niva: \"C\",\r\n poang: [0,2,0],\r\n underkategori: [\"formler\"],\r\n kod: \"a27\"\r\n},\r\n{\r\n fraga: `Visa att ekvationen \\\\(sin(cosx)=1\\\\) saknar l\u00f6sningar.\\\\(x\\\\) ges i radianer `,\r\n svar: `Visa att eftersom \\\\(cosx\\\\) g\u00e5r mellan v\u00e4rdena -1 och 1 kommer likheten aldrig tillfredst\u00e4llas.`,\r\n niva: \"A\",\r\n poang: [0,1,1],\r\n underkategori: [\"ekvationer\"],\r\n kod: \"a28\"\r\n},\r\n{\r\n fraga: `<p>Visa att f\u00f6ljande uttryck \u00e4r konstant oavsett v\u00e4rde p\u00e5 \\\\(x\\\\) <\/p> \\\\(\\\\frac{(\\\\sqrt2cosx-1)(\\\\sqrt2cosx+1)}{cos2x}\\\\)`,\r\n svar: `Skriv om t\u00e4ljaren och visa att den g\u00e5r att skriva som cosinus f\u00f6r dubbla vinkeln.`,\r\n niva: \"C\",\r\n poang: [1,1,0],\r\n underkategori: [\"formler\"],\r\n kod: \"a29\"\r\n},\r\n{\r\n fraga: `L\u00f6s ekvationen \\\\(tan2x=\\\\sqrt{3}\\\\). Svara i radianer. `,\r\n svar: `\\\\(x=\\\\frac{\\\\pi}{6}+\\\\frac{\\\\pi}{2} \\\\cdot n\\\\)`,\r\n niva: \"C\",\r\n poang: [1,1,0],\r\n underkategori: [\"ekvationer\"],\r\n kod: \"a30\"\r\n},\r\n{\r\n fraga: `L\u00f6s ekvationen \\\\(sinxcosx=\\\\frac{1}{4}\\\\). Svara i radianer. `,\r\n svar: `\\\\(x=\\\\frac{\\\\pi}{12}+\\\\pi \\\\cdot n\\\\) eller \\\\(x=\\\\frac{5\\\\pi}{12}+\\\\pi \\\\cdot n\\\\)`,\r\n niva: \"C\",\r\n poang: [2,1,0],\r\n underkategori: [\"ekvationer\"],\r\n kod: \"a31\"\r\n},\r\n{\r\n fraga: `L\u00f6s ekvationen \\\\(cos3x=\\\\frac{1}{2}\\\\). Svara i radianer. `,\r\n svar: `\\\\(x=\\\\pm \\\\frac{\\\\pi}{9}+\\\\frac{2\\\\pi}{3} \\\\cdot n\\\\)`,\r\n niva: \"E\",\r\n poang: [3,0,0],\r\n underkategori: [\"ekvationer\"],\r\n kod: \"a32\"\r\n},\r\n{\r\n fraga: `L\u00f6s ekvationen \\\\(tanx=1\\\\). Svara i grader. `,\r\n svar: `\\\\(x= 30^{\\\\circ}+180^{\\\\circ} \\\\cdot n\\\\)`,\r\n niva: \"E\",\r\n poang: [2,0,0],\r\n underkategori: [\"ekvationer\"],\r\n kod: \"a33\"\r\n},\r\n{\r\n fraga: `L\u00f6s ekvationen \\\\(2sin2x=-1\\\\). Svara i radianer. `,\r\n svar: `\\\\(x=\\\\frac{7\\\\pi}{12}+\\\\pi \\\\cdot n\\\\) eller \\\\(x=\\\\frac{11\\\\pi}{12}+\\\\pi \\\\cdot n\\\\)`,\r\n niva: \"C\",\r\n poang: [2,1,0],\r\n underkategori: [\"ekvationer\"],\r\n kod: \"a34\"\r\n},\r\n{\r\n fraga: `L\u00f6s ekvationen \\\\(sinx=\\\\frac{-\\\\sqrt{3}}{2}\\\\). Svara i radianer. `,\r\n svar: `\\\\(x=\\\\frac{4\\\\pi}{3}+2\\\\pi \\\\cdot n\\\\) eller \\\\(x=\\\\frac{5\\\\pi}{3}+2\\\\pi \\\\cdot n\\\\)`,\r\n niva: \"E\",\r\n poang: [3,0,0],\r\n underkategori: [\"ekvationer\"],\r\n kod: \"a35\"\r\n},\r\n{\r\n fraga: `L\u00f6s ekvationen \\\\(cosx=-\\\\frac{\\\\sqrt{3}}{2}\\\\). Svara i radianer. `,\r\n svar: `\\\\(x=\\\\pm \\\\frac{5\\\\pi}{6}+2\\\\pi \\\\cdot n\\\\) `,\r\n niva: \"E\",\r\n poang: [2,0,0],\r\n underkategori: [\"ekvationer\"],\r\n kod: \"a36\"\r\n},\r\n{\r\n fraga: `L\u00f6s ekvationen \\\\(cos^2x-sin^2x+sin^2x+cos^2x=2\\\\). Svara i grader. `,\r\n svar: `\\\\(x=\\\\pi \\\\cdot n\\\\)`,\r\n niva: \"C\",\r\n poang: [2,1,0],\r\n underkategori: [\"ekvationer\"],\r\n kod: \"a37\"\r\n},\r\n{\r\n fraga: `L\u00f6s ekvationen \\\\(sinxcos(35^{\\\\circ})=1-cosxsin(35^{\\\\circ})\\\\). Svara i grader. `,\r\n svar: `\\\\(x=55^{\\\\circ} + 360^{\\\\circ}\\\\cdot n\\\\)`,\r\n niva: \"A\",\r\n poang: [0,1,1],\r\n underkategori: [\"ekvationer\"],\r\n kod: \"a38\"\r\n},\r\n{\r\n fraga: `Best\u00e4m ett exakt v\u00e4rde f\u00f6r f\u00f6ljande uttryck \\\\(cos(27^{\\\\circ})cos(3^{\\\\circ})-sin(27^{\\\\circ})sin(3^{\\\\circ})\\\\)`,\r\n svar: `\\\\(\\\\frac{\\\\sqrt{3}}{2}`,\r\n niva: \"C\",\r\n poang: [0,1,0],\r\n underkategori: [\"formler\"],\r\n kod: \"a39\"\r\n},\r\n{\r\n fraga: `Best\u00e4m ett exakt f\u00f6r \\\\(sin(15^{\\\\circ})\\\\)`,\r\n svar: `\\\\(\\\\frac{\\\\sqrt{6}-\\\\sqrt{2}}{4}\\\\)`,\r\n niva: \"A\",\r\n poang: [0,0,2],\r\n underkategori: [\"formler\"],\r\n kod: \"a40\"\r\n},\r\n{\r\n fraga: `Best\u00e4m ett exakt v\u00e4rde f\u00f6r f\u00f6ljande uttryck \\\\(cos^2(75^{\\\\circ})-sin^2(75^{\\\\circ})\\\\)`,\r\n svar: `\\\\(-\\\\frac{\\\\sqrt{3}}{2}\\\\)`,\r\n niva: \"C\",\r\n poang: [1,1,0],\r\n underkategori: [\"formler\"],\r\n kod: \"a41\"\r\n},\r\n{\r\n fraga: `Best\u00e4m ett exakt v\u00e4rde f\u00f6r f\u00f6ljande uttryck \\\\(sin(11\\\\pi)\\\\)`,\r\n svar: `0`,\r\n niva: \"C\",\r\n poang: [0,1,0],\r\n underkategori: [\"formler\"],\r\n kod: \"a42\"\r\n},\r\n{\r\n fraga: `Motivera varf\u00f6r \\\\(cos(30^{\\\\circ})=sin(120^{\\\\circ})\\\\)`,\r\n svar: `Anv\u00e4nd enhetscirkeln och motivera genom det.`,\r\n niva: \"E\",\r\n poang: [1,0,0],\r\n underkategori: [\"formler\"],\r\n kod: \"a43\"\r\n},\r\n{\r\n fraga: `L\u00f6s ekvationen \\\\(cos2x=cos(x+\\\\frac{\\\\pi}{3})\\\\). Svara i radianer.`,\r\n svar: `\\\\(x=\\\\frac{\\\\pi}{3}+2\\\\pi \\\\cdot n\\\\) eller \\\\(x=-\\\\frac{\\\\pi}{9}+\\\\frac{2\\\\pi}{3} \\\\cdot n\\\\)`,\r\n niva: \"A\",\r\n poang: [0,0,2],\r\n underkategori: [\"ekvationer\"],\r\n kod: \"a44\"\r\n},\r\n{\r\n fraga: `Visa att f\u00f6ljande likhet st\u00e4mmer \\\\(\\\\frac{2sinxcosx}{(1-\\\\sqrt{2}sinx)(1+\\\\sqrt{2}sinx)}=tan2x\\\\)`,\r\n svar: `Skriv om t\u00e4ljare och n\u00e4rmnare till sinus och cosinus f\u00f6r dubbla vinkeln.`,\r\n niva: \"C\",\r\n poang: [1,1,0],\r\n underkategori: [\"formler\"],\r\n kod: \"a45\"\r\n},\r\n{\r\n fraga: `Best\u00e4m ett exakt v\u00e4rde f\u00f6r uttrycket \\\\(sin^2(\\\\frac{\\\\pi}{13})+cos^2(\\\\frac{\\\\pi}{13})+sin(\\\\pi)\\\\)`,\r\n svar: `0`,\r\n niva: \"E\",\r\n poang: [2,0,0],\r\n underkategori: [\"formler\"],\r\n kod: \"a46\"\r\n},\r\n{\r\n fraga: `Best\u00e4m ett exakt v\u00e4rde f\u00f6r f\u00f6ljande uttryck \\\\(cos^2(347^{\\\\circ})+sin^2(167^{\\\\circ})\\\\)`,\r\n svar: `1, anv\u00e4nd trignomotriettan. `,\r\n niva: \"C\",\r\n poang: [0,2,0],\r\n underkategori: [\"formler\"],\r\n kod: \"a47\"\r\n},\r\n{\r\n fraga: `Best\u00e4m ett exakt v\u00e4rde f\u00f6r uttrycket \\\\(cos(345^{\\\\circ})sin(165^{\\\\circ})\\\\)`,\r\n svar: `\\\\(\\\\frac{1}{4}\\\\)`,\r\n niva: \"A\",\r\n poang: [0,0,2],\r\n underkategori: [\"formler\"],\r\n kod: \"a48\"\r\n},\r\n{\r\n fraga: `Best\u00e4m ett exakt v\u00e4rde f\u00f6r f\u00f6ljande uttryck \\\\(sin^2(124^{\\\\circ})+sin^2(34^{\\\\circ})\\\\)`,\r\n svar: `1`,\r\n niva: \"A\",\r\n poang: [0,0,1],\r\n underkategori: [\"formler\"],\r\n kod: \"a49\"\r\n},\r\n{\r\n fraga: `Visa att f\u00f6ljande likhet st\u00e4mmer \\\\(\\\\frac{sin^2x}{1-cos^2x}=1\\\\)`,\r\n svar: `Anv\u00e4nd trigettan p\u00e5 n\u00e4mnaren `,\r\n niva: \"E\",\r\n poang: [2,0,0],\r\n underkategori: [\"formler\"],\r\n kod: \"a50\"\r\n},\r\n{\r\n fraga: `Visa att f\u00f6ljande likhet st\u00e4mmer \\\\(\\\\frac{sin^2x}{1-sin^2x}=tan^2x\\\\)`,\r\n svar: `Anv\u00e4nd trigettan p\u00e5 n\u00e4mnaren `,\r\n niva: \"E\",\r\n poang: [2,0,0],\r\n underkategori: [\"formler\"],\r\n kod: \"a51\"\r\n},\r\n{\r\n fraga: `Visa att f\u00f6ljande likhet st\u00e4mmer \\\\(\\\\frac{1-cos2x}{sin2x}=tanx\\\\)`,\r\n svar: `Anv\u00e4nd sinus och cosinus f\u00f6r dubbla vinkeln.`,\r\n niva: \"C\",\r\n poang: [0,2,0],\r\n underkategori: [\"formler\"],\r\n kod: \"a52\"\r\n},\r\n{\r\n fraga: `Visa att f\u00f6ljande likhet st\u00e4mmer \\\\(\\\\frac{sinx}{1+cosx}=\\\\frac{1-cosx}{sinx}\\\\)`,\r\n svar: `Tips: F\u00f6rl\u00e4ng h\u00f6ger och v\u00e4nsterledet med respektives konjugat. `,\r\n niva: \"C\",\r\n poang: [0,2,0],\r\n underkategori: [\"formler\"],\r\n kod: \"a53\"\r\n},\r\n{\r\n fraga: `Visa att f\u00f6ljande likhet st\u00e4mmer \\\\(\\\\frac{sinx}{cosx}+\\\\frac{cosx}{sinx}=\\\\frac{1}{sinxcosx}\\\\)`,\r\n svar: `F\u00f6rl\u00e4ng VL och skriv p\u00e5 gemensamt br\u00e5k.`,\r\n niva: \"C\",\r\n poang: [1,1,0],\r\n underkategori: [\"formler\"],\r\n kod: \"a54\"\r\n},\r\n{\r\n fraga: `Visa att f\u00f6ljande likhet st\u00e4mmer \\\\(\\\\frac{1-cos2x}{1+cos2x}=tan^2x\\\\)`,\r\n svar: `Skriv om cosinus f\u00f6r dubbla vinkeln.`,\r\n niva: \"C\",\r\n poang: [0,2,0],\r\n underkategori: [\"formler\"],\r\n kod: \"a55\"\r\n},\r\n{\r\n fraga: `Visa att f\u00f6ljande likhet st\u00e4mmer \\\\(\\\\frac{sinx+cosx}{sinx-cosx}=\\\\frac{1+sin2x}{cos2x}\\\\)`,\r\n svar: `Anv\u00e4nd relevanta trigregler och skriv om VL med en f\u00f6rl\u00e4gning eller HL med dubbla vinkeln. `,\r\n niva: \"A\",\r\n poang: [0,1,1],\r\n underkategori: [\"formler\"],\r\n kod: \"a56\"\r\n},\r\n{\r\n fraga: `L\u00f6s ekvationen \\\\(sin2x=sinx\\\\). Svara med radianer. `,\r\n svar: `\\\\(x=0+2\\\\pi \\\\cdot n\\\\) eller \\\\(x=\\\\pi+2\\\\pi \\\\cdot n\\\\) eller \\\\(x=\\\\pm \\\\frac{\\\\pi}{3}+2\\\\pi \\\\cdot n\\\\)`,\r\n niva: \"A\",\r\n poang: [0,1,1],\r\n underkategori: [\"ekvationer\"],\r\n kod: \"a57\"\r\n},\r\n{\r\n fraga: `L\u00f6s ekvationen \\\\(\\\\frac{sin2x}{1+cos2x}=1\\\\) Svara med grader. `,\r\n svar: `\\\\(x=45^{\\\\circ}+180^{\\\\circ} \\\\cdot n\\\\)`,\r\n niva: \"C\",\r\n poang: [0,2,0],\r\n underkategori: [\"ekvationer\"],\r\n kod: \"a58\"\r\n},\r\n{\r\n fraga: `L\u00f6s ekvationen \\\\(sinx=\\\\sqrt{3}cosx\\\\). Svara med radianer. `,\r\n svar: `\\\\(x=\\\\frac{\\\\pi}{3}+\\\\pi \\\\cdot n\\\\)`,\r\n niva: \"C\",\r\n poang: [0,2,0],\r\n underkategori: [\"ekvationer\"],\r\n kod: \"a59\"\r\n},\r\n{\r\n fraga: `Skriv 6 grader i radianer.`,\r\n svar: `\\\\(\\\\frac{\\\\pi}{30}\\\\)`,\r\n niva: \"C\",\r\n poang: [0,1,0],\r\n underkategori: [\"formler\"],\r\n kod: \"a60\"\r\n},\r\n{\r\n fraga: `Skriv 15 grader i radianer.`,\r\n svar: `\\\\(\\\\frac{\\\\pi}{12}\\\\)`,\r\n niva: \"E\",\r\n poang: [1,0,0],\r\n underkategori: [\"formler\"],\r\n kod: \"a61\"\r\n},\r\n{\r\n fraga: `Skriv \\\\(\\\\frac{4\\\\pi}{15}\\\\) i grader.`,\r\n svar: `48 grader`,\r\n niva: \"C\",\r\n poang: [0,1,0],\r\n underkategori: [\"formler\"],\r\n kod: \"a62\"\r\n},\r\n{\r\n fraga: `Visa att \\\\(sin(3x)=3sinx-4sin^3x\\\\)`,\r\n svar: `Skriv om VL med hj\u00e4lp av additionsformler. Se l\u00f6sningsf\u00f6rslag `,\r\n niva: \"A\",\r\n poang: [0,0,2],\r\n underkategori: [\"formler\"],\r\n losningsforslag: `<img decoding=\"async\" src=\"https:\/\/mahifi.se\/wp-content\/uploads\/2026\/07\/Skarmavbild-2026-07-23-kl.-22.20.20.png\" alt=\"L\u00f6sningsf\u00f6rslag\">`,\r\n kod: \"a63\"\r\n},\r\n{\r\n fraga: `Visa att f\u00f6ljande likhet st\u00e4mmer \\\\(\\\\frac{1-cos2x+sin2x}{1+cos2x+sin2x}=tanx\\\\)`,\r\n svar: `Skriv om VL med hj\u00e4lp av sinus och cosinus f\u00f6r dubbla vinkeln.`,\r\n niva: \"C\",\r\n poang: [0,2,0],\r\n underkategori: [\"formler\"],\r\n kod: \"a64\"\r\n},\r\n{\r\n fraga: `<p> Under kursens g\u00e5ng har ni v\u00e4rt er formlerna f\u00f6r dubbla vinkeln f\u00f6r sinus och cosinus. Det finns \u00e4ven en formel f\u00f6r dubbla vinkeln f\u00f6r tangens. <\/p> <p> \\\\(tan2x=\\\\frac{2tanx}{1-tan^2x}\\\\) <\/p> Visa att likheten st\u00e4mmer. `,\r\n svar: `Skriv om VL med hj\u00e4lp av sinus och cosinus f\u00f6r dubbla vinkeln.`,\r\n niva: \"A\",\r\n poang: [0,0,2],\r\n underkategori: [\"formler\"],\r\n kod: \"a65\"\r\n}\r\n ],\r\n\r\n funktioner: [\r\n {\r\n fraga: `Best\u00e4m amplituden f\u00f6r funktionen \\\\(f(x)=4sin(2x)\\\\).`,\r\n svar: `Amplituden=4`,\r\n niva: \"E\",\r\n poang: [1,0,0],\r\n underkategori: [\"funktioner\"],\r\n kod: \"f1\"\r\n },\r\n {\r\n fraga: `Best\u00e4m perioden i radianer f\u00f6r f\u00f6ljande funktion \\\\(f(x)=3cos(2x)\\\\).`,\r\n svar: `Perioden=\\\\(\\\\pi\\\\)`,\r\n niva: \"E\",\r\n poang: [1,0,0],\r\n underkategori: [\"funktioner\"],\r\n kod: \"f2\"\r\n },\r\n {\r\n fraga: `Best\u00e4m \\\\(f(\\\\pi)\\\\) f\u00f6r f\u00f6ljande funktion \\\\(f(x)=5cos(2x)\\\\)`,\r\n svar: `\\\\(f(\\\\pi)=5\\\\)`,\r\n niva: \"E\",\r\n poang: [1,0,0],\r\n underkategori: [\"funktioner\"],\r\n kod: \"f3\"\r\n },\r\n {\r\n fraga: `L\u00f6s ekvationen \\\\(f(x)=2\\\\) f\u00f6r f\u00f6ljande funktion \\\\(f(x)=4cos(x)\\\\). Svara i radianer`,\r\n svar: `\\\\(x=\\\\pm \\\\frac{\\\\pi}{3}+2\\\\pi \\\\cdot n\\\\)`,\r\n niva: \"E\",\r\n poang: [3,0,0],\r\n underkategori: [\"funktioner\"],\r\n kod: \"f4\"\r\n },\r\n {\r\n fraga: `F\u00f6r vilket v\u00e4rde p\u00e5 \\\\(y\\\\) sk\u00e4r funktionen \\\\(f(x)=2sin(x+\\\\frac{\\\\pi}{6})\\\\) \\\\(y\\\\)-axeln`,\r\n svar: `1`,\r\n niva: \"E\",\r\n poang: [2,0,0],\r\n underkategori: [\"funktioner\"],\r\n kod: \"f5\"\r\n },\r\n {\r\n fraga: `Best\u00e4m v\u00e4rdem\u00e4ngden f\u00f6r funktionen \\\\(f(x)=acosx+a\\\\) uttryckt i \\\\(a\\\\)`,\r\n svar: `\\\\(0 \\\\le y \\\\le 2a\\\\) `,\r\n niva: \"C\",\r\n poang: [0,1,0],\r\n underkategori: [\"funktioner\"],\r\n kod: \"f6\"\r\n },\r\n {\r\n fraga: `Best\u00e4m det minsta v\u00e4rdet f\u00f6ljande funktion kan anta \\\\(f(x)=e^{2sinx}\\\\)`,\r\n svar: `Minst v\u00e4rdet \u00e4r \\\\(y=\\\\frac{1}{e^2}\\\\)`,\r\n niva: \"A\",\r\n poang: [0,0,1],\r\n underkategori: [\"funktioner\"],\r\n kod: \"f7\"\r\n },\r\n {\r\n fraga: `F\u00f6r en funktion p\u00e5 formen \\\\(f(x)=asinkx+b\\\\) vet du att dess st\u00f6rsta v\u00e4rde \u00e4r 10 och dess minsta v\u00e4rde \u00e4r 6. Funktionens perioden \u00e4r \\\\(4\\\\pi\\\\). Best\u00e4m konstanterna a, b och k`,\r\n svar: `\\\\(a=2, b=8, k=\\\\frac{1}{2}\\\\)`,\r\n niva: \"C\",\r\n poang: [2,1,0],\r\n underkategori: [\"problemfunk\"],\r\n kod: \"f8\"\r\n },\r\n {\r\n fraga: `Best\u00e4m det st\u00f6rsta v\u00e4rdet f\u00f6r funktionen \\\\(f(x)=sinx+cosx\\\\)`,\r\n svar: `St\u00f6rsta v\u00e4rdet \u00e4r \\\\(\\\\sqrt2\\\\)`,\r\n niva: \"C\",\r\n poang: [0,2,0],\r\n underkategori: [\"funktioner\"],\r\n kod: \"f9\"\r\n },\r\n {\r\n fraga: `Best\u00e4m det st\u00f6rsta v\u00e4rdet f\u00f6r funktionen \\\\(f(x)=sin^{100}x+cos^{100}x\\\\)`,\r\n svar: `St\u00f6rsta v\u00e4rdet \u00e4r 1`,\r\n niva: \"A\",\r\n poang: [0,0,2],\r\n underkategori: [\"funktioner\"],\r\n kod: \"f10\"\r\n },\r\n {\r\n fraga: `Best\u00e4m perioden f\u00f6r f\u00f6ljande funktion \\\\(f(x)=cos(3x+4\\\\pi)\\\\). Svara i radianer`,\r\n svar: `Perioden \u00e4r \\\\(\\\\frac{2 \\\\pi}{3}\\\\)`,\r\n niva: \"C\",\r\n poang: [0,1,0],\r\n underkategori: [\"funktioner\"],\r\n kod: \"f11\"\r\n },\r\n {\r\n fraga: `<p> Funktionen \\\\(f(x)=cosx\\\\) g\u00e5r att definieras som \\\\(f(x)=\\\\sqrt{\\\\frac{1}{2}+\\\\frac{1}{2}cos2x}\\\\) i till exempel intervallet \\\\(0 \\\\le x \\\\le \\\\frac{\\\\pi}{2}\\\\) <\/p> <p> a) Visa att \\\\(cosx=\\\\sqrt{\\\\frac{1}{2}+\\\\frac{1}{2}cos2x}\\\\) <\/p> b) Ange ett annat intervall d\u00e4r likheten st\u00e4mmer `,\r\n svar: `a) Skriv om h\u00f6gerledet! b) Till exempel \\\\(\\\\frac{3\\\\pi}{2} \\\\le x \\\\le 2\\\\pi\\\\)`,\r\n niva: \"A\",\r\n poang: [0,2,1],\r\n underkategori: [\"funktioner\"],\r\n kod: \"f12\"\r\n },\r\n {\r\n fraga: `Visa att f\u00f6ljande likhet aldrig kommer g\u00e4lla \\\\(\\\\frac{3cos(2,4x+\\\\frac{\\\\pi}{13})-4}{sin^2x}=0\\\\)`,\r\n svar: `Visa att t\u00e4ljaren aldrig blir noll. Det blir den aldrig eftersom t\u00e4ljaruttrycket aldrig sk\u00e4r x-axeln`,\r\n niva: \"C\",\r\n poang: [0,2,0],\r\n underkategori: [\"funktioner\"],\r\n kod: \"f13\"\r\n },\r\n {\r\n fraga: `<p> F\u00f6r en funktion p\u00e5 formen \\\\(f(x)=sin(Ax+B)+C\\\\) vet du f\u00f6ljande <\/p>\r\n <p> *\\\\(f(x)\\\\) har perioden \\\\(\\\\pi\\\\) <\/p> <p> *\\\\(f(x)\\\\) har sin f\u00f6rsta maximipunkt d\u00e5 \\\\(0 < x\\\\) infaller i \\\\(x=\\\\frac{\\\\pi}{2}\\\\) <\/p> <p> *\\\\(f(\\\\pi)=4\\\\) <\/p> Best\u00e4m konstanterna \\\\(A, B\\\\) och \\\\(C\\\\)`,\r\n svar: `\\\\(A=2, B=-\\\\frac{\\\\pi}{2}\\\\) och \\\\(C=5\\\\)`,\r\n niva: \"A\",\r\n poang: [0,0,2],\r\n underkategori: [\"problemfunk\"],\r\n kod: \"f14\"\r\n },\r\n {\r\n fraga: `Best\u00e4m hur m\u00e5nga g\u00e5nger funktionen \\\\(f(x)=cos2x\\\\) sk\u00e4r \\\\(x\\\\)-axeln i intervallet \\\\(0 \\\\le x \\\\le 2\\\\pi\\\\)`,\r\n svar: `4 g\u00e5nger`,\r\n niva: \"C\",\r\n poang: [1,1,0],\r\n underkategori: [\"funktioner\"],\r\n kod: \"f15\"\r\n },\r\n {\r\n fraga: `Vi definierar funktionen \\\\(f(x)=asinx+bcosx\\\\) och \\\\(g(x)=asinx-bcosx\\\\). Du vet att \\\\(f(\\\\frac{\\\\pi}{2})=1\\\\) samt att \\\\(g(0)=2\\\\). Best\u00e4m \\\\(a\\\\) och \\\\(b\\\\)`,\r\n svar: `\\\\(a=1, b=-2\\\\)`,\r\n niva: \"C\",\r\n poang: [1,1,0],\r\n underkategori: [\"funktioner\"],\r\n kod: \"f16\"\r\n },\r\n {\r\n fraga: `F\u00f6r en funktion p\u00e5 formen \\\\(f(x)=sin(x+A)\\\\) vet du att den har en maximipunkt i \\\\(x=\\\\frac{\\\\pi}{4}\\\\). Best\u00e4m ett exempel p\u00e5 \\\\(A\\\\)`,\r\n svar: `Till exempel \\\\(A=\\\\frac{\\\\pi}{4}\\\\)`,\r\n niva: \"C\",\r\n poang: [0,1,0],\r\n underkategori: [\"funktioner\"],\r\n kod: \"f17\"\r\n },\r\n {\r\n fraga: `Best\u00e4m det st\u00f6rsta v\u00e4rdet f\u00f6r funktionen \\\\(f(x)=(sin^2x+cos^2x)^{100}+2cos2x\\\\)`,\r\n svar: `St\u00f6rsta v\u00e4rdet \u00e4r \\\\(y=3\\\\)`,\r\n niva: \"C\",\r\n poang: [0,1,0],\r\n underkategori: [\"funktioner\"],\r\n kod: \"f18\"\r\n },\r\n {\r\n fraga: `Best\u00e4m v\u00e4rdem\u00e4ngden f\u00f6r funktionen \\\\(f(x)=3cos3x+2\\\\)`,\r\n svar: `\\\\(-1 \\\\le y \\\\le 5\\\\)`,\r\n niva: \"C\",\r\n poang: [0,1,0],\r\n underkategori: [\"funktioner\"],\r\n kod: \"f19\"\r\n },\r\n {\r\n fraga: `Best\u00e4m perioden f\u00f6r funktionen \\\\(f(x)=sin(\\\\pi x)\\\\), \\\\(x\\\\) \u00e4r skrivet i radianer`,\r\n svar: `Perioden \u00e4r 2`,\r\n niva: \"E\",\r\n poang: [1,0,0],\r\n underkategori: [\"funktioner\"],\r\n kod: \"f20\"\r\n },\r\n {\r\n fraga: `Best\u00e4m f\u00f6ljande funktion p\u00e5 formen \\\\(f(x)=Asin(x+B)+C\\\\) d\u00e4r \\\\(A, B\\\\) och \\\\(C\\\\) \u00e4r konstanter. Svara i radianer. <img decoding=\"async\" src=\"https:\/\/mahifi.se\/wp-content\/uploads\/2025\/11\/Skarmbild-2025-11-12-155132.png\" alt=\"L\u00f6sningsf\u00f6rslag\">`,\r\n svar: `\\\\(f(x)=2sin(x+\\\\pi)+1\\\\)`,\r\n niva: \"C\",\r\n poang: [2,1,0],\r\n underkategori: [\"grafer\"],\r\n kod: \"f21\"\r\n },\r\n {\r\n fraga: `Best\u00e4m f\u00f6ljande funktion p\u00e5 formen \\\\(f(x)=Acos(kx)+B\\\\) d\u00e4r \\\\(A, k\\\\) och \\\\(B\\\\) \u00e4r konstanter. <img decoding=\"async\" src=\"https:\/\/mahifi.se\/wp-content\/uploads\/2025\/11\/Skarmbild-2025-11-12-155157.png\" alt=\"L\u00f6sningsf\u00f6rslag\">`,\r\n svar: `\\\\(f(x)=2cos(\\\\frac{x}{3})-1\\\\)`,\r\n niva: \"C\",\r\n poang: [2,1,0],\r\n underkategori: [\"grafer\"],\r\n kod: \"f22\"\r\n },\r\n {\r\n fraga: `Vi definierar funktionen \\\\(f(x)=3a \\\\cdot sinx +4a \\\\cdot cosx\\\\) Om du vet att funktionen har sitt st\u00f6rsta v\u00e4rde \\\\(y=15\\\\) best\u00e4m konstanten \\\\(a\\\\)`,\r\n svar: `\\\\(a=3\\\\)`,\r\n niva: \"C\",\r\n poang: [0,2,0],\r\n underkategori: [\"funktioner\"],\r\n kod: \"f23\"\r\n },\r\n {\r\n fraga: `Best\u00e4m f\u00f6ljande funktion p\u00e5 formen \\\\(f(x)=Asin(k(x-v))+B\\\\) d\u00e4r \\\\(A, k, v\\\\) och \\\\(B\\\\) \u00e4r konstanter. <img decoding=\"async\" src=\"https:\/\/mahifi.se\/wp-content\/uploads\/2025\/11\/Skarmbild-2025-11-12-155218.png\" alt=\"L\u00f6sningsf\u00f6rslag\">`,\r\n svar: `\\\\(f(x)=2cos(3(x-20^{\\\\circ}))+1\\\\)`,\r\n niva: \"C\",\r\n poang: [2,1,0],\r\n underkategori: [\"grafer\"],\r\n kod: \"f24\"\r\n },\r\n {\r\n fraga: `Best\u00e4m koordinaterna f\u00f6r samtliga extrempunkter f\u00f6r funktionen \\\\(f(x)=-sin(\\\\frac{x}{2})+1\\\\) i intervallet \\\\(0 < x < 4\\\\pi\\\\)`,\r\n svar: `\\\\((\\\\pi, 0)\\\\) och \\\\((\\\\frac{3\\\\pi}{2}, 2)\\\\)`,\r\n niva: \"C\",\r\n poang: [1,2,0],\r\n underkategori: [\"funktioner\"],\r\n kod: \"f25\"\r\n },\r\n {\r\n fraga: `<p>Temperaturen i en pool varierar under dygnets timmar. Den h\u00f6gsta temperaturen f\u00f6r\r\npoolen \u00e4r 32 \u00b0C och den l\u00e4gsta temperaturen \u00e4r 24 \u00b0C, vilket den \u00e4r klockan 04.00 p\u00e5\r\nmorgonen. Temperaturen under ett dygn kan beskrivas som en sinusfunktion p\u00e5\r\nformen \\\\(T(x)=Asin(B(x+C))+D\\\\). D\u00e4r \ud835\udc65 \u00e4r antalet timmar fr\u00e5n 00.00. <\/p>\r\nBest\u00e4m konstanterna \\\\(A, B, C\\\\) och \\\\(D\\\\)\r\n`,\r\n svar: `\\\\(A=4, B=\\\\frac{\\\\pi}{12}, C=-10, D=28\\\\)`,\r\n niva: \"A\",\r\n poang: [1,1,1],\r\n underkategori: [\"problemfunk\"],\r\n kod: \"f26\",\r\n digital: \"true\"\r\n },\r\n {\r\n fraga: `<p>Funktionen \\\\(d(t)=10-8sin(\\\\frac{\\\\pi t}{6})\\\\) beskriver havsdjupet i meter vid en position p\u00e5\r\nstranden i kuststaden Playa Jide. \\\\(t\\\\) st\u00e5r f\u00f6r tiden i timmar fr\u00e5n 07.00 <\/p> <p> a) Best\u00e4m och tolka \\\\(d(4)\\\\) <\/p> <p> b) Best\u00e4m och tolka \\\\(d(t)=10\\\\)`,\r\n svar: `a) Kl 11.00 \u00e4r havsdjupet ungef\u00e4r 3 meter. b) kl 13.00 \u00e4r havsdjupet 10 meter och sedan \u00e4r det det var 12 timme.`,\r\n niva: \"C\",\r\n poang: [1,1,0],\r\n underkategori: [\"problemfunk\"],\r\n kod: \"f27\",\r\n digital: \"true\"\r\n },\r\n {\r\n fraga: `<p> H\u00f6jden f\u00f6r en vagn i ett parishjul g\u00e5r att beskriva med funktionen \\\\(f(x)=80sin(\\\\frac{\\\\pi x}{25}-\\\\frac{9 \\\\pi}{25})+84\\\\) d\u00e4r \\\\(x\\\\) \u00e4r antalet minuter fr\u00e5n start. Best\u00e4m f\u00f6ljande <\/p> <p> a) Hur h\u00f6gt g\u00e5r parishjulet som h\u00f6gst? <\/p> <p>\r\nb) Hur l\u00e5ng tid tar det f\u00f6r parishjulet att snurra ett varv? <\/p>`,\r\n svar: `a) 164 meter b) 50 minuter`,\r\n niva: \"C\",\r\n poang: [1,1,0],\r\n underkategori: [\"problemfunk\"],\r\n kod: \"f28\",\r\n digital: \"true\"\r\n },\r\n {\r\n fraga: `<p> I ett naturomr\u00e5de g\u00e5r antalet lejon i tusental att beskriva med funktionen \\\\(L(t)=9+6sin(\\\\frac{\\\\pi t}{4})\\\\) och antalet gnuer i tusental g\u00e5r att beskriva med funktionen \\\\(G(t)=27+19cos(\\\\frac{\\\\pi t}{4})\\\\) d\u00e4r \\\\(t\\\\) \u00e4r antalet \u00e5r fr\u00e5n 2010.<\/p> Best\u00e4m efter hur m\u00e5nga \u00e5r f\u00f6rh\u00e5llandet mellan lejon och gnuer \u00e4r som st\u00f6rst om vi ser 10 \u00e5r fram\u00f6ver. `,\r\n svar: `3,64 \u00e5r`,\r\n niva: \"C\",\r\n poang: [0,2,0],\r\n underkategori: [\"problemfunk\"],\r\n kod: \"f29\",\r\n digital: \"true\"\r\n },\r\n {\r\n fraga: `<p> M\u00e5nens synlighet i procent \\\\(P(t)\\\\) fr\u00e5n nym\u00e5ne (tillst\u00e5ndet d\u00e4r m\u00e5nen inte syns\r\n\u00f6verhuvudtaget) g\u00e5r att modellera med funktionen <\/p> <p> \\\\(P(t)=50-50cos(\\\\frac{2\\\\pi t}{29,53})\\\\) <\/p> <p> a) Efter hur m\u00e5nga dagar fr\u00e5n nym\u00e5ne syns hela m\u00e5nen? Svara med\r\nen decimal. Endast svar kr\u00e4vs. <\/p> b) Under hur m\u00e5nga dagar syns m\u00e5nen mer \u00e4n 80%? Svara med en decimal. <img decoding=\"async\" src=\"https:\/\/mahifi.se\/wp-content\/uploads\/2026\/07\/Skarmavbild-2026-07-21-kl.-17.35.46.png\" alt=\"L\u00f6sningsf\u00f6rslag\">\r\n`,\r\n svar: `a) 14,8 dagar b) 8,7 dagar`,\r\n niva: \"C\",\r\n poang: [1,1,0],\r\n underkategori: [\"problemfunk\"],\r\n kod: \"f30\",\r\n digital: \"true\"\r\n },\r\n {\r\n fraga: `Nedan ser du en funktion p\u00e5 formen \\\\(f(x)=2sin(kx+v)+1\\\\) Best\u00e4m konstanterna \\\\(k\\\\) och \\\\(v\\\\). <img decoding=\"async\" src=\"https:\/\/mahifi.se\/wp-content\/uploads\/2026\/07\/Skarmavbild-2026-07-21-kl.-17.46.29.png\" alt=\"L\u00f6sningsf\u00f6rslag\">\r\n`,\r\n svar: `\\\\(k=\\\\pi\\\\) och \\\\(v=-\\\\frac{\\\\pi}{2}\\\\)`,\r\n niva: \"A\",\r\n poang: [0,1,1],\r\n underkategori: [\"funktioner\"],\r\n kod: \"f31\"\r\n },\r\n {\r\n fraga: `<p> En fl\u00e4kts varvtal (i varv per minut, rpm) varierar harmoniskt \u00f6ver tid enligt modellen: <\/p> <p>\r\n\r\n\\\\(f(t) = A \\\\cdot sin(B \\\\cdot t) + D\\\\) <\/p> <p> Vid \\\\(t=2\\\\) har fl\u00e4kten 1200 varv\/minut <\/p> <p> Vid \\\\(t=6\\\\) har fl\u00e4kten 400 varv\/minut vilket \u00e4r det l\u00e4gsta varvantalet. <\/p> Best\u00e4m konstanterna \\\\(A, B\\\\) och \\\\(D\\\\).\r\n`,\r\n svar: `\\\\(A=400, B=\\\\frac{\\\\pi}{4}\\\\) och \\\\(D=800\\\\)`,\r\n niva: \"A\",\r\n poang: [0,1,1],\r\n underkategori: [\"problemfunk\"],\r\n kod: \"f32\"\r\n }\r\n ],\r\n\r\n derivata: [\r\n {\r\n fraga: `Derivera funktionen \\\\(f(x)=sin2x\\\\).`,\r\n svar: `\\\\(f'(x)=2cos2x\\\\)`,\r\n niva: \"E\",\r\n poang: [1,0,0],\r\n underkategori: [\"deriveringsregler\"],\r\n kod: \"d1\"\r\n },\r\n {\r\n fraga: `Derivera funktionen \\\\(f(x)=x\\\\cdot e^x\\\\).`,\r\n svar: `\\\\(f'(x)=e^x+xe^x\\\\)`,\r\n niva: \"E\",\r\n poang: [1,0,0],\r\n underkategori: [\"deriveringsregler\"],\r\n kod: \"d2\"\r\n },\r\n {\r\n fraga: `Derivera funktionen \\\\(f(x)=x^2\\\\cdot e^{2x}\\\\).`,\r\n svar: `\\\\(f'(x)=2xe^{2x}+2x^2e^{2x}\\\\)`,\r\n niva: \"E\",\r\n poang: [2,0,0],\r\n underkategori: [\"deriveringsregler\"],\r\n kod: \"d3\"\r\n },\r\n {\r\n fraga: `Derivera funktionen \\\\(f(x)=2cos3x\\\\).`,\r\n svar: `\\\\(f'(x)=-6sin3x\\\\)`,\r\n niva: \"E\",\r\n poang: [1,0,0],\r\n underkategori: [\"deriveringsregler\"],\r\n kod: \"d4\"\r\n },\r\n {\r\n fraga: `Derivera funktionen \\\\(f(x)=lnx\\\\).`,\r\n svar: `\\\\(f'(x)=\\\\frac{1}{x}\\\\)`,\r\n niva: \"E\",\r\n poang: [1,0,0],\r\n underkategori: [\"deriveringsregler\"],\r\n kod: \"d5\"\r\n },\r\n {\r\n fraga: `Derivera funktionen \\\\(f(x)=cos^2x\\\\).`,\r\n svar: `\\\\(f'(x)=-2cosxsinx\\\\)`,\r\n niva: \"C\",\r\n poang: [0,1,0],\r\n underkategori: [\"deriveringsregler\"],\r\n kod: \"d6\"\r\n },\r\n {\r\n fraga: `Derivera funktionen \\\\(f(x)=sin^2x\\\\).`,\r\n svar: `\\\\(f'(x)=2cosxsinx\\\\)`,\r\n niva: \"C\",\r\n poang: [0,1,0],\r\n underkategori: [\"deriveringsregler\"],\r\n kod: \"d7\"\r\n },\r\n {\r\n fraga: `Derivera funktionen \\\\(f(x)=\\\\sqrt{2x+1}\\\\).`,\r\n svar: `\\\\(f'(x)=\\\\frac{1}{\\\\sqrt{2x+1}}\\\\)`,\r\n niva: \"C\",\r\n poang: [0,1,0],\r\n underkategori: [\"deriveringsregler\"],\r\n kod: \"d8\"\r\n },\r\n {\r\n fraga: `Derivera funktionen \\\\(f(x)=x\\\\cdot sinx\\\\).`,\r\n svar: `\\\\(f'(x)=sinx+x\\\\cdot cosx\\\\)`,\r\n niva: \"E\",\r\n poang: [2,0,0],\r\n underkategori: [\"deriveringsregler\"],\r\n kod: \"d11\"\r\n },\r\n {\r\n fraga: `Visa att derivatafunktionen f\u00f6r funktionen \\\\(f(x)=lnax\\\\) blir samma oavsett v\u00e4rde p\u00e5 \\\\(a\\\\).`,\r\n svar: `Visa att oavsett v\u00e4rde p\u00e5 \\\\(a\\\\) blir derivatan alltid \\\\(f'(x)=\\\\frac{1}{a}\\\\)`,\r\n niva: \"C\",\r\n poang: [1,1,0],\r\n underkategori: [\"deriveringsregler\"],\r\n kod: \"d12\"\r\n },\r\n {\r\n fraga: `Derivera funktionen \\\\(f(x)=\\\\frac{1}{sinx}\\\\).`,\r\n svar: `\\\\(f'(x)=\\\\frac{-cosx}{sin^2x}\\\\)`,\r\n niva: \"C\",\r\n poang: [0,1,0],\r\n underkategori: [\"deriveringsregler\"],\r\n kod: \"d13\"\r\n },\r\n {\r\n fraga: `Visa att derivatan till \\\\(f(x)=tanx\\\\) \u00e4r \\\\(f(x)=\\\\frac{1}{cos^2x}\\\\).`,\r\n svar: `\\\\(f'(x)=\\\\frac{-cosx}{sin^2x}\\\\)`,\r\n niva: \"C\",\r\n poang: [0,2,0],\r\n underkategori: [\"deriveringsregler\"],\r\n kod: \"d14\"\r\n },\r\n {\r\n fraga: `Funktionen \\\\(f(x)=x \\\\cdot e^{-x}\\\\) har en extrempunkt. Best\u00e4m \\\\(x\\\\)-koordinaten till den punkten.`,\r\n svar: `\\\\(x=1\\\\)`,\r\n niva: \"C\",\r\n poang: [1,1,0],\r\n underkategori: [\"derivatagraf\"],\r\n kod: \"d15\"\r\n },\r\n {\r\n fraga: `Vi definierar funktionen \\\\(f(x)=e^{h(x)}\\\\) d\u00e4r \\\\(h(x)\\\\) \u00e4r definierad f\u00f6r alla \\\\(x\\\\). Visa att om \\\\(h(x)\\\\) har en extrempunkt eller terrasspunkt \u00e4ven \\\\(f(x)\\\\) det.`,\r\n svar: `Visa att n\u00e4r \\\\(h'(x)=0\\\\) \u00e4r \\\\(f'(x)=0\\\\).` ,\r\n niva: \"A\",\r\n poang: [0,1,1],\r\n underkategori: [\"problemderivata\"],\r\n kod: \"d16\"\r\n },\r\n {\r\n fraga: `Under grafen till \\\\(f(x)=e^{kx}\\\\) placeras en rektangel som avgr\u00e4nsas av de positiva koordinataxlarna. F\u00f6r vilket v\u00e4rde p\u00e5 konstanten k kommer den rektangel\r\nsom har st\u00f6rsta arean att vara en kvadrat? <img decoding=\"async\" src=\"https:\/\/mahifi.se\/wp-content\/uploads\/2025\/11\/Skarmbild-2025-11-15-143121.png\" alt=\"L\u00f6sningsf\u00f6rslag\">`,\r\n svar: `\\\\(k=-e\\\\)` ,\r\n niva: \"A\",\r\n poang: [0,0,3],\r\n underkategori: [\"problemderivata\"],\r\n kod: \"d18\"\r\n },\r\n {\r\n fraga: `En isskulptur i form av ett klot sm\u00e4lter p\u00e5 ett s\u00e5dant s\u00e4tt att den hela tiden \u00e4r ett\r\nklot. Volymen p\u00e5 klotet minskar med en konstant hastighet p\u00e5 2\u03c0 kubikmeter\r\nper timme. Med vilken hastighet f\u00f6r\u00e4ndras arean p\u00e5 klotet i det \u00f6gonblick d\u00e5\r\nskulpturens radie \u00e4r 5 meter?`,\r\n svar: `\\\\(\\\\frac{-4\\\\pi}{5}\\\\)` ,\r\n niva: \"A\",\r\n poang: [0,0,3],\r\n underkategori: [\"problemderivata\"],\r\n kod: \"d19\",\r\n losningsforslag: `<img decoding=\"async\" src=\"https:\/\/mahifi.se\/wp-content\/uploads\/2025\/11\/Skarmbild-2025-11-15-143451.png\" alt=\"L\u00f6sningsf\u00f6rslag\">`,\r\n digital: \"true\" \r\n },\r\n {\r\n fraga: `Nedan ser du grafen till funktionen \\\\(f(x)=sin4x\\\\). Under grafen till funktionen\r\nplacerar vi en rektangel vars omkrets beror p\u00e5 var man s\u00e4tter punkten \ud835\udc34. Vid vilken\r\npunkt \ud835\udc34 blir omkretsen maximal? <img decoding=\"async\" src=\"https:\/\/mahifi.se\/wp-content\/uploads\/2025\/11\/Skarmbild-2025-11-15-144046.png\" alt=\"L\u00f6sningsf\u00f6rslag\">`,\r\n svar: `\\\\(x=\\\\frac{\\\\pi}{12}\\\\)` ,\r\n niva: \"A\",\r\n poang: [0,0,2],\r\n underkategori: [\"problemderivata\"],\r\n kod: \"d20\",\r\n digital: `true` \r\n },\r\n {\r\n fraga: `Vi definierar funktionen \\\\(h(x)=ln(f(x))\\\\). Nedan ser du grafen till \\\\(f(x)\\\\). Best\u00e4m f\u00f6r vilket\/vilka \\\\(x\\\\) \\\\(h'(x)\\\\) \u00e4r odefinierat om \\\\(f(x)\\\\) \u00e4r definierad \\\\(0 \\\\le x \\\\le 5\\\\) <img decoding=\"async\" src=\"https:\/\/mahifi.se\/wp-content\/uploads\/2025\/11\/Skarmbild-2025-11-15-144646.png\" alt=\"L\u00f6sningsf\u00f6rslag\">`,\r\n svar: `\\\\(x_1=0, x_2=2\\\\)` ,\r\n niva: \"A\",\r\n poang: [0,1,1],\r\n underkategori: [\"problemderivata\"],\r\n kod: \"d22\" \r\n },\r\n {\r\n fraga: `Joakim menar att funktionen \\\\(f(x)=\\\\frac{x}{e^x}\\\\) saknar extrempunkter. Unders\u00f6k om han har r\u00e4tt. `,\r\n svar: `Joakim har fel. Funktionen har en extrempunkt i \\\\(x=1\\\\)` ,\r\n niva: \"C\",\r\n poang: [1,2,0],\r\n underkategori: [\"derivatagraf\"],\r\n kod: \"d23\" \r\n },\r\n {\r\n fraga: `Visa att de tangenteran som tangerar funktionen \\\\(f(x)=sinx\\\\) i dess f\u00f6rsta positiva nollst\u00e4llen \u00e4r vinkelr\u00e4ta.`,\r\n svar: `Visa att \\\\(k\\\\)-v\u00e4rdet f\u00f6r tangenterna multiplicerade blir \\\\(-1\\\\)` ,\r\n niva: \"C\",\r\n poang: [1,1,0],\r\n underkategori: [\"derivatagraf\"],\r\n kod: \"d24\" \r\n },\r\n {\r\n fraga: `Derivera funktionen \\\\(f(x)=x^x\\\\) med hj\u00e4lp av ditt digitala hj\u00e4lpmedel.`,\r\n svar: `\\\\(f'(x)=x^x\\\\cdot lnx+x^x\\\\)` ,\r\n niva: \"E\",\r\n poang: [1,0,0],\r\n underkategori: [\"deriveringsregler\"],\r\n kod: \"d25\",\r\n digital: \"true\"\r\n },\r\n {\r\n fraga: `Derivera funktionen \\\\(f(x)=e^{x^2}\\\\) med hj\u00e4lp av ditt digitala hj\u00e4lpmedel.`,\r\n svar: `\\\\(f(x)=2xe^{x^2}\\\\)` ,\r\n niva: \"E\",\r\n poang: [1,0,0],\r\n underkategori: [\"deriveringsregler\"],\r\n kod: \"d26\",\r\n digital: \"true\"\r\n },\r\n {\r\n fraga: `Best\u00e4m \\\\(f'(1)\\\\) f\u00f6r funktionen \\\\(f(x)=cos^3x\\\\) med hj\u00e4lp av ditt digitala hj\u00e4lpmedel. Svara med tv\u00e5 decimaler.`,\r\n svar: `\\\\(f'(1)=-0,74\\\\)` ,\r\n niva: \"E\",\r\n poang: [1,0,0],\r\n underkategori: [\"deriveringsregler\"],\r\n kod: \"d27\",\r\n digital: \"true\"\r\n },\r\n {\r\n fraga: `Best\u00e4m \\\\(f'(2)\\\\) f\u00f6r funktionen \\\\(f(x)=ln(sinx)\\\\) med hj\u00e4lp av ditt digitala hj\u00e4lpmedel. Svara med tv\u00e5 decimaler.`,\r\n svar: `\\\\(f'(1)=-0,46\\\\)` ,\r\n niva: \"E\",\r\n poang: [1,0,0],\r\n underkategori: [\"deriveringsregler\"],\r\n kod: \"d28\",\r\n digital: \"true\"\r\n },\r\n {\r\n fraga: `Derivera f\u00f6ljande funktion \\\\(f(x)=(2+x^5)^{10}\\\\)`,\r\n svar: `\\\\(f'(x)=50x^4(2+x^5)^9\\\\)` ,\r\n niva: \"C\",\r\n poang: [0,1,0],\r\n underkategori: [\"deriveringsregler\"],\r\n kod: \"d29\"\r\n },\r\n {\r\n fraga: `Best\u00e4m ekvationen f\u00f6r tangenten som tangerar funktionen \\\\(f(x)=\\\\sqrt{3x-2}\\\\) i \\\\(x=1\\\\)`,\r\n svar: `\\\\(y=\\\\frac{3}{2}x-\\\\frac{1}{2}\\\\)` ,\r\n niva: \"C\",\r\n poang: [1,2,0],\r\n losningsforslag: `<img decoding=\"async\" src=\"https:\/\/mahifi.se\/wp-content\/uploads\/2025\/11\/Skarmbild-2025-11-15-151550.png\" alt=\"L\u00f6sningsf\u00f6rslag\">`,\r\n underkategori: [\"derivatagraf\"],\r\n kod: \"d30\"\r\n },\r\n {\r\n fraga: `Visa att \\\\(f(x)=ln(ax)+x\\\\) har en extrempunkt i \\\\(x=1\\\\) oavsett v\u00e4rdet p\u00e5 \\\\(a\\\\)`,\r\n svar: `Visa att derivatafunktionen inte beror p\u00e5 \\\\(a\\\\) och visa sedan att extrempunkten \u00e4r i \\\\(x=1\\\\)` ,\r\n niva: \"C\",\r\n poang: [0,2,0],\r\n underkategori: [\"derivatagraf\"],\r\n kod: \"d31\"\r\n },\r\n {\r\n fraga: `<p>F\u00f6r en funktion \\\\(h(x)\\\\) vet du att <\/p> <p>\\\\(h(x)>0\\\\) samt \\\\(h'(0)>0\\\\) f\u00f6r alla \\\\(x\\\\) inom definitionsm\u00e4ngden. <\/p> Visa att \\\\(f(x)=\\\\frac{1}{(h(x))^2}\\\\) \u00e4r avtagande f\u00f6r alla \\\\(x\\\\) inom definitionsm\u00e4ngden.`,\r\n svar: `Visa att derivatafunktionen inte beror p\u00e5 \\\\(a\\\\) och visa sedan att extrempunkten \u00e4r i \\\\(x=1\\\\)` ,\r\n niva: \"A\",\r\n poang: [0,0,2],\r\n losningsforslag: `<img decoding=\"async\" src=\"https:\/\/mahifi.se\/wp-content\/uploads\/2025\/11\/Skarmbild-2025-11-15-152916.png\" alt=\"L\u00f6sningsf\u00f6rslag\">`,\r\n underkategori: [\"problemderivata\"],\r\n kod: \"d32\"\r\n },\r\n {\r\n fraga: `Best\u00e4m koordinaterna f\u00f6r samtliga extrempunkter f\u00f6r funktionen \\\\(f(x)=-sin(\\\\frac{x}{2}+1)\\\\) i intervallet \\\\(0 \\\\le x \\\\le 4\\\\pi\\\\)`,\r\n svar: `\\\\((\\\\pi, 0)\\\\) och \\\\((\\\\frac{3\\\\pi}{2}, 2)\\\\)` ,\r\n niva: \"C\",\r\n poang: [1,1,0],\r\n underkategori: [\"derivatagraf\"],\r\n kod: \"d33\"\r\n },\r\n {\r\n fraga: `<p> Den trigonometriska funktionen \\\\(T(t)=4cos(\\\\frac{\\\\pi}{12}t+4)+24\\\\) beskriver temperaturen\r\nen fin sommardag i Lund fr\u00e5n kl 07.00 d\u00e4r \ud835\udc61 \u00e4r tid i timmar.<\/p> <p>\r\na) Best\u00e4m vilken tid p\u00e5 dygnet det \u00e4r som varmast respektive kallast p\u00e5 dagen.<\/p> \r\nb) Best\u00e4m vilken tid p\u00e5 dygnet temperaturen v\u00e4xer som mest. `,\r\n svar: `a) Kallast: ungef\u00e4r 3,30 p\u00e5 natten, varmast: 15.30 p\u00e5 dagen b) V\u00e4xer som snabbast 9.30 ` ,\r\n niva: \"C\",\r\n poang: [1,2,0],\r\n underkategori: [\"problemderivata\"],\r\n kod: \"d34\",\r\n digital: \"true\"\r\n },\r\n {\r\n fraga: `<p> Funktionen \\\\(d(t)=10-8sin(\\\\frac{\\\\pi t}{6})\\\\) beskriver havsdjupet i meter vid en position p\u00e5\r\nstranden i kuststaden Playa Jide. \ud835\udc61 st\u00e5r f\u00f6r tiden i timmar fr\u00e5n 07.00 <\/p> <p>\r\na) Best\u00e4m och tolka \\\\(d(4)\\\\) <\/p> <p>\r\nb) Best\u00e4m och tolka \\\\(d(t)=10\\\\) <\/p>\r\nc) Best\u00e4m och tolka \\\\(d'(12)\\\\) `,\r\n svar: `a) Kl 11.00 \u00e4r vattendjupet ungef\u00e4r 3 meter. b) Kl 13.00 och sedan var 12 timme \u00e4r vattendjupet 10 meter. c) Vid kl 19.00 minskar havsdjupet med -4,19 meter per timme.` ,\r\n niva: \"C\",\r\n poang: [2,1,0],\r\n underkategori: [\"problemderivata\"],\r\n kod: \"d35\",\r\n digital: \"true\"\r\n },\r\n {\r\n fraga: `<p>Kroppstemperaturen f\u00f6r en person kan beskrivas med funktionen \\\\(f(x)=1,1sin(\\\\frac{\\\\pi}{12}x-\\\\pi)+36\\\\) d\u00e4r \\\\(x=0\\\\) \u00e4r timmen d\u00e5 personen somnar.<\/p> <p>\r\na) Efter hur m\u00e5nga timmar fr\u00e5n insomning \u00e4r kroppstemperaturen som l\u00e4gst? <\/p> <p>\r\nb) Efter hur m\u00e5nga timmar fr\u00e5n insomning \u00e4r kroppstemperaturen som h\u00f6gst? <\/p>\r\nc) Efter hur m\u00e5nga timmar \u00f6kar kroppstemperaturen som snabbast? `,\r\n svar: `a) 6 timmar b) 18 timmar c) 12 timmar ` ,\r\n niva: \"C\",\r\n poang: [2,1,0],\r\n underkategori: [\"problemderivata\"],\r\n kod: \"d36\",\r\n digital: \"true\"\r\n },\r\n {\r\n fraga: `Joakim har f\u00e5tt sitt blodsocker unders\u00f6kts av en duktig l\u00e4kare. Det visar sig att Joakims\r\nblodsocker efter en m\u00e5ltid g\u00e5r att beskriva med funktionen \\\\(f(x)=0,032x^2e^{-0,07x}+4\\\\) d\u00e4r \\\\(f(x)\\\\) \u00e4r sockerhalten i millimolar och \ud835\udc65 \u00e4r antalet\r\nminuter efter m\u00e5ltiden. Best\u00e4m n\u00e4r Joakims blodsocker \u00f6kar som snabbast. `,\r\n svar: `Ungef\u00e4r 8,4 minuter efter m\u00e5ltiden` ,\r\n niva: \"C\",\r\n poang: [0,2,0],\r\n underkategori: [\"problemderivata\"],\r\n kod: \"d37\",\r\n digital: \"true\"\r\n },\r\n {\r\n fraga: `En sf\u00e4risk sn\u00f6boll sm\u00e4lter s\u00e5 att radien minskar med 2,0 cm\/h. Med vilken hastighet\r\n\u00e4ndras sn\u00f6bollens volym d\u00e5 dess radie \u00e4r 3,0 cm?`,\r\n svar: `Minskar med 226 cm\/timme` ,\r\n niva: \"A\",\r\n poang: [0,2,1],\r\n underkategori: [\"problemderivata\"],\r\n kod: \"d38\",\r\n digital: \"true\"\r\n },\r\n {\r\n fraga: `<p>F\u00f6r en funktion \\\\(h(x)=sin(g(x))\\\\) vet du f\u00f6ljande<\/p> <p>\r\n *\\\\(h(\\\\pi)=\\\\frac{\\\\sqrt{3}}{2}\\\\)<\/p> <p>\r\n *\\\\(g'(\\\\pi)=2\\\\) <\/p> Best\u00e4m de t\u00e4nkbara v\u00e4rdena f\u00f6r \\\\(h'(\\\\pi)\\\\)`,\r\n svar: `\\\\(h'(\\\\pi)=1\\\\) eller \\\\(h'(\\\\pi)=-1\\\\)` ,\r\n niva: \"A\",\r\n poang: [0,0,3],\r\n underkategori: [\"problemderivata\"],\r\n losningsforslag: `<img decoding=\"async\" src=\"https:\/\/mahifi.se\/wp-content\/uploads\/2025\/11\/Skarmbild-2025-11-15-162348.png\" alt=\"L\u00f6sningsf\u00f6rslag\">`,\r\n kod: \"d40\"\r\n },\r\n {\r\n fraga: `Visa att derivatan till funktionen \\\\(f(x)=2sinxcosx\\\\) \u00e4r \\\\(f'(x)=2cos2x\\\\)`,\r\n svar: `Skriv om \\\\(f(x)\\\\) till \\\\(sin2x\\\\) eller skriv om derivatafunktionen.` ,\r\n niva: \"C\",\r\n poang: [0,1,0],\r\n underkategori: [\"deriveringsregler\"],\r\n kod: \"d41\"\r\n } ,\r\n {\r\n fraga: `Roger \u00e4r p\u00e5 en marknad och ska k\u00f6pa en ballong som \u00e4r\r\nklotformad. N\u00e4r f\u00f6rs\u00e4ljaren bl\u00e5ser upp ballongen n\u00e5r den en\r\nvolym p\u00e5 6500 \\\\(cm^3\\\\)\r\n. Ballongen bl\u00e5ses upp s\u00e5 att dess volym \u00f6kar\r\nmed konstant hastighet och n\u00e4r ballongens radie \u00e4r 5,6 cm \u00f6kar\r\nradien med 4,4 cm\/s. Best\u00e4m hur m\u00e5nga sekunder det tar att\r\nbl\u00e5sa upp ballongen om den \u00e4r tom fr\u00e5n b\u00f6rjan.`,\r\n svar: `3,75 sekunder` ,\r\n niva: \"A\",\r\n poang: [0,2,1],\r\n digital: true,\r\n underkategori: [\"problemderivata\"],\r\n kod: \"d42\"\r\n },\r\n {\r\n fraga: `Visa att funktionen \\\\(f(x)=x \\\\cdot e^{-\\\\frac{x}{a}}\\\\) har en extrempunkt i \\\\(x=a\\\\) f\u00f6ruts\u00e4tt att \\\\(a \\\\neq 0\\\\).`,\r\n svar: `Derivera funktionen och visa att derivatafunktionen \u00e4r lika med noll d\u00e5 \\\\(x=a\\\\)` ,\r\n niva: \"C\",\r\n poang: [0,2,0],\r\n underkategori: [\"derivatagraf\"],\r\n kod: \"d43\"\r\n },\r\n {\r\n fraga: `Derivera funktionen \\\\(f(x)=(e^{2x}+3)^{10}\\\\) ` ,\r\n svar: `\\\\(f'(x)=20(e^{2x}+3)^9\\\\)`,\r\n niva: \"E\",\r\n poang: [2,0,0],\r\n underkategori: [\"deriveringsregler\"],\r\n kod: \"d44\"\r\n },\r\n {\r\n fraga: `<p> a) Derivera f\u00f6ljande funktion \\\\(f(x)=x^x\\\\) genom att f\u00f6rst skriva om funktionen till \\\\(f(x)=(e^{lnx})^x\\\\) <\/p> <p> b) Observera f\u00f6ljande m\u00f6nster, h\u00f6gerleden \u00e4r n\u00e4rmev\u00e4rden till uttrycken. <\/p> <p> \\\\(\\\\sqrt[2]{2}=1,414\\\\) <\/p> <p> \\\\(\\\\sqrt[3]{3}=1,442\\\\) <\/p> <p> \\\\(\\\\sqrt[4]{4}=1,414\\\\) <\/p> <p> \\\\(\\\\sqrt[5]{5}=1,379\\\\) <\/p> <p> \\\\(\\\\sqrt[a]{a}\\\\) <\/p> F\u00f6r vilket v\u00e4rde p\u00e5 \\\\(a\\\\) maximeras uttryckets v\u00e4rde, svara exakt.` ,\r\n svar: `a) \\\\(f'(x)=x^x(lnx+1)\\\\) b) \\\\(a=e\\\\) ger det st\u00f6rsta v\u00e4rdet p\u00e5 uttrycket.`,\r\n niva: \"A\",\r\n poang: [0,1,3],\r\n underkategori: [\"problemderivata\", \"deriveringsregler\"],\r\n kod: \"d45\"\r\n },\r\n {\r\n fraga: `L\u00e5t \\\\(f(x)=e^{2x}-e^x+e\\\\). Visa att \\\\(f(x) > 2\\\\) f\u00f6r alla \\\\(x\\\\)` ,\r\n svar: `Anv\u00e4nd derivata f\u00f6r att visa att funktionen inte n\u00e5r ett v\u00e4rde h\u00f6gre \u00e4n \\\\(y=2\\\\)`,\r\n niva: \"A\",\r\n poang: [0,1,2],\r\n underkategori: [\"problemderivata\"],\r\n kod: \"d46\"\r\n },\r\n {\r\n fraga: `Best\u00e4m ekvationen f\u00f6r den tangent som tangerar funktion \\\\(f(x)=sin2x\\\\) i \\\\((\\\\frac{\\\\pi}{2},0)\\\\)`,\r\n svar: `\\\\(y=-2x+\\\\pi\\\\)`,\r\n niva: \"C\",\r\n poang: [1,1,0],\r\n underkategori: [\"derivatagraf\"],\r\n kod: \"d47\"\r\n },\r\n {\r\n fraga: `F\u00f6r vilket v\u00e4rde p\u00e5 \\\\(x\\\\) har funktionen \\\\(f(x)=x \\\\cdot e^x\\\\) har en extrempunkt`,\r\n svar: `\\\\(x=-1\\\\)`,\r\n niva: \"C\",\r\n poang: [1,1,0],\r\n underkategori: [\"derivatagraf\"],\r\n kod: \"d48\"\r\n },\r\n {\r\n fraga: `Best\u00e4m ekvationen f\u00f6r den tangent som tangerar funktion \\\\(f(x)=x \\\\cdot e^{2x}\\\\) i \\\\(x=1\\\\)`,\r\n svar: `\\\\(y=2ex-e\\\\)`,\r\n niva: \"C\",\r\n poang: [1,1,0],\r\n underkategori: [\"derivatagraf\"],\r\n kod: \"d49\"\r\n },\r\n {\r\n fraga: `F\u00f6r vilka v\u00e4rden i intervallet \\\\(0 < x < 2\\\\pi\\\\) \u00e4r funktionen \\\\(sin^3x\\\\) avtagande?`,\r\n svar: `\\\\(\\\\frac{\\\\pi}{2} < x < \\\\frac{3\\\\pi}{2}\\\\)`,\r\n niva: \"C\",\r\n poang: [0,2,0],\r\n underkategori: [\"derivatagraf\"],\r\n kod: \"d50\"\r\n },\r\n {\r\n fraga: `Visa att funktionen \\\\(f(x)=ln(ax)+x\\\\) har en extrempunkt i \\\\(x=-1\\\\) oavsett v\u00e4rde p\u00e5 \\\\(a\\\\) d\u00e4r \\\\(a \\\\neq 0\\\\)`,\r\n svar: `Derivera funktionen och visa att \\\\(f'(x)=0\\\\) har samma l\u00f6sning eftersom derivatan \u00e4r oberoende av \\\\(a\\\\)`,\r\n niva: \"C\",\r\n poang: [0,2,0],\r\n underkategori: [\"derivatagraf\"],\r\n kod: \"d51\"\r\n },\r\n {\r\n fraga: `Best\u00e4m det st\u00f6rsta v\u00e4rdet f\u00f6r funktionen \\\\(f(x)=ln(\\\\sqrt{x})-\\\\frac{x}{2}\\\\). Svara exakt.`,\r\n svar: `Anv\u00e4nd derivata f\u00f6r att visa att funktionens st\u00f6rsta v\u00e4rde \u00e4r \\\\(y=-\\\\frac{1}{2}\\\\)`,\r\n niva: \"C\",\r\n poang: [0,3,0],\r\n underkategori: [\"derivatagraf\"],\r\n kod: \"d52\"\r\n },\r\n {\r\n fraga: `Best\u00e4m ekvationen f\u00f6r den r\u00e4ta linje som tangerar funktionen \\\\(\\\\sqrt{3x-2}\\\\) i \\\\(x=1\\\\)`,\r\n svar: `\\\\(y=\\\\frac{3x}{2}-\\\\frac{1}{2}\\\\)`,\r\n niva: \"C\",\r\n poang: [1,2,0],\r\n underkategori: [\"derivatagraf\"],\r\n kod: \"d53\"\r\n },\r\n {\r\n fraga: `En konisk beh\u00e5llare har spetsen ned\u00e5t och lika stor radie som h\u00f6jd. Beh\u00e5llaren l\u00e4cker\r\nmed 300 \\\\(cm^3\/minut\\\\). Hur f\u00f6r\u00e4ndras v\u00e4tskeniv\u00e5n i konen vid l\u00e4get att h\u00f6jden \u00e4r 15\r\ncm? <img decoding=\"async\" src=\"https:\/\/mahifi.se\/wp-content\/uploads\/2026\/07\/Skarmavbild-2026-07-21-kl.-21.35.15.png\" alt=\"L\u00f6sningsf\u00f6rslag\">`,\r\n svar: `0,42 cm\/minut`,\r\n niva: \"C\",\r\n poang: [0,2,0],\r\n underkategori: [\"problemderivata\"],\r\n kod: \"d54\"\r\n },\r\n {\r\n fraga: `Visa att derivatans till \\\\(f(x)=tanx\\\\) \u00e4r \\\\(f'(x)=\\\\frac{1}{cos^2x}\\\\)`,\r\n svar: `Skriv om funktionen till \\\\(f(x)=\\\\frac{sinx}{cosx}\\\\) derivera sedan funktion och f\u00f6renkla uttrycket. `,\r\n niva: \"C\",\r\n poang: [1,2,0],\r\n underkategori: [\"deriveringsregler\"],\r\n kod: \"d55\"\r\n },\r\n {\r\n fraga: `<p> F\u00f6r funktionen \\\\(h(x)=f(g(x))\\\\) vet du att <\/p> <p> \\\\(g(1)=3\\\\) <\/p> <p> \\\\(g'(1)=2\\\\) <\/p> <p> \\\\(f'(1)=7\\\\) <\/p> Best\u00e4m \\\\(h'(1)\\\\)`,\r\n svar: `\\\\(h'(1)=14\\\\) `,\r\n niva: \"A\",\r\n poang: [0,1,1],\r\n underkategori: [\"problemderivata\"],\r\n kod: \"d56\"\r\n },\r\n {\r\n fraga: `Derivera f\u00f6ljande funktion \\\\(f(x)=e^{\\\\sqrt{2x^2+1}}\\\\)`,\r\n svar: `\\\\(f'(x)=\\\\frac{2xe^{\\\\sqrt{2x^2+1}}}{\\\\sqrt{2x^2+1}}\\\\)`,\r\n niva: \"A\",\r\n poang: [0,0,1],\r\n underkategori: [\"deriveringsregler\"],\r\n kod: \"d57\"\r\n },\r\n {\r\n fraga: `Derivera f\u00f6ljande funktion \\\\(f(x)=sin(\\\\frac{x}{2})\\\\)`,\r\n svar: `\\\\(f'(x)=\\\\frac{cos(\\\\frac{x}{2})}{2}\\\\)`,\r\n niva: \"E\",\r\n poang: [1,0,0],\r\n underkategori: [\"deriveringsregler\"],\r\n kod: \"d58\"\r\n },\r\n {\r\n fraga: `Derivera f\u00f6ljande funktion \\\\(f(x)=e^{x^2}\\\\)`,\r\n svar: `\\\\(f'(x)=2xe^{x^2}\\\\)`,\r\n niva: \"C\",\r\n poang: [0,1,0],\r\n underkategori: [\"deriveringsregler\"],\r\n kod: \"d59\"\r\n },\r\n {\r\n fraga: `Derivera f\u00f6ljande funktion \\\\(f(x)=2 \\\\cdot lnx\\\\)`,\r\n svar: `\\\\(f'(x)=\\\\frac{2}{x}\\\\)`,\r\n niva: \"E\",\r\n poang: [1,0,0],\r\n underkategori: [\"deriveringsregler\"],\r\n kod: \"d60\"\r\n },\r\n {\r\n fraga: `Best\u00e4m samtliga asymptoter f\u00f6r f\u00f6ljande funktion \\\\(f(x)=\\\\frac{1}{x-2}+3\\\\)`,\r\n svar: `V\u00e5gr\u00e4t asymptot i \\\\(x=2\\\\) och lodr\u00e4t asymptot i \\\\(y=3\\\\)`,\r\n niva: \"E\",\r\n poang: [2,0,0],\r\n underkategori: [\"analysgraf\"],\r\n kod: \"d61\"\r\n },\r\n {\r\n fraga: `Best\u00e4m de v\u00e5gr\u00e4ta asymptoterna f\u00f6r f\u00f6ljande funktion \\\\(f(x)=\\\\frac{4}{x^2-9}\\\\)`,\r\n svar: `V\u00e5gr\u00e4ta asymptoter i \\\\(x=\\\\pm 3\\\\)`,\r\n niva: \"E\",\r\n poang: [2,0,0],\r\n underkategori: [\"analysgraf\"],\r\n kod: \"d62\"\r\n },\r\n {\r\n fraga: `Best\u00e4m samtliga asymptoter till f\u00f6ljande funktion \\\\(f(x)=\\\\frac{4}{e^{2x}-3}+2x+1\\\\)`,\r\n svar: `V\u00e5gr\u00e4t asymptot i \\\\(x=\\\\frac{ln3}{2}\\\\) och sned asymptot i \\\\(y=2x+1\\\\)`,\r\n niva: \"C\",\r\n poang: [0,2,0],\r\n underkategori: [\"analysgraf\"],\r\n kod: \"d63\"\r\n },\r\n {\r\n fraga: `Best\u00e4m den sneda asymptoten f\u00f6r f\u00f6ljande funktion \\\\(f(x)=\\\\frac{2x+2}{x^3}-3x+7\\\\)`,\r\n svar: `Sned asymptot i \\\\(y=-3x+7\\\\)`,\r\n niva: \"C\",\r\n poang: [0,1,0],\r\n underkategori: [\"analysgraf\"],\r\n kod: \"d64\"\r\n },\r\n {\r\n fraga: `Derivera f\u00f6ljande funktion \\\\(f(x)=e^{\\\\sqrt{x}}\\\\)`,\r\n svar: `\\\\(f'(x)=\\\\frac{e^{\\\\sqrt{x}}}{2\\\\sqrt{x}}\\\\)`,\r\n niva: \"C\",\r\n poang: [0,1,0],\r\n underkategori: [\"deriveringsregler\"],\r\n kod: \"d65\"\r\n },\r\n {\r\n fraga: `Derivera f\u00f6ljande funktion \\\\(f(x)=sin(\\\\sqrt{x})\\\\)`,\r\n svar: `\\\\(f'(x)=\\\\frac{cos(\\\\sqrt{x})}{2\\\\sqrt{x}}\\\\)`,\r\n niva: \"C\",\r\n poang: [0,1,0],\r\n underkategori: [\"deriveringsregler\"],\r\n kod: \"d66\"\r\n },\r\n {\r\n fraga: `<p> En matematiker har tagit fram statistik f\u00f6r kommunen Joakimk\u00f6ping. Hon har dels\r\ntagit fram en funktion som visar p\u00e5 hur mycket skatt (i miljoner) kommunen kan ta in\r\nberoende p\u00e5 hur stor befolkningen \\\\(b\\\\) \u00e4r i tusental \\\\(s(b)=9,7\\\\sqrt{0,5b}\\\\) Samtidigt har hon\r\ntagit fram statistisk f\u00f6r den f\u00f6rv\u00e4ntade befolkningsf\u00f6r\u00e4ndringen \r\nsom g\u00e5r att beskriva med funktionen \\\\(b(t)=35+3,5sin(\\\\frac{\\\\pi t}{6})+1,6t\\\\) d\u00e4r \\\\(t\\\\) \u00e4r antalet \u00e5r fr\u00e5n 2024. <\/p> <p> a) Best\u00e4m vilket \u00e5r skatteint\u00e4kterna \u00f6verstiger 45 miljoner i Joakimk\u00f6ping. <\/p>\r\nb) Vilket\/vilka \u00e5r \u00f6kar skatteint\u00e4kterna med 0,91 miljoner per \u00e5r i under de f\u00f6rsta 11\r\n\u00e5ren fr\u00e5n 2024?`,\r\n svar: `a) Ungef\u00e4r 2027 b) Unged\u00e4r 2027 och 2032`,\r\n niva: \"A\",\r\n poang: [0,1,3],\r\n underkategori: [\"problemderivata\"],\r\n kod: \"d67\",\r\n digital: true\r\n }, \r\n {\r\n fraga: `Derivera f\u00f6ljande funktion \\\\(f(x)=(2x+1)^4\\\\)`,\r\n svar: `\\\\(f'(x)=8(2x+1)^3\\\\)`,\r\n niva: \"E\",\r\n poang: [1,0,0],\r\n underkategori: [\"deriveringsregler\"],\r\n kod: \"d70\"\r\n },\r\n {\r\n fraga: `Best\u00e4m extrempunkten och dess karakt\u00e4r f\u00f6r f\u00f6ljande funktion \\\\(f(x)=x \\\\cdot e^{-x}\\\\)`,\r\n svar: `Extrempunkt i \\\\((1, \\\\frac{1}{e})\\\\) det \u00e4r en maximipunkt`,\r\n niva: \"C\",\r\n poang: [1,2,0],\r\n underkategori: [\"derivatagraf\"],\r\n kod: \"d71\"\r\n },\r\n {\r\n fraga: `Unders\u00f6k f\u00f6r vilket v\u00e4rde p\u00e5 \\\\(a\\\\) som funktionen \\(f(x)=ex \\\\cdot e^{-x}\\\\) har en extrempunkt i \\\\((1,1)\\\\)`,\r\n svar: `D\u00e5 \\\\(a=e\\\\)`,\r\n niva: \"C\",\r\n poang: [0,3,0],\r\n underkategori: [\"derivatagraf\"],\r\n kod: \"d72\"\r\n },\r\n {\r\n fraga: `Vi definierar funktionen \\\\(f(x)=ln(g(x))\\\\), du vet att funktionen \\\\(g(x)\\\\) har en tangent i \\\\(x=4\\\\) vars ekvation \u00e4r \\\\(y=-2x+10\\\\). Best\u00e4m \\\\(f'(4)\\\\)`,\r\n svar: `\\\\(f'(4)=-1\\\\)`,\r\n niva: \"A\",\r\n poang: [0,1,1],\r\n underkategori: [\"problemderivata\"],\r\n kod: \"d73\"\r\n },\r\n {\r\n fraga: `Vi definierar funktionen \\\\(f(x)=e^{h(x)}\\\\). Funktionen \\\\(h(x)\\\\) har en tangent \\\\(y=3x-2\\\\) i \\\\(x=1\\\\). Best\u00e4m ekvationen f\u00f6r tangenten som tangerar \\\\(f(x)\\\\) i \\\\(x=1\\\\)`,\r\n svar: `\\\\(y=3ex-2e\\\\)`,\r\n niva: \"A\",\r\n poang: [0,0,2],\r\n underkategori: [\"problemderivata\"],\r\n kod: \"d74\"\r\n }\r\n ],\r\n\r\n integraler: [\r\n {\r\n fraga: `Ber\u00e4kna integralen \\\\(\\\\int_{0}^{2\\\\pi} sinx \\\\: dx\\\\).`,\r\n svar: `0`,\r\n niva: \"E\",\r\n poang: [2,0,0],\r\n underkategori: [\"primintegral\"],\r\n kod: \"i1\"\r\n }, \r\n {\r\n fraga: `Ber\u00e4kna integralen \\\\(\\\\int_{0}^{\\\\frac{\\\\pi}{2}} sin2x \\\\: dx\\\\).`,\r\n svar: `1`,\r\n niva: \"E\",\r\n poang: [2,0,0],\r\n underkategori: [\"primintegral\"],\r\n kod: \"i2\"\r\n }, \r\n {\r\n fraga: `Ber\u00e4kna integralen \\\\(\\\\int_{1}^{e} \\\\frac{1}{x} \\\\: dx\\\\).`,\r\n svar: `1`,\r\n niva: \"E\",\r\n poang: [2,0,0],\r\n underkategori: [\"primintegral\"],\r\n kod: \"i3\"\r\n }, \r\n {\r\n fraga: `Ber\u00e4kna integralen \\\\(\\\\int_{-1}^{1} e^{x^2} \\\\: dx\\\\). Svara med tv\u00e5 decimaler`,\r\n svar: `2,93`,\r\n niva: \"E\",\r\n poang: [1,0,0],\r\n underkategori: [\"primintegral\"],\r\n digital: \"true\",\r\n kod: \"i4\"\r\n }, \r\n {\r\n fraga: `L\u00f6s ekvationen \\\\(\\\\int_{0}^{t} cosx \\\\: dx=0\\\\). Algebraiska och grafiska motiveringar godtas.`,\r\n svar: `\\\\(t=\\\\pi \\\\cdot n\\\\)`,\r\n niva: \"C\",\r\n poang: [0,2,0],\r\n underkategori: [\"primintegral\"],\r\n kod: \"i5\"\r\n }, \r\n {\r\n fraga: `Best\u00e4m samtliga t\u00e4nkbara v\u00e4rdet p\u00e5 \\\\(k\\\\) f\u00f6r att f\u00f6ljande likhet ska st\u00e4mma \\\\(\\\\int_{0}^{\\\\frac{\\\\pi}{2}} k\\\\cdot coskx \\\\: dx=1\\\\).`,\r\n svar: `\\\\(k=1 + 4\\\\cdot n\\\\)`,\r\n niva: \"C\",\r\n poang: [0,2,0],\r\n underkategori: [\"primintegral\"],\r\n kod: \"i6\"\r\n }, \r\n {\r\n fraga: `Ber\u00e4kna integralen med ditt digitala hj\u00e4lpmedel \\\\(\\\\int_{0}^{\\\\pi} sin^2x \\\\: dx\\\\).`,\r\n svar: `1,57`,\r\n niva: \"C\",\r\n poang: [0,1,0],\r\n underkategori: [\"primintegral\"],\r\n digital: \"true\",\r\n kod: \"i7\"\r\n },\r\n {\r\n fraga: `Ber\u00e4kna integralen \\\\(\\\\int_{0}^{2\\\\pi} sin^3x dx\\\\).`,\r\n svar: `0, se l\u00f6sningsf\u00f6rslag f\u00f6r motivering`,\r\n losningsforslag: `<img decoding=\"async\" src=\"https:\/\/mahifi.se\/wp-content\/uploads\/2025\/11\/Skarmbild-2025-11-16-074656.png\" alt=\"L\u00f6sningsf\u00f6rslag\">`,\r\n niva: \"A\",\r\n poang: [0,0,2],\r\n underkategori: [\"primintegral\"],\r\n kod: \"i8\"\r\n },\r\n {\r\n fraga: `Joakim vill att integralen \\\\(\\\\int_{a}^{b} sinx+c \\\\: dx\\\\) ska vara st\u00f6rre eller lika med noll. F\u00f6r vilka v\u00e4rdet p\u00e5 \\\\(c\\\\).`,\r\n svar: `\\\\(1 \\\\le a\\\\)`,\r\n niva: \"C\",\r\n poang: [1,1,0],\r\n underkategori: [\"primintegral\"],\r\n kod: \"i9\"\r\n },\r\n {\r\n fraga: `<p> Ber\u00e4kna f\u00f6ljande integraluttryck med hj\u00e4lp av graferna nedan. <\/p> \\\\(\\\\int_{0}^{1} f'(x)g(x) \\\\: dx + \\\\int_{0}^{1} f(x)g'(x) \\\\: dx\\\\) <img decoding=\"async\" src=\"https:\/\/mahifi.se\/wp-content\/uploads\/2026\/07\/Skarmavbild-2026-07-22-kl.-21.07.03.png\" alt=\"L\u00f6sningsf\u00f6rslag\">`,\r\n svar: `4`,\r\n niva: \"A\",\r\n poang: [0,1,1],\r\n underkategori: [\"grafintegral\"],\r\n kod: \"i10\"\r\n },\r\n {\r\n fraga: `Nedanf\u00f6r visas graferna till de\r\nprimitiva funktionerna till \\\\(f(x)\\\\) och\r\n\\\\(g(x)\\\\). Best\u00e4m v\u00e4rdet p\u00e5 integralen \\\\(\\\\int_{0}^{2} f(G(x))g(x) \\\\: dx \\\\) <img decoding=\"async\" src=\"https:\/\/mahifi.se\/wp-content\/uploads\/2026\/07\/Skarmavbild-2026-07-22-kl.-21.22.21.png\" alt=\"L\u00f6sningsf\u00f6rslag\">`,\r\n svar: `2`,\r\n niva: \"A\",\r\n poang: [0,0,2],\r\n underkategori: [\"grafintegral\"],\r\n kod: \"i11\"\r\n },\r\n {\r\n fraga: `Nedan ser du tv\u00e5 funktioner \\\\(f(x)\\\\) och\r\n\\\\(g(x)\\\\) och en avgr\u00e4nsad area som \u00e4r\r\n25 areaenheter. Du vet att \\\\(\\\\int_{0}^{a} f(x) \\\\: dx=18 \\\\) Best\u00e4m v\u00e4rdet p\u00e5 integralen \\\\(\\\\int_{0}^{a} g(x) \\\\: dx\\\\) <img decoding=\"async\" src=\"https:\/\/mahifi.se\/wp-content\/uploads\/2026\/07\/Skarmavbild-2026-07-22-kl.-21.24.18.png\" alt=\"L\u00f6sningsf\u00f6rslag\">`,\r\n svar: `-7`,\r\n niva: \"C\",\r\n poang: [0,2,0],\r\n underkategori: [\"grafintegral\"],\r\n kod: \"i12\"\r\n },\r\n {\r\n fraga: `Nedan ser du tv\u00e5 funktioner \\\\(f(x)=e^{\\\\frac{x}{2}}\\\\) och\r\n\\\\(g(x)=lnx\\\\) och en avgr\u00e4nsad area. Best\u00e4m arean f\u00f6r den arean. Svara med tv\u00e5 decimaler. <img decoding=\"async\" src=\"https:\/\/mahifi.se\/wp-content\/uploads\/2026\/07\/Skarmavbild-2026-07-22-kl.-21.32.25.png\" alt=\"L\u00f6sningsf\u00f6rslag\">`,\r\n svar: `4,37 a.e`,\r\n niva: \"E\",\r\n poang: [2,0,0],\r\n underkategori: [\"grafintegral\"],\r\n kod: \"i13\",\r\n digital: true\r\n },\r\n {\r\n fraga: `Ber\u00e4kna integralen \\\\(\\\\int_{0}^{\\\\pi} sin^2x \\\\: dx\\\\).`,\r\n svar: `\\\\(\\\\frac{\\\\pi}{2}. Tips skriv om funktionsuttrycket med trigonometriska identiteter`,\r\n niva: \"C\",\r\n poang: [0,3,0],\r\n underkategori: [\"primintegral\"],\r\n kod: \"i14\"\r\n },\r\n {\r\n fraga: `<p> a) Ber\u00e4kna integralen \\\\(\\\\int_{1}^{e} \\\\frac{4}{3x} \\\\: dx\\\\). <\/p> b) Visa att integralens v\u00e4rde alltid kommer vara \\\\(a\\\\) f\u00f6r f\u00f6ljande integral \\\\(\\\\int_{1}^{e} \\\\frac{a}{x} \\\\: dx\\\\)`,\r\n svar: `a) \\\\(\\\\frac{4}{3}\\\\) b) Ber\u00e4kna integraluttrycket och visa att uttrycket alltid blir \\\\(a\\\\)`,\r\n niva: \"C\",\r\n poang: [2,2,0],\r\n underkategori: [\"primintegral\"],\r\n kod: \"i15\"\r\n },\r\n {\r\n fraga: `Nedanf\u00f6r visas graferna \\\\(f(x)=sinx\\\\) och\r\n\\\\(g(x)=cosx\\\\) och en sk\u00e4rningspunkt till graferna. Best\u00e4m arean av den markerade arean, svara exakt. <img decoding=\"async\" src=\"https:\/\/mahifi.se\/wp-content\/uploads\/2026\/07\/Skarmavbild-2026-07-23-kl.-15.31.12.png\" alt=\"L\u00f6sningsf\u00f6rslag\">`,\r\n svar: `\\\\frac{2-\\\\sqrt2}{sqrt2}`,\r\n niva: \"E\",\r\n poang: [3,0,0],\r\n underkategori: [\"grafintegral\"],\r\n kod: \"i16\"\r\n },\r\n {\r\n fraga: `Nedan ser du tv\u00e5 funktioner d\u00e4r den ena \u00e4r en derivatafunktion och den andra\r\nursprungsfunktionen. Best\u00e4m den gr\u00f6na arean. <img decoding=\"async\" src=\"https:\/\/mahifi.se\/wp-content\/uploads\/2026\/07\/Skarmavbild-2026-07-23-kl.-15.38.24.png\" alt=\"L\u00f6sningsf\u00f6rslag\">`,\r\n svar: `2 a.e`,\r\n niva: \"A\",\r\n poang: [0,0,1],\r\n underkategori: [\"grafintegral\"],\r\n kod: \"i17\"\r\n },\r\n {\r\n fraga: `Nedan ser du funktionerna \\\\(f(x)=sinx+ax\\\\) och \\\\(g(x)=ax\\\\) d\u00e4r \\\\(a > 0\\\\). Visa att den\r\nbl\u00e5markerade arean \u00e4r konstant oavsett v\u00e4rde p\u00e5 \\\\(a\\\\) <img decoding=\"async\" src=\"https:\/\/mahifi.se\/wp-content\/uploads\/2026\/07\/Skarmavbild-2026-07-23-kl.-15.41.12.png\" alt=\"L\u00f6sningsf\u00f6rslag\">`,\r\n svar: `S\u00e4tt in funktionerna i en integral med den f\u00f6rsta sk\u00e4rningspunkterna d\u00e5 \\\\(x>0\\\\). Visa sedan att den bl\u00e5markerade \u00e4r oberoende av \\\\(a\\\\)`,\r\n niva: \"A\",\r\n poang: [0,1,1],\r\n underkategori: [\"grafintegral\"],\r\n kod: \"i18\"\r\n },\r\n {\r\n fraga: `Ber\u00e4kna integralen \\\\(\\\\int_{0}^{\\\\frac{\\\\pi}{4}} 4cos2x \\\\: dx\\\\)`,\r\n svar: `2`,\r\n niva: \"C\",\r\n poang: [1,1,0],\r\n underkategori: [\"primintegral\"],\r\n kod: \"i19\"\r\n },\r\n {\r\n fraga: `Ber\u00e4kna integralen \\\\(\\\\int_{0}^{1} xe^x+e^x \\\\: dx\\\\). Svara exakt.`,\r\n svar: `e`,\r\n niva: \"A\",\r\n poang: [0,0,2],\r\n underkategori: [\"primintegral\"],\r\n kod: \"i20\"\r\n },\r\n {\r\n fraga: `Ber\u00e4kna integralen \\\\(\\\\int_{0}^{\\\\frac{\\\\pi}{2}} cosx \\\\: dx\\\\). Svara exakt.`,\r\n svar: `1`,\r\n niva: \"E\",\r\n poang: [2,0,0],\r\n underkategori: [\"primintegral\"],\r\n kod: \"i21\"\r\n },\r\n {\r\n fraga: `Ber\u00e4kna integralen \\\\(\\\\int_{0}^{\\\\frac{\\\\pi}{4}} 3cos2x \\\\: dx\\\\). Svara exakt.`,\r\n svar: `\\\\(\\\\frac{3}{2}\\\\)`,\r\n niva: \"E\",\r\n poang: [3,0,0],\r\n underkategori: [\"primintegral\"],\r\n kod: \"i22\"\r\n },\r\n {\r\n fraga: `Best\u00e4m 2 olika v\u00e4rden p\u00e5 \\\\(a\\\\) d\u00e4r \\\\(a \\\\neq 0 \\\\) d\u00e4r du f\u00e5r f\u00f6ljande likhet \\\\(\\\\int_{0}^{a} sinx \\\\: dx=0\\\\).`,\r\n svar: `Till exempel \\\\(2\\\\pi\\\\) och \\\\(4\\\\pi\\\\)`,\r\n niva: \"C\",\r\n poang: [1,1,0],\r\n underkategori: [\"primintegral\"],\r\n kod: \"i23\"\r\n },\r\n {\r\n fraga: `Best\u00e4m samtliga primitiva funktioner till \\\\(f(x)=2cosx\\\\)`,\r\n svar: `\\\\(F(x)=2sinx+C\\\\)`,\r\n niva: \"E\",\r\n poang: [2,0,0],\r\n underkategori: [\"primitiv\"],\r\n kod: \"i24\"\r\n },\r\n {\r\n fraga: `Best\u00e4m samtliga primitiva funktioner till \\\\(f(x)=cos2x\\\\)`,\r\n svar: `\\\\(F(x)=\\\\frac{sin2x}{2}+C\\\\)`,\r\n niva: \"E\",\r\n poang: [2,0,0],\r\n underkategori: [\"primitiv\"],\r\n kod: \"i25\"\r\n },\r\n {\r\n fraga: `Best\u00e4m samtliga primitiva funktioner till \\\\(f(x)=sinx+2x\\\\)`,\r\n svar: `\\\\(F(x)=-cosx+x^2+C\\\\)`,\r\n niva: \"E\",\r\n poang: [2,0,0],\r\n underkategori: [\"primitiv\"],\r\n kod: \"i26\"\r\n },\r\n {\r\n fraga: `Best\u00e4m den primitiva funktion till \\\\(f(x)=2cos2x\\\\) som uppfyller kravet \\\\(F(\\\\pi)=4\\\\)`,\r\n svar: `\\\\(F(x)=sin2x+4\\\\)`,\r\n niva: \"E\",\r\n poang: [3,0,0],\r\n underkategori: [\"primitiv\"],\r\n kod: \"i27\"\r\n },\r\n {\r\n fraga: `Best\u00e4m samtliga primitiva funktioner till \\\\(f(x)=sin^2xcos^2x\\\\)`,\r\n svar: `\\\\(F(x)=\\\\frac{x}{8}-\\\\frac{sin4x}{32}+C\\\\)`,\r\n niva: \"A\",\r\n poang: [0,1,2],\r\n underkategori: [\"primitiv\"],\r\n kod: \"i28\"\r\n },\r\n {\r\n fraga: `Best\u00e4m samtliga primitiva funktioner till \\\\(f(x)=cos2x+e^{2x}\\\\)`,\r\n svar: `\\\\(F(x)=\\\\frac{sin2x}{2}+\\\\frac{e^{2x}{2}+C\\\\)`,\r\n niva: \"E\",\r\n poang: [2,0,0],\r\n underkategori: [\"primitiv\"],\r\n kod: \"i29\"\r\n }\r\n ],\r\n\r\n tillampintegral: [\r\n {\r\n fraga: `Joakimtremonie \u00e4r en bakterie som f\u00f6rekommer i sushi. Den kan vara mycket farlig\r\noch d\u00e4rf\u00f6r vill man inte att bakterieantalet ska \u00f6verstiga 100 000 i en sushi. Joakim har\r\ni sitt labb unders\u00f6kt olika sushi och kommit fram till att de flesta har 1000 bakterier i\r\nsig fr\u00e5n b\u00f6rjan n\u00e4r de serveras och sedan \u00f6kar bakterieantalet med hastigheten \\\\(v(t)=64,5e^{0,215t}\\\\) d\u00e4r \ud835\udc61 \u00e4r antalet minuter efter servering. Efter hur m\u00e5nga minuter \u00e4r inte\r\nsushin l\u00e4ngre \u00e4tbar enligt modellen?`,\r\n svar: \"Ungef\u00e4r 27 minuter\",\r\n niva: \"A\",\r\n poang: [0,2,1],\r\n digital: true,\r\n underkategori: [\"integralproblem\"],\r\n kod: \"b1\"\r\n },\r\n {\r\n fraga: `En tom tunna fylls med vatten med en hastighet \\\\(cm^3\\\\)\r\n\/minut som g\u00e5r att beskriva med\r\nfunktionen \\\\(f(x)=2cos(\\\\frac{\\\\pi x}{30})+3\\\\) d\u00e4r \\\\(x\\\\) \u00e4r antalet minuter. Samtidigt l\u00e4mnar vatten\r\ntunnan med en konstant hastighet p\u00e5 1,5 \\\\(cm^3\\\\)\r\n\/minut. Hur m\u00e5nga \\\\(cm^3\\\\) har tunnan i sig\r\nefter 1 timme?`,\r\n svar: \"Ungef\u00e4r 90 \\\\(cm^3\\\\)\",\r\n niva: \"A\",\r\n poang: [0,2,1],\r\n digital: true,\r\n underkategori: [\"integralproblem\"],\r\n kod: \"b2\"\r\n },\r\n {\r\n fraga: `<p>I ett samh\u00e4lle kan f\u00f6rdelningen av \u00e5ldrar beskrivas med t\u00e4thetsfunktionen nedan. Man\r\nvet att den \u00e4ldsta personer i samh\u00e4llet \u00e4r 100 \u00e5r gammal.<\/p> <p>\r\na) Best\u00e4m t\u00e4thetsfunktions ekvation. <\/p>\r\nb) Vad \u00e4r sannolikheten att man v\u00e4ljer tre slumpm\u00e4ssigt valda personer som \u00e4r \u00f6ver 80\r\n\u00e5r? <img decoding=\"async\" src=\"https:\/\/mahifi.se\/wp-content\/uploads\/2026\/07\/Skarmavbild-2026-07-23-kl.-17.13.53.png\" alt=\"L\u00f6sningsf\u00f6rslag\">`,\r\n svar: \"a) \\\\(f(x)=\\\\frac{-x}{500}+\\\\frac{1}{50}\\\\) b) 0,000064\",\r\n niva: \"A\",\r\n poang: [0,2,1],\r\n digital: true,\r\n underkategori: [\"normal\"],\r\n kod: \"b3\"\r\n },\r\n {\r\n fraga: `Ett omr\u00e5de begr\u00e4nsas av kurvan \\\\(y=x^2-4\\\\)och linjen \\\\(y=5\\\\). Best\u00e4m volymen som\r\nbildas n\u00e4r detta omr\u00e5de roterar runt linjen \\\\(y=5\\\\)`,\r\n svar: \"814 volymenheter\",\r\n niva: \"A\",\r\n poang: [0,0,2],\r\n digital: true,\r\n underkategori: [\"rotation\"],\r\n kod: \"b4\"\r\n },\r\n {\r\n fraga: `Visa med hj\u00e4lp av rotationsvolymer att volymen av en kon \u00e4r \\\\(V=\\\\frac{\\\\pi r^2 h}{3}\\\\)`,\r\n svar: \"S\u00e4tt upp ett integraluttryck f\u00f6r volymen med hj\u00e4lp av en r\u00e4t linje. Se l\u00f6sningsf\u00f6rslag\",\r\n niva: \"A\",\r\n poang: [0,1,2],\r\n losningsforslag: `<img decoding=\"async\" src=\"https:\/\/mahifi.se\/wp-content\/uploads\/2026\/07\/Skarmavbild-2026-07-23-kl.-17.28.19.png\" alt=\"L\u00f6sningsf\u00f6rslag\">`,\r\n underkategori: [\"rotation\"],\r\n kod: \"b5\"\r\n },\r\n {\r\n fraga: `Joakim har designat ett glas som har modellen ungef\u00e4r som visas nedan. Glaset rymmer 300 \\\\(cm^3\\\\) och dess cirkul\u00e4ra botten har en diameter som \u00e4r 6 cm.\r\nBest\u00e4m diametern p\u00e5 \u00f6ppningen om h\u00f6jden p\u00e5 glaset \u00e4r 10 cm. Svara med en\r\ndecimal. <img decoding=\"async\" src=\"https:\/\/mahifi.se\/wp-content\/uploads\/2026\/07\/Skarmavbild-2026-07-23-kl.-17.34.34.png\" alt=\"L\u00f6sningsf\u00f6rslag\">`,\r\n svar: \"6,8 cm\",\r\n niva: \"A\",\r\n poang: [0,0,3],\r\n losningsforslag: `<img decoding=\"async\" src=\"https:\/\/mahifi.se\/wp-content\/uploads\/2026\/07\/Skarmavbild-2026-07-23-kl.-17.35.50.png\" alt=\"L\u00f6sningsf\u00f6rslag\">`,\r\n digital: true,\r\n underkategori: [\"rotation\"],\r\n kod: \"b6\"\r\n },\r\n {\r\n fraga: `<p> Joakim har designat ett vinglas. Ytterkanten\r\ntill vinglaset g\u00e5r att beskriva med\r\nfunktionen \\\\(f(x)=a \\\\cdot sin(0,16x)\\\\) d\u00e4r \\\\(a\\\\) \u00e4r\r\nen konstant. \\\\(x\\\\) avger h\u00f6jden i glaset och \\\\(y\\\\) \u00e4r\r\nbredden p\u00e5 glaset fr\u00e5n mitten med samtliga\r\nstr\u00e4ckor i cm (se bilden till nedanf\u00f6r). Om man\r\nskulle rotera funktionen \\\\(f(x)\\\\) runt \\\\(x\\\\)-axeln\r\nskulle vinglasets design uppst\u00e5. Joakim vill\r\natt h\u00f6jden p\u00e5 glaset ska vara 13 cm och\r\nglaset ska rymma 250 \\\\(cm^3\\\\) <\/p> <p> a) Anv\u00e4nd funktionen f\u00f6r att best\u00e4mma glasets st\u00f6rsta bredd? <\/p>\r\nb) En restaurang \u00e4r intresserade att k\u00f6pa in glaset och vill veta hur h\u00f6gt upp i glaset man\r\nska fylla f\u00f6r att det ska vara fyllt till 70%. Hj\u00e4lp restaurangen!\r\n. <img decoding=\"async\" src=\"https:\/\/mahifi.se\/wp-content\/uploads\/2026\/07\/Skarmavbild-2026-07-23-kl.-21.40.07.png\" alt=\"L\u00f6sningsf\u00f6rslag\">`,\r\n svar: \"a) Bredden \u00e4r 8,2 cm b) Ungef\u00e4r 10,9 cm.\",\r\n niva: \"A\",\r\n poang: [0,1,3],\r\n digital: true,\r\n underkategori: [\"rotation\"],\r\n kod: \"b7\"\r\n },\r\n {\r\n fraga: `Grafen till funktionen \\\\(f(x)=x^2\\\\) roteras runt \\\\(x\\\\)-axeln i intervallet \\\\(x=0\\\\) till \\\\(x=2\\\\). Best\u00e4m volymen f\u00f6r kroppen som uppst\u00e5r. Svara exakt`,\r\n svar: \"\\\\(\\\\frac{32pi}{5} volymenheter\",\r\n niva: \"E\",\r\n poang: [3,0,0],\r\n underkategori: [\"rotation\"],\r\n kod: \"b8\"\r\n },\r\n {\r\n fraga: `Grafen till funktionen \\\\(f(x)=e^x\\\\) roteras runt \\\\(x\\\\)-axeln i intervallet \\\\(x=0\\\\) till \\\\(x=1\\\\). Best\u00e4m volymen f\u00f6r kroppen som uppst\u00e5r. Svara exakt`,\r\n svar: \"\\\\(\\\\pi \\\\cdot \\\\frac{e^2-1}{2}\\\\) volymenheter\",\r\n niva: \"E\",\r\n poang: [3,0,0],\r\n underkategori: [\"rotation\"],\r\n kod: \"b9\"\r\n },\r\n {\r\n fraga: `Grafen till funktionen \\\\(f(x)=\\\\frac{1}{\\\\sqrt{x}}\\\\) roteras runt \\\\(x\\\\)-axeln i intervallet \\\\(x=1\\\\) till \\\\(x=e\\\\). Best\u00e4m volymen f\u00f6r kroppen som uppst\u00e5r. Svara exakt`,\r\n svar: \"\\\\(\\\\pi\\\\) volymenheter\",\r\n niva: \"C\",\r\n poang: [2,1,0],\r\n underkategori: [\"rotation\"],\r\n kod: \"b10\"\r\n },\r\n {\r\n fraga: `<p> Under en sn\u00f6ig dag i Lund faller det sn\u00f6 med en hastighet \\\\(S(t)\\\\) cm\/timme enligt\r\nfunktionen d\u00e4r \\\\(t\\\\) \u00e4r timmar fr\u00e5n kl.12.00 <\/p> <p> \\\\(S(t)=4+2sin(\\\\frac{\\\\pi t}{6}) \\\\cdot e^{-0,1t}\\\\) <\/p> <p> a) Hur m\u00e5nga timmar efter 12.00 sn\u00f6ar det som mest? Endast svar kr\u00e4vs. <\/p>\r\nb) Hur m\u00e5nga cm sn\u00f6ar det totalt mellan klockan 12:00 och 18:00?`,\r\n svar: \"a) 2,63 timmar efter 12.00 b) 29,7 cm.\",\r\n niva: \"C\",\r\n poang: [1,2,0],\r\n digital: true, \r\n underkategori: [\"integralproblem\"],\r\n kod: \"b11\"\r\n },\r\n {\r\n fraga: `Grafen till funktionen \\\\(f(x)=x^2\\\\) roteras runt \\\\(y\\\\)-axeln. i intervallet \\\\(y=1\\\\) och \\\\(y=3\\\\). Best\u00e4m volymen av den kropp som uppst\u00e5r. `,\r\n svar: \"\\\\(4\\\\pi\\\\) volymenheter\",\r\n niva: \"C\",\r\n poang: [1,1,0],\r\n underkategori: [\"rotation\"],\r\n kod: \"b12\"\r\n },\r\n {\r\n fraga: `Betrakta funktionen \\\\(y=ax^2\\\\)\r\n, \\\\(0 \\\\leq x \\\\leq a\\\\), d\u00e4r \\\\(a > 0\\\\) \u00e4r en konstant. Om grafen till funktionen\r\nroterar runt x-axeln respektive y-axeln skapas tv\u00e5 olika rotationskroppar. Det finns ett v\u00e4rde p\u00e5\r\n\\\\(a\\\\) s\u00e5 att de b\u00e5da kropparna f\u00e5r samma volym. Best\u00e4m detta \\\\(a\\\\).`,\r\n svar: \"\\\\(a=\\\\sqrt{\\\\frac{5}{2}}=\\\\frac{sqrt{10}}{2}\\\\)\",\r\n niva: \"A\",\r\n poang: [0,0,2],\r\n underkategori: [\"rotation\"],\r\n kod: \"b13\"\r\n },\r\n {\r\n fraga: `Grafen till funktionen \\\\(f(x)=\\\\sqrt{x}\\\\) roteras runt \\\\(y\\\\)-axeln. i intervallet \\\\(y=1\\\\) och \\\\(y=3\\\\). Best\u00e4m volymen av den kropp som uppst\u00e5r. `,\r\n svar: \"\\\\(\\\\frac{242\\\\pi}{5}\\\\) volymenheter\",\r\n niva: \"C\",\r\n poang: [0,2,0],\r\n underkategori: [\"rotation\"],\r\n kod: \"b14\"\r\n },\r\n {\r\n fraga: `Ett omr\u00e5de begr\u00e4nsas av grafen till funktionen \\\\(f(x)=\\\\frac{1}{\\\\sqrt{x}}\\\\) och intevallet \\\\(1 \\\\leq x \\\\leq 9\\\\). Detta omr\u00e5de roteras runt \\\\(x\\\\)-axeln och skapar en rotationskropp \\\\(R\\\\). F\u00f6r vilket v\u00e4rde p\u00e5 \\\\(a\\\\) konstrueras en rotationskropp f\u00f6r intervallet \\\\(1 \\\\leq x \\\\leq a\\\\) som f\u00e5r volymen \\\\(\\\\frac{R}{2}\\\\)? Svara exakt.`,\r\n svar: \"\\\\(a=3\\\\)\",\r\n niva: \"A\",\r\n poang: [0,0,3],\r\n underkategori: [\"rotation\"],\r\n kod: \"b15\"\r\n },\r\n {\r\n fraga: `Grafen till funktionen \\\\(f(x)=x^2\\\\) roteras kring \\\\(x\\\\)-axeln i intervallet \\\\(0 \\\\leq x \\\\leq 2\\\\). Best\u00e4m volymen f\u00f6r den rotationsvolymen.`,\r\n svar: \"\\\\(\\\\frac{32\\\\pi}{5}\\\\)\",\r\n niva: \"E\",\r\n poang: [3,0,0],\r\n underkategori: [\"rotation\"],\r\n kod: \"b16\"\r\n }\r\n ],\r\n\r\n komplex: [\r\n {\r\n fraga: `<p>Vi definierar de komplexa talen \\\\(z_1=3+2i\\\\) och \\\\(z_2=-3+4i\\\\). Best\u00e4m f\u00f6ljande <\/p>\r\n <p> a) \\\\(Re(z_1)\\\\) <\/p> <p> b) \\\\(z_1+z_2\\\\) <\/p> c) \\\\(\\\\mid z_2 \\\\mid \\\\)`,\r\n svar: \"a) \\\\(Re(z_1)=3\\\\) b) \\\\(6i\\\\) c)\\\\(\\\\mid z_2 \\\\mid =5 \\\\) \",\r\n niva: \"E\",\r\n poang: [3,0,0],\r\n underkategori: [\"komplexa\"],\r\n kod: \"k1\"\r\n },\r\n {\r\n fraga: `<p>Vi definierar det komplexa talet \\\\(z=4(cos(\\\\frac{\\\\pi}{4})+isin(\\\\frac{\\\\pi}{4}))\\\\) Best\u00e4m f\u00f6ljande <\/p>\r\n <p> a) \\\\(arg(z)\\\\) <\/p> <p> b) \\\\(\\\\mid z \\\\mid \\\\)`,\r\n svar: \"a) \\\\(arg(z)=\\\\frac{\\\\pi}{4}\\\\) b) \\\\(\\\\mid z \\\\mid =4 \\\\) \",\r\n niva: \"E\",\r\n poang: [2,0,0],\r\n underkategori: [\"komplexa\"],\r\n kod: \"k2\"\r\n },\r\n {\r\n fraga: `<p>Vi definierar det komplexa talet \\\\(z=2(cos(\\\\frac{\\\\pi}{3})+isin(\\\\frac{\\\\pi}{3}))\\\\) Best\u00e4m f\u00f6ljande <\/p>\r\n <p> a) \\\\(arg(z)\\\\) <\/p> <p> b) \\\\(\\\\mid z \\\\mid \\\\) <\/p> c) \\\\(z^5\\\\)`,\r\n svar: \"a) \\\\(arg(z)=\\\\frac{\\\\pi}{3}\\\\) b) \\\\(\\\\mid z \\\\mid =2 \\\\) c) \\\\(z^5=32(cos(\\\\frac{5\\\\pi}{3})+isin(\\\\frac{5\\\\pi}{3}))\\\\)\",\r\n niva: \"C\",\r\n poang: [2,1,0],\r\n underkategori: [\"komplexa\"],\r\n kod: \"k3\"\r\n },\r\n {\r\n fraga: `Skriv f\u00f6ljande komplexa tal \\\\(z=1+\\\\sqrt3i\\\\) p\u00e5 pol\u00e4r form `,\r\n svar: \"\\\\(z=2(cos(\\\\frac{\\\\pi}{3})+i sin(\\\\frac{\\\\pi}{3}))\\\\)\",\r\n niva: \"C\",\r\n poang: [1,1,0],\r\n underkategori: [\"komplexa\"],\r\n kod: \"k4\"\r\n },\r\n {\r\n fraga: `Skriv f\u00f6ljande komplexa tal \\\\(z=\\\\sqrt{5}e^{\\\\frac{5\\\\pi}{12}i}\\\\) p\u00e5 pol\u00e4r form `,\r\n svar: \"\\\\(z=\\\\sqrt{5}(cos(\\\\frac{5\\\\pi}{12})+isin(\\\\frac{5\\\\pi}{12}))\\\\)\",\r\n niva: \"C\",\r\n poang: [0,1,0],\r\n underkategori: [\"komplexa\"],\r\n kod: \"k5\"\r\n },\r\n {\r\n fraga: `F\u00f6r ett komplext tal \\\\(z\\\\) vet du f\u00f6ljande \\\\(z+i+4=7i-2\\\\)`,\r\n svar: \"\\\\(z=6i-6\\\\)\",\r\n niva: \"E\",\r\n poang: [2,0,0],\r\n underkategori: [\"komplexa\"],\r\n kod: \"k6\"\r\n },\r\n {\r\n fraga: `Best\u00e4m \\\\(\\\\mid z \\\\mid\\\\) f\u00f6r \\\\(z=2(cos(\\\\frac{\\\\pi}{3})+isin(\\\\frac{\\\\pi}{6}))\\\\)`,\r\n svar: \"\\\\(\\\\mid z \\\\mid = \\\\sqrt2\",\r\n niva: \"A\",\r\n poang: [0,0,1],\r\n underkategori: [\"komplexa\"],\r\n kod: \"k6\"\r\n },\r\n {\r\n fraga: `Visa att \\\\(arg(\\\\frac{1}{z})=-arg(z)\\\\)`,\r\n svar: \"Se l\u00f6sningsf\u00f6rslag\",\r\n niva: \"A\",\r\n poang: [0,0,1],\r\n losningsforslag: `<img decoding=\"async\" src=\"https:\/\/mahifi.se\/wp-content\/uploads\/2025\/11\/Skarmbild-2025-11-17-073657.png\" alt=\"L\u00f6sningsf\u00f6rslag\">`,\r\n underkategori: [\"komplexa\"],\r\n kod: \"k7\"\r\n },\r\n {\r\n fraga: `Visa att \\\\(z\\\\cdot \\\\bar{z} = \\\\mid z \\\\mid^2\\\\)`,\r\n svar: \"Se l\u00f6sningsf\u00f6rlag\",\r\n niva: \"C\",\r\n poang: [0,2,0],\r\n losningsforslag: `<img decoding=\"async\" src=\"https:\/\/mahifi.se\/wp-content\/uploads\/2025\/11\/Skarmbild-2025-11-17-075631.png\" alt=\"L\u00f6sningsf\u00f6rslag\">`,\r\n underkategori: [\"komplexa\"],\r\n kod: \"k8\"\r\n },\r\n {\r\n fraga: `<p>Vi definierar det komplexa talet \\\\(z=6+8i\\\\) Best\u00e4m f\u00f6ljande <\/p> <p> a) \\\\(\\\\bar{z}\\\\) <\/p> b) \\\\(\\\\mid z \\\\mid\\\\) `,\r\n svar: \"a) \\\\(6-8i\\\\) b) 10\",\r\n niva: \"E\",\r\n poang: [2,0,0],\r\n underkategori: [\"komplexa\"],\r\n kod: \"k9\"\r\n },\r\n {\r\n fraga: `Best\u00e4m talet \\\\(z\\\\) om f\u00f6ljande likhet g\u00e4ller \\\\(z(1+i)+i=1+4i\\\\) `,\r\n svar: \"\\\\(z=2+i\\\\)\",\r\n niva: \"C\",\r\n poang: [1,1,0],\r\n underkategori: [\"komplexa\"],\r\n kod: \"k10\"\r\n },\r\n {\r\n fraga: `L\u00f6s ekvationen \\\\(z\\\\cdot (cos(30^{\\\\circ})+isin(30^{\\\\circ}))=i\\\\) `,\r\n svar: \"\\\\(z=cos(60^{\\\\circ})+isin(60^{\\\\circ})\\\\)\",\r\n niva: \"C\",\r\n poang: [0,2,0],\r\n underkategori: [\"komplexa\"],\r\n kod: \"k11\"\r\n },\r\n {\r\n fraga: `Cosinus g\u00e5r att definiera p\u00e5 flera olika s\u00e4tt. Visa att man kan definiera cosv med hj\u00e4lp av komplexa tal med hj\u00e4lp av f\u00f6ljande likhet \\\\(cosv=\\\\frac{e^{vi}+e^{-vi}}{2}\\\\) `,\r\n svar: \"Se l\u00f6sningsf\u00f6rslag\",\r\n niva: \"A\",\r\n poang: [0,0,2],\r\n losningsforslag: `<img decoding=\"async\" src=\"https:\/\/mahifi.se\/wp-content\/uploads\/2025\/11\/Skarmbild-2025-11-17-085646.png\" alt=\"L\u00f6sningsf\u00f6rslag\">`,\r\n underkategori: [\"komplexa\"],\r\n kod: \"k12\"\r\n },\r\n {\r\n fraga: `Skriv det komplexa talet \\\\(z=1-\\\\sqrt3i\\\\) p\u00e5 exponentialform `,\r\n svar: \"\\\\(z=2e^{-\\\\frac{\\\\pi}{3}i}\\\\) eller \\\\(z=2e^{\\\\frac{5\\\\pi}{3}i}\\\\)\",\r\n niva: \"C\",\r\n poang: [0,2,0],\r\n underkategori: [\"komplexa\"],\r\n kod: \"k13\"\r\n },\r\n {\r\n fraga: `L\u00f6s ekvationen \\\\(x^2-4x+5=0\\\\)`,\r\n svar: \"\\\\(x=2\\\\pm i\\\\)\",\r\n niva: \"E\",\r\n poang: [1,0,0],\r\n underkategori: [\"komplexa\"],\r\n kod: \"k14\"\r\n },\r\n {\r\n fraga: `L\u00f6s ekvationen \\\\(z^3=-1\\\\) fullst\u00e4ndigt.`,\r\n svar: \"<p> \\\\(z_1=cos(\\\\frac{\\\\pi}{3})+isin(\\\\frac{\\\\pi}{3})\\\\) <\/p> <p> \\\\(z_2=cos(\\\\pi)+isin(\\\\pi)\\\\) <\/p> \\\\(z_3=cos(\\\\frac{5\\\\pi}{3})+isin(\\\\frac{5\\\\pi}{3})\\\\)\",\r\n niva: \"C\",\r\n poang: [0,3,0],\r\n losningsforslag: `<img decoding=\"async\" src=\"https:\/\/mahifi.se\/wp-content\/uploads\/2025\/11\/Skarmbild-2025-11-17-090412.png\" alt=\"L\u00f6sningsf\u00f6rslag\">`,\r\n underkategori: [\"komplexa\"],\r\n kod: \"k15\"\r\n },\r\n {\r\n fraga: `Visa att om man multiplicerar ett komplext tal med dess konjugat f\u00e5r man alltid ett reellt tal.`,\r\n svar: \"Skriv upp \\\\(z=a+bi\\\\) och \\\\(\\\\bar{z}=a-bi\\\\). Multiplicera sedan talen och visa att vi f\u00e5r \\\\(a^2+b^2\\\\) som alltid \u00e4r reellt.\",\r\n niva: \"C\",\r\n poang: [1,1,0],\r\n underkategori: [\"komplexa\"],\r\n kod: \"k16\"\r\n },\r\n {\r\n fraga: ` Visa att \\\\((cos(\\\\frac{\\\\pi}{3})+isin(\\\\frac{\\\\pi}{3}))^6=1\\\\)`,\r\n svar: \"Se l\u00f6sningsf\u00f6rslag\",\r\n niva: \"C\",\r\n poang: [1,1,0],\r\n losningsforslag: `<img decoding=\"async\" src=\"https:\/\/mahifi.se\/wp-content\/uploads\/2025\/11\/Skarmbild-2025-11-17-091656.png\" alt=\"L\u00f6sningsf\u00f6rslag\">`,\r\n underkategori: [\"komplexa\"],\r\n kod: \"k17\"\r\n },\r\n {\r\n fraga: `<p> F\u00f6r vilka v\u00e4rden p\u00e5 \\\\(k\\\\) blir f\u00f6ljande uttryck reellt?<\/p> \\\\((cos(\\\\frac{\\\\pi}{3})+isin(\\\\frac{\\\\pi}{3}))^k\\\\)`,\r\n svar: \"\\\\(k=6\\\\cdot n\\\\) eller \\\\(k=3+6\\\\cdot n\\\\)\",\r\n niva: \"A\",\r\n poang: [0,1,1],\r\n losningsforslag: `<img decoding=\"async\" src=\"https:\/\/mahifi.se\/wp-content\/uploads\/2025\/11\/Skarmbild-2025-11-17-091712.png\" alt=\"L\u00f6sningsf\u00f6rslag\">`,\r\n underkategori: [\"komplexa\"],\r\n kod: \"k18\"\r\n },\r\n {\r\n fraga: `<p>Vi definierar de komplexa talen \\\\(z_1=2(cos(\\\\frac{\\\\pi}{3})+isin(\\\\frac{\\\\pi}{3}))\\\\) och \\\\(z_2=4(cos(\\\\frac{\\\\pi}{2})+isin(\\\\frac{\\\\pi}{2}))\\\\). Best\u00e4m f\u00f6ljande <\/p> <p>a) \\\\(z_1 \\\\cdot z_2\\\\) <\/p> <p> \\\\(\\\\frac{z_2}{z_1}\\\\)`,\r\n svar: \"a) \\\\(z_1 \\\\cdot z_2=8(cos(\\\\frac{5\\\\pi}{6})+isin(\\\\frac{5\\\\pi}{6}))\\\\) b) \\\\(2(cos(\\\\frac{\\\\pi}{6})+isin(\\\\frac{\\\\pi}{6}))\\\\)\",\r\n niva: \"E\",\r\n poang: [4,0,0],\r\n underkategori: [\"komplexa\"],\r\n kod: \"k19\"\r\n },\r\n {\r\n fraga: `Best\u00e4m \\\\((1+i)^6\\\\) p\u00e5 valfri form`,\r\n svar: \"\\\\((1+i)^6=8(cos(\\\\frac{3\\\\pi}{2})+isin(\\\\frac{3\\\\pi}{2}))=8e^{\\\\frac{3\\\\pi}{2}}\\\\)\",\r\n niva: \"C\",\r\n poang: [0,3,0],\r\n underkategori: [\"komplexa\"],\r\n kod: \"k20\"\r\n },\r\n {\r\n fraga: `Visa att \\\\(i^i\\\\) \u00e4r ett reellt tal`,\r\n svar: \"Tips: Skriv om \\\\(i=e^{\\\\frac{\\\\pi}{2}i}\\\\)\",\r\n niva: \"A\",\r\n poang: [0,0,2],\r\n underkategori: [\"komplexa\"],\r\n kod: \"k21\"\r\n },\r\n {\r\n fraga: `Best\u00e4m \\\\(\\\\mid z \\\\mid\\\\) f\u00f6r f\u00f6ljande komplexa tal \\\\(z=2isin(\\\\frac{\\\\pi}{2})+cos(\\\\frac{\\\\pi}{2})+i+1\\\\)`,\r\n svar: \"\\\\(\\\\mid z \\\\mid=5\\\\)\",\r\n niva: \"A\",\r\n poang: [0,0,1],\r\n underkategori: [\"komplexa\"],\r\n kod: \"k22\"\r\n },\r\n {\r\n fraga: `Best\u00e4m \\\\(\\\\mid z \\\\mid\\\\) f\u00f6r f\u00f6ljande komplexa tal \\\\(z=(\\\\frac{1}{2}+\\\\frac{\\\\sqrt3i}{2})^{100}\\\\)`,\r\n svar: \"\\\\(\\\\mid z \\\\mid=1\\\\) Tips: Skriv om \",\r\n niva: \"A\",\r\n poang: [0,1,1],\r\n underkategori: [\"komplexa\"],\r\n kod: \"k23\"\r\n },\r\n {\r\n fraga: `Nedan ser du en av l\u00f6sningarna till ekvationen \\\\(z^{36}=a\\\\) i det komplexa talplanet, d\u00e4r \ud835\udc4e\r\n\u00e4r n\u00e5got komplext tal. Best\u00e4m hur m\u00e5nga l\u00f6sningar till ekvationen som ligger i\r\nintervallet \\\\(100^{\\\\circ} < arg(z) < 130^{\\\\circ}\\\\). Motivera. <img decoding=\"async\" src=\"https:\/\/mahifi.se\/wp-content\/uploads\/2025\/11\/Skarmbild-2025-11-17-094320.png\" alt=\"L\u00f6sningsf\u00f6rslag\">`,\r\n svar: \"3 st. \\\\(102^{\\\\circ}, 112^{\\\\circ}, 122^{\\\\circ}\\\\)\",\r\n niva: \"A\",\r\n poang: [0,1,1],\r\n losningsforslag: `<img decoding=\"async\" src=\"https:\/\/mahifi.se\/wp-content\/uploads\/2025\/11\/Skarmbild-2025-11-17-094344.png\" alt=\"L\u00f6sningsf\u00f6rslag\">`,\r\n underkategori: [\"komplexa\"],\r\n kod: \"k24\"\r\n },\r\n {\r\n fraga: `<p>Joakim definierar f\u00f6ljande komplexa tal \\\\(z=cos(\\\\frac{\\\\pi}{2})+isin(\\\\frac{\\\\pi}{2})\\\\). Best\u00e4m f\u00f6ljande <\/p> <p> a) \\\\(arg(z)\\\\) <\/p> b) Joakim menar att \\\\(z^{25}=z\\\\) unders\u00f6k om han har r\u00e4tt.`,\r\n svar: \"a) \\\\(arg(z)=\\\\frac{\\\\pi}{2} b) Det st\u00e4mmer. \",\r\n niva: \"C\",\r\n poang: [1,2,0],\r\n losningsforslag:`<img decoding=\"async\" src=\"https:\/\/mahifi.se\/wp-content\/uploads\/2025\/11\/Skarmbild-2025-11-17-095341.png\" alt=\"L\u00f6sningsf\u00f6rslag\">`,\r\n underkategori: [\"komplexa\"],\r\n kod: \"k25\"\r\n },\r\n {\r\n fraga: `F\u00f6renkla f\u00f6ljande uttryck \\\\((2+i)(1-i)\\\\)`,\r\n svar: \"\\\\(3-i\\\\)\",\r\n niva: \"E\",\r\n poang: [1,0,0],\r\n underkategori: [\"komplexa\"],\r\n kod: \"k26\"\r\n },\r\n {\r\n fraga: `<p>Vi definierar f\u00f6ljande komplexa tal \\\\(z=4e^{\\\\frac{4\\\\pi}{3}i}\\\\). Best\u00e4m f\u00f6ljande <\/p> <p> a) \\\\(\\\\mid z \\\\mid \\\\) <\/p> b) \\\\(arg(z)\\\\)`,\r\n svar: \"a) \\\\(\\\\mid z \\\\mid =4\\\\) b) \\\\(arg(z)=\\\\frac{4\\\\pi}{3}\\\\)\",\r\n niva: \"E\",\r\n poang: [2,0,0],\r\n underkategori: [\"komplexa\"],\r\n kod: \"k27\"\r\n },\r\n {\r\n fraga: `<p>Som f\u00f6ljande uttryck som ett komplexa tal p\u00e5 formen \\\\(z=a+bi\\\\) <\/p> \\\\(\\\\frac{7-i}{3+i}\\\\) `,\r\n svar: \"\\\\(2-i\\\\)\",\r\n niva: \"E\",\r\n poang: [2,0,0],\r\n underkategori: [\"komplexa\"],\r\n kod: \"k28\"\r\n },\r\n {\r\n fraga: `Joakim p\u00e5st\u00e5r att om \\\\(a\\\\) \u00e4r ett reellt tal st\u00f6rre \u00e4n 1 kommer \\\\(a^i\\\\) alltid resultera i ett komplext tal. Unders\u00f6k om han har r\u00e4tt.`,\r\n svar: \"Se l\u00f6sningsf\u00f6rslag\",\r\n niva: \"A\",\r\n losningsforslag: `<img decoding=\"async\" src=\"https:\/\/mahifi.se\/wp-content\/uploads\/2025\/11\/Skarmbild-2025-11-17-101512.png\" alt=\"L\u00f6sningsf\u00f6rslag\">`,\r\n poang: [0,0,3],\r\n underkategori: [\"komplexa\"],\r\n kod: \"k29\"\r\n },\r\n {\r\n fraga: `Joakim vill veta vilka komplexa tal som har f\u00f6ljande samband. Rita upp dem i det komplexa talplanet \\\\(\\\\mid z+\\\\bar{z} \\\\mid =\\\\mid z-\\\\bar{z} \\\\mid\\\\)`,\r\n svar: \"Se l\u00f6sningsf\u00f6rslag\",\r\n niva: \"A\",\r\n losningsforslag: `<img decoding=\"async\" src=\"https:\/\/mahifi.se\/wp-content\/uploads\/2025\/11\/Skarmbild-2025-11-17-101832.png\" alt=\"L\u00f6sningsf\u00f6rslag\">`,\r\n poang: [0,0,2],\r\n underkategori: [\"komplexa\"],\r\n kod: \"k30\"\r\n },\r\n {\r\n fraga: `Visa att \\\\(z_2-z_1=0\\\\) f\u00f6r f\u00f6ljande komplexa tal \\\\(z_1=1+i\\\\) och \\\\(z_2=\\\\sqrt2(cos(\\\\frac{\\\\pi}{4})+isin(\\\\frac{\\\\pi}{4}))\\\\)`,\r\n svar: \"Skriv om d\u00e5 de komplexa talen \u00e4r p\u00e5 samma form och visa sedan att de \u00e4r samma.\",\r\n niva: \"C\",\r\n poang: [0,1,0],\r\n underkategori: [\"komplexa\"],\r\n kod: \"k31\"\r\n },\r\n {\r\n fraga: `Best\u00e4m f\u00f6r vilka komplexa tal som f\u00f6ljande likhet st\u00e4mmer. Skriv det som ett f\u00f6renklat samband eller rita upp det i det komplexa talplanet \\\\(\\\\mid z \\\\mid =Im(z)+1\\\\)`,\r\n svar: \"Om det komplexa talet skrivs p\u00e5 formen \\\\(z=a+bi\\\\) kommer f\u00f6ljande likhet g\u00e4lla de aktuella komplexa talen \\\\(b=\\\\frac{a^2-1}{2}\\\\)\",\r\n niva: \"A\",\r\n poang: [0,0,2],\r\n underkategori: [\"komplexa\"],\r\n kod: \"k32\"\r\n },\r\n {\r\n fraga: `Best\u00e4m samtliga komplexa tal \\\\(z\\\\) d\u00e4r f\u00f6ljande kvot blir reell \\\\(\\\\frac{z-2i}{z+2i}\\\\)`,\r\n svar: \"Det komplexa talet \\\\(z\\\\) f\u00e5r inte ha n\u00e5gon reell dvs. \\\\(Re(z)=0\\\\)\",\r\n niva: \"A\",\r\n poang: [0,0,2],\r\n underkategori: [\"komplexa\"],\r\n kod: \"k33\"\r\n },\r\n {\r\n fraga: `<p> Vi definierar det komplexa talet \\\\(z=5-2i\\\\) Best\u00e4m f\u00f6ljande <\/p> <p> a) \\\\(Im(z)\\\\)<\/p> b) Best\u00e4m \\\\(z^2\\\\) p\u00e5 formen \\\\(z=a+bi\\\\)`,\r\n svar: \"a) \\\\(Im(z)=-2\\\\) b) \\\\(z=21-20i\\\\)\",\r\n niva: \"E\",\r\n poang: [3,0,0],\r\n underkategori: [\"komplexa\"],\r\n kod: \"k34\"\r\n }\r\n ],\r\n geogebra: [\r\n{\r\n fraga: `<p>F\u00f6ljande funktion kan du inte derivera algebraiskt i den h\u00e4r kursen \\\\(f(x)=x\\\\cdot e^x\\\\).<\/p> Best\u00e4m \\\\(f'(1)\\\\) med hj\u00e4lp av geogebra, svara med tv\u00e5 decimaler.`,\r\n svar: `\\\\(f'(1)=5,43\\\\)`,\r\n niva: \"E\",\r\n poang: [1,0,0],\r\n digital: true,\r\n underkategori: [\"geogebraderivata\"],\r\n kod: \"g1\"\r\n}\r\n ]\r\n };\r\n \r\n let visaKoder = false;\r\n\r\n\/\/ ---- Filterfunktion ----\r\nfunction getValdaFragor() {\r\n const valdaAmnen = Object.keys(fragor).filter(a => document.getElementById(a)?.checked);\r\n const valdaNivaer = [\"E\",\"C\",\"A\"].filter(n => document.getElementById(n).checked);\r\n let lista = [];\r\n for (let amne of valdaAmnen) {\r\n lista = lista.concat(\r\n fragor[amne].filter(q=>valdaNivaer.includes(q.niva)).map(q=>({...q, amne}))\r\n );\r\n }\r\n return lista;\r\n}\r\n\r\n\/\/ ---- Ut\u00f6kad filterfunktion ----\r\nfunction getValdaFragorUtokad() {\r\n const valdaUnderkategorier = Array.from(document.querySelectorAll(\".underkategori:checked\")).map(cb => cb.value);\r\n const valdaNivaer = [\"E\",\"C\",\"A\"].filter(n => document.getElementById(\"utokad\" + n).checked);\r\n \r\n let lista = [];\r\n for (let amne in fragor) {\r\n lista = lista.concat(\r\n fragor[amne].filter(q =>\r\n valdaNivaer.includes(q.niva) &&\r\n (Array.isArray(q.underkategori)\r\n ? q.underkategori.some(u => valdaUnderkategorier.includes(u))\r\n : valdaUnderkategorier.includes(q.underkategori))\r\n ).map(q => ({...q, amne}))\r\n );\r\n }\r\n return lista;\r\n}\r\n\r\n\/\/ ---- Sortering ----\r\nfunction sorteraEfterNivaOchPoang(lista) {\r\n const ordning = { \"E\": 1, \"C\": 2, \"A\": 3 };\r\n return lista.sort((a,b)=>{\r\n const nivaSkillnad = ordning[a.niva]-ordning[b.niva];\r\n if(nivaSkillnad!==0) return nivaSkillnad;\r\n const poangA = a.poang.reduce((acc,x)=>acc+x,0);\r\n const poangB = b.poang.reduce((acc,x)=>acc+x,0);\r\n return poangA-poangB;\r\n });\r\n}\r\n\r\n\/\/ ---- Rendera fr\u00e5gor ----\r\nfunction renderFragor(lista) {\r\n const container = document.getElementById(\"uppgifterContainer\");\r\n container.innerHTML = \"\";\r\n\r\n lista.forEach((q, i) => {\r\n const div = document.createElement(\"div\");\r\n div.className = \"uppgift\";\r\n div.setAttribute(\"data-amne\", q.amne);\r\n div.setAttribute(\"data-niva\", q.niva);\r\n div.innerHTML = `\r\n <b class=\"byt-uppgift\" onclick=\"bytUppgift(${i})\">Uppgift ${i + 1}:<\/b> ${q.fraga}\r\n <span class=\"poang\">(${q.poang.join(\"\/\")})<\/span>\r\n ${visaKoder && q.kod ? `<span style=\"color:#555; font-size:13px; margin-left:8px;\">[${q.kod}]<\/span>` : \"\"}\r\n ${q.digital ? '<span class=\"digital-icon\">\ud83e\uddee<\/span>' : \"\"}\r\n\r\n <div>\r\n <button onclick=\"toggleSvar(this)\">Visa facit<\/button>\r\n <div class=\"facit\" style=\"display:none;\">${q.svar}<\/div>\r\n ${\r\n q.losningsforslag\r\n ? `\r\n <button onclick=\"toggleLosning(this)\">Visa l\u00f6sningsf\u00f6rslag<\/button>\r\n <div class=\"losningsforslag\" style=\"display:none;\">${q.losningsforslag}<\/div>`\r\n : \"\"\r\n }\r\n <\/div>\r\n `;\r\n container.appendChild(div);\r\n });\r\n \r\n\r\n \/\/ --- visa kod l\u00e4ngst ner ---\r\n const kodLista = lista.map(q => q.kod);\r\n const kodStrang = kodLista.join(\"\");\r\n\r\n const kodDiv = document.createElement(\"div\");\r\n kodDiv.style.marginTop = \"20px\";\r\n kodDiv.innerHTML = `\r\n <b>Kod f\u00f6r denna upps\u00e4ttning:<\/b>\r\n <span id=\"kodStrang\" style=\"font-family:monospace;\">${kodStrang}<\/span>\r\n <button id=\"kopieraKodBtn\" style=\"margin-left:8px;padding:4px 8px;font-size:12px;\">Kopiera<\/button>\r\n `;\r\n container.appendChild(kodDiv);\r\n\r\n \/\/ Kopieringsfunktion\r\n document.getElementById(\"kopieraKodBtn\").addEventListener(\"click\", () => {\r\n const kodText = document.getElementById(\"kodStrang\").textContent;\r\n navigator.clipboard.writeText(kodText);\r\n const btn = document.getElementById(\"kopieraKodBtn\");\r\n btn.textContent = \"Kopierad!\";\r\n setTimeout(() => (btn.textContent = \"Kopiera\"), 1500);\r\n });\r\n\r\n MathJax.typesetPromise();\r\n}\r\n\r\n\/\/ ---- Byt uppgift ----\r\nfunction bytUppgift(index) {\r\n const container=document.getElementById(\"uppgifterContainer\");\r\n const div=container.children[index];\r\n const amne=div.getAttribute(\"data-amne\");\r\n const niva=div.getAttribute(\"data-niva\");\r\n let kandidater=fragor[amne].filter(q=>q.niva===niva);\r\n if(kandidater.length===0) return;\r\n const ny=kandidater[Math.floor(Math.random()*kandidater.length)];\r\n div.innerHTML=`\r\n <b class=\"byt-uppgift\" onclick=\"bytUppgift(${index})\">Uppgift ${index+1}:<\/b> ${ny.fraga}\r\n <span class=\"poang\">(${ny.poang.join(\"\/\")})<\/span>\r\n ${ny.digital ? '<span class=\"digital-icon\">\ud83e\uddee<\/span>' : \"\"}\r\n <div>\r\n <button onclick=\"toggleSvar(this)\">Visa facit<\/button>\r\n <div class=\"facit\" style=\"display:none;\">${ny.svar}<\/div>\r\n <\/div>\r\n `;\r\n MathJax.typesetPromise();\r\n}\r\n\r\n\/\/ ---- Toggle ----\r\nfunction toggleSvar(btn){const f=btn.nextElementSibling;f.style.display=(f.style.display===\"none\"?\"block\":\"none\");btn.textContent=f.style.display===\"none\"?\"Visa facit\":\"D\u00f6lj facit\";}\r\nfunction toggleLosning(btn){const l=btn.nextElementSibling;l.style.display=(l.style.display===\"none\"?\"block\":\"none\");btn.textContent=l.style.display===\"none\"?\"Visa l\u00f6sningsf\u00f6rslag\":\"D\u00f6lj l\u00f6sningsf\u00f6rslag\";}\r\n\r\n\/\/ ---- Knapph\u00e4ndelser ----\r\ndocument.getElementById(\"genereraBtn\").addEventListener(\"click\",()=>{\r\n let lista=getValdaFragor();\r\n if(lista.length===0){alert(\"V\u00e4lj minst en kategori och en sv\u00e5righetsgrad.\");return;}\r\n const antal=Math.min(document.getElementById(\"antal\").value,lista.length);\r\n const slumpade=lista.sort(()=>0.5-Math.random()).slice(0,antal);\r\n renderFragor(sorteraEfterNivaOchPoang(slumpade));\r\n});\r\ndocument.getElementById(\"genereraAllaBtn\").addEventListener(\"click\",()=>{\r\n let lista=getValdaFragor();\r\n if(lista.length===0){alert(\"V\u00e4lj minst en kategori och en sv\u00e5righetsgrad.\");return;}\r\n renderFragor(sorteraEfterNivaOchPoang(lista));\r\n});\r\n\r\n\/\/ ---- Popup f\u00f6r diagnos ----\r\ndocument.getElementById(\"skapaDiagnosBtn\").addEventListener(\"click\",()=>{document.getElementById(\"diagnosPopup\").style.display=\"block\";});\r\ndocument.getElementById(\"stangPopupBtn\").addEventListener(\"click\",()=>{document.getElementById(\"diagnosPopup\").style.display=\"none\";});\r\n\r\n\/\/ ---- Ut\u00f6kad s\u00f6kning ----\r\ndocument.getElementById(\"utokadBtn\").addEventListener(\"click\",()=>{\r\n document.getElementById(\"utokadPopup\").style.display=\"block\";\r\n});\r\ndocument.getElementById(\"stangUtokadBtn\").addEventListener(\"click\",()=>{\r\n document.getElementById(\"utokadPopup\").style.display=\"none\";\r\n});\r\ndocument.getElementById(\"visaKoderUtokad\").addEventListener(\"change\", (e) => {\r\n visaKoder = e.target.checked;\r\n});\r\n\r\ndocument.getElementById(\"startUtokadBtn\").addEventListener(\"click\",()=>{\r\n let lista = getValdaFragorUtokad();\r\n if(lista.length===0){alert(\"V\u00e4lj minst en underkategori och en sv\u00e5righetsgrad.\");return;}\r\n const antal = Math.min(document.getElementById(\"utokadAntal\").value, lista.length);\r\n const slumpade = lista.sort(()=>0.5-Math.random()).slice(0, antal);\r\n renderFragor(sorteraEfterNivaOchPoang(slumpade));\r\n document.getElementById(\"utokadPopup\").style.display=\"none\";\r\n});\r\n\/\/ ---- Visa alla uppgifter i ut\u00f6kad s\u00f6kning ----\r\ndocument.getElementById(\"visaAllaUtokadBtn\").addEventListener(\"click\",()=>{\r\n let lista = getValdaFragorUtokad();\r\n if(lista.length===0){\r\n alert(\"V\u00e4lj minst en underkategori och en sv\u00e5righetsgrad.\");\r\n return;\r\n }\r\n renderFragor(sorteraEfterNivaOchPoang(lista));\r\n document.getElementById(\"utokadPopup\").style.display=\"none\";\r\n});\r\n\r\n\/\/ ---- Starta diagnos ----\r\ndocument.getElementById(\"laddaDiagnosKodBtn\").addEventListener(\"click\", () => {\r\n const kodInput = document.getElementById(\"diagnosKod\").value.trim();\r\n if (!kodInput) {\r\n alert(\"Ange en diagnoskod f\u00f6rst.\");\r\n return;\r\n }\r\n\r\n \/\/ Extrahera koder (t.ex. f1f2a4 \u2192 [\"f1\", \"f2\", \"a4\"])\r\n const koder = kodInput.match(\/[a-z]+\\d+\/gi);\r\n if (!koder || koder.length === 0) {\r\n alert(\"Ogiltig kod. Anv\u00e4nd format som f1f3f7a2.\");\r\n return;\r\n }\r\n\r\n let lista = [];\r\n\r\n \/\/ Hitta uppgifterna i fr\u00e5gebiblioteket\r\n for (const kod of koder) {\r\n for (const kategori in fragor) {\r\n const hittad = fragor[kategori].find(q => q.kod?.toLowerCase() === kod.toLowerCase());\r\n if (hittad) lista.push({ ...hittad, amne: kategori });\r\n }\r\n }\r\n\r\n if (lista.length === 0) {\r\n alert(\"Inga uppgifter hittades f\u00f6r den angivna koden.\");\r\n return;\r\n }\r\n\r\n \/\/ Sortera efter niv\u00e5 och rendera som diagnos\r\n lista = sorteraEfterNivaOchPoang(lista);\r\n\r\n const container = document.getElementById(\"uppgifterContainer\");\r\n container.innerHTML = \"\";\r\n lista.forEach((q, i) => {\r\n const div = document.createElement(\"div\");\r\n div.className = \"uppgift\";\r\n div.dataset.facit = q.svar;\r\n if (q.losningsforslag) div.dataset.losning = q.losningsforslag;\r\n div.setAttribute(\"data-amne\", q.amne);\r\n div.setAttribute(\"data-niva\", q.niva);\r\n\r\n div.innerHTML = `\r\n <b>Uppgift ${i + 1}:<\/b> ${q.fraga}\r\n <span class=\"poang\">(${q.poang.join(\"\/\")})<\/span>\r\n ${q.digital ? '<span class=\"digital-icon\">\ud83e\uddee<\/span>' : \"\"}\r\n `;\r\n container.appendChild(div);\r\n });\r\n\r\n \/\/ Visa knappen f\u00f6r facit\/l\u00f6sning\r\n const klarContainer = document.getElementById(\"diagnosKlarContainer\");\r\n klarContainer.innerHTML = `<button id=\"diagnosKlarBtn\">Visa facit\/Klar<\/button>`;\r\n document.getElementById(\"diagnosKlarBtn\").addEventListener(\"click\", () => {\r\n document.querySelectorAll(\".uppgift\").forEach(div => {\r\n const f = document.createElement(\"div\");\r\n f.className = \"facit\";\r\n f.style.display = \"block\";\r\n f.innerHTML = \"<b>Facit:<\/b> \" + div.dataset.facit;\r\n div.appendChild(f);\r\n if (div.dataset.losning) {\r\n const l = document.createElement(\"div\");\r\n l.className = \"losningsforslag\";\r\n l.style.display = \"block\";\r\n l.innerHTML = \"<b>L\u00f6sningsf\u00f6rslag:<\/b> \" + div.dataset.losning;\r\n div.appendChild(l);\r\n }\r\n });\r\n klarContainer.innerHTML = \"\";\r\n MathJax.typesetPromise();\r\n });\r\n\r\n document.getElementById(\"diagnosPopup\").style.display = \"none\";\r\n MathJax.typesetPromise();\r\n});\r\n\r\n\r\ndocument.getElementById(\"startDiagnosBtn\").addEventListener(\"click\",()=>{\r\n const epoang=+document.getElementById(\"epoang\").value;\r\n const cpoang=+document.getElementById(\"cpoang\").value;\r\n const apoang=+document.getElementById(\"apoang\").value;\r\n const slump=document.getElementById(\"slumpmasigt\").checked;\r\n document.getElementById(\"diagnosPopup\").style.display=\"none\";\r\n\r\n let lista=getValdaFragor();if(lista.length===0){alert(\"V\u00e4lj minst en kategori och en sv\u00e5righetsgrad.\");return;}\r\n let diagnosLista=[];\r\n\r\n if(slump){\r\n const antal=Math.min(document.getElementById(\"antal\").value,lista.length);\r\n diagnosLista=lista.sort(()=>0.5-Math.random()).slice(0,antal);\r\n } else {\r\n [\"E\",\"C\",\"A\"].forEach(niva=>{\r\n const m\u00e5l=niva===\"E\"?epoang:niva===\"C\"?cpoang:apoang;\r\n let uppgifter=lista.filter(q=>q.niva===niva);\r\n uppgifter.sort(()=>0.5-Math.random());\r\n let po\u00e4ngSum=0;\r\n for(let q of uppgifter){\r\n if(po\u00e4ngSum>=m\u00e5l) break;\r\n diagnosLista.push(q);\r\n po\u00e4ngSum += q.poang.reduce((s,x)=>s+x,0);\r\n }\r\n \/\/ \ud83d\udd39 Skapa diagnoskod baserat p\u00e5 uppgifterna som valts\r\n const diagnosKod = diagnosLista.map(q => q.kod || \"\").join(\"\");\r\n document.getElementById(\"diagnosKodVisning\").innerHTML = `\r\n <b>Diagnoskod:<\/b> <span style=\"font-family:monospace;\">${diagnosKod}<\/span>\r\n <button onclick=\"navigator.clipboard.writeText('${diagnosKod}')\" style=\"margin-left:8px;\">\ud83d\udccb Kopiera<\/button>\r\n `;\r\n\r\n });\r\n }\r\n\r\n diagnosLista=sorteraEfterNivaOchPoang(diagnosLista);\r\n\r\n \/\/ Rendera utan facit\/l\u00f6sning\r\n const container=document.getElementById(\"uppgifterContainer\");\r\n container.innerHTML=\"\";\r\n diagnosLista.forEach((q,i)=>{\r\n const div=document.createElement(\"div\");\r\n div.className=\"uppgift\";\r\n div.dataset.facit=q.svar;\r\n if(q.losningsforslag) div.dataset.losning=q.losningsforslag;\r\n div.innerHTML=`<b>Uppgift ${i+1}:<\/b> ${q.fraga} <span class=\"poang\">(${q.poang.join(\"\/\")})<\/span> ${q.digital ? '<span class=\"digital-icon\">\ud83e\uddee<\/span>' : \"\"}`;\r\n container.appendChild(div);\r\n });\r\n\r\n const klarContainer=document.getElementById(\"diagnosKlarContainer\");\r\n klarContainer.innerHTML=`<button id=\"diagnosKlarBtn\">Visa facit\/Klar<\/button>`;\r\n document.getElementById(\"diagnosKlarBtn\").addEventListener(\"click\",()=>{\r\n document.querySelectorAll(\".uppgift\").forEach(div=>{\r\n const f=document.createElement(\"div\");\r\n f.className=\"facit\";\r\n f.style.display=\"block\";\r\n f.innerHTML=\"<b>Facit:<\/b> \"+div.dataset.facit;\r\n div.appendChild(f);\r\n if(div.dataset.losning){\r\n const l=document.createElement(\"div\");\r\n l.className=\"losningsforslag\";\r\n l.style.display=\"block\";\r\n l.innerHTML=\"<b>L\u00f6sningsf\u00f6rslag:<\/b> \"+div.dataset.losning;\r\n div.appendChild(l);\r\n }\r\n });\r\n klarContainer.innerHTML=\"\";\r\n MathJax.typesetPromise();\r\n });\r\n\r\n MathJax.typesetPromise();\r\n});\r\n\/\/ ---- ADAPTIV DIAGNOS STATUS ----\r\nlet adaptivStatus = {\r\n underkategorier: [],\r\n nuvarandeNiva: \"E\",\r\n rattIRad: 0,\r\n felIRad: 0,\r\n minRattForUpp: 3,\r\n maxFelForNer: 3,\r\n avsluta: false\r\n};\r\n\r\n\/\/ ---- H\u00e4mta n\u00e4sta adaptiva uppgift ----\r\nfunction getN\u00e4staAdaptivUppgift() {\r\n let lista = [];\r\n for (let amne in fragor) {\r\n lista = lista.concat(\r\n fragor[amne].filter(q =>\r\n q.niva === adaptivStatus.nuvarandeNiva &&\r\n (Array.isArray(q.underkategori)\r\n ? q.underkategori.some(u => adaptivStatus.underkategorier.includes(u))\r\n : adaptivStatus.underkategorier.includes(q.underkategori)\r\n )\r\n )\r\n );\r\n }\r\n if (lista.length === 0) return null;\r\n return lista[Math.floor(Math.random() * lista.length)];\r\n}\r\n\r\n\/\/ ---- Visa n\u00e4sta uppgift ----\r\nfunction visaN\u00e4staAdaptivUppgift() {\r\n if (adaptivStatus.avsluta) return;\r\n\r\n const uppgift = getN\u00e4staAdaptivUppgift();\r\n if (!uppgift) {\r\n alert(\"Inga fler uppgifter p\u00e5 denna niv\u00e5 och kategori.\");\r\n return;\r\n }\r\n\r\n const container = document.getElementById(\"uppgifterContainer\");\r\n container.innerHTML = `\r\n <div class=\"adaptiv-uppgift\">\r\n <div class=\"adaptiv-header\">Adaptiv diagnos<\/div>\r\n <div class=\"adaptiv-info\">\r\n Niv\u00e5: <b>${adaptivStatus.nuvarandeNiva}<\/b> |\r\n R\u00e4tt i rad: ${adaptivStatus.rattIRad} |\r\n Fel i rad: ${adaptivStatus.felIRad}\r\n <\/div>\r\n <div class=\"adaptiv-fraga\">\r\n ${uppgift.fraga}\r\n ${uppgift.digital ? '<span class=\"digital-icon\">\ud83e\uddee<\/span>' : \"\"}\r\n<\/div>\r\n <div style=\"margin-top:10px;\">\r\n <button class=\"adaptiv-btn klarBtn\">Visa facit<\/button>\r\n <\/div>\r\n <div class=\"adaptiv-facit\" style=\"display:none;\">\r\n <b>Facit:<\/b> ${uppgift.svar}\r\n ${\r\n uppgift.losningsforslag\r\n ? `<div class=\"adaptiv-l\u00f6sning\"><b>L\u00f6sningsf\u00f6rslag:<\/b> ${uppgift.losningsforslag}<\/div>`\r\n : \"\"\r\n }\r\n <div style=\"margin-top:8px;\">\r\n <button class=\"adaptiv-btn rattBtn\">R\u00e4tt<\/button>\r\n <button class=\"adaptiv-btn felBtn\">Fel<\/button>\r\n <\/div>\r\n <\/div>\r\n <\/div>\r\n`;\r\n\r\n\r\n \/\/ Visa facit\r\n container.querySelector(\".klarBtn\").addEventListener(\"click\", () => {\r\n container.querySelector(\".adaptiv-facit\").style.display = \"block\";\r\n });\r\n\r\n \/\/ R\u00e4tt p\u00e5 uppgift\r\n container.querySelector(\".rattBtn\").addEventListener(\"click\", () => {\r\n adaptivStatus.rattIRad++;\r\n adaptivStatus.felIRad = 0;\r\n if (adaptivStatus.rattIRad >= adaptivStatus.minRattForUpp) {\r\n if (adaptivStatus.nuvarandeNiva === \"E\") adaptivStatus.nuvarandeNiva = \"C\";\r\n else if (adaptivStatus.nuvarandeNiva === \"C\") adaptivStatus.nuvarandeNiva = \"A\";\r\n adaptivStatus.rattIRad = 0;\r\n }\r\n visaN\u00e4staAdaptivUppgift();\r\n });\r\n\r\n \/\/ Fel p\u00e5 uppgift\r\ncontainer.querySelector(\".felBtn\").addEventListener(\"click\", () => {\r\n adaptivStatus.felIRad++;\r\n adaptivStatus.rattIRad = 0;\r\n if (adaptivStatus.felIRad >= adaptivStatus.maxFelForNer) {\r\n if (adaptivStatus.nuvarandeNiva === \"A\") adaptivStatus.nuvarandeNiva = \"C\";\r\n else if (adaptivStatus.nuvarandeNiva === \"C\") adaptivStatus.nuvarandeNiva = \"E\";\r\n adaptivStatus.felIRad = 0;\r\n }\r\n visaN\u00e4staAdaptivUppgift();\r\n});\r\n\r\n\/\/ \ud83d\udd39 L\u00e4gg till detta l\u00e4ngst ner\r\nMathJax.typesetPromise();\r\n}\r\n\r\n\r\n\/\/ ---- DOMContentLoaded: eventregistrering ----\r\ndocument.addEventListener(\"DOMContentLoaded\", () => {\r\n \/\/ \u00d6ppna popup\r\n document.getElementById(\"startAdaptivBtn\").addEventListener(\"click\", () => {\r\n document.getElementById(\"adaptivPopup\").style.display = \"block\";\r\n });\r\n\r\n \/\/ St\u00e4ng popup\r\n document.getElementById(\"stangAdaptivBtn\").addEventListener(\"click\", () => {\r\n document.getElementById(\"adaptivPopup\").style.display = \"none\";\r\n });\r\n\r\n \/\/ ---- Uppdatera totalpo\u00e4ng ----\r\n function uppdateraTotal() {\r\n const e = +document.getElementById(\"epoang\").value;\r\n const c = +document.getElementById(\"cpoang\").value;\r\n const a = +document.getElementById(\"apoang\").value;\r\n document.getElementById(\"totalPoang\").textContent = \"Totalt valt: \" + (e + c + a);\r\n }\r\n\r\n document.getElementById(\"epoang\").addEventListener(\"input\", uppdateraTotal);\r\n document.getElementById(\"cpoang\").addEventListener(\"input\", uppdateraTotal);\r\n document.getElementById(\"apoang\").addEventListener(\"input\", uppdateraTotal);\r\n\r\n \/\/ ---- Ladda uppgifter fr\u00e5n kod ----\r\n document.getElementById(\"laddaKodBtn\").addEventListener(\"click\", () => {\r\n const kod = document.getElementById(\"kodInput\").value.trim();\r\n if (!kod) { alert(\"Skriv in en kod f\u00f6rst.\"); return; }\r\n\r\n const delar = kod.match(\/[a-z]\\d+\/g);\r\n if (!delar) { alert(\"Ogiltig kod.\"); return; }\r\n\r\n let lista = [];\r\n for (let kodDel of delar) {\r\n for (let amne in fragor) {\r\n const uppg = fragor[amne].find(q => q.kod === kodDel);\r\n if (uppg) lista.push({ ...uppg, amne });\r\n }\r\n }\r\n\r\n if (lista.length === 0) {\r\n alert(\"Inga uppgifter hittades f\u00f6r denna kod.\");\r\n return;\r\n }\r\n\r\n renderFragor(lista);\r\n });\r\n\/\/ \u2705 H\u00e4r st\u00e4nger du DOMContentLoaded p\u00e5 riktigt\r\n\r\n \/\/ Starta adaptiv diagnos\r\n document.getElementById(\"startAdaptivUppgifterBtn\").addEventListener(\"click\", () => {\r\n const valda = Array.from(document.querySelectorAll(\"#adaptivPopup .underkategori:checked\"))\r\n .map(cb => cb.value);\r\n\r\n if (valda.length === 0) {\r\n alert(\"V\u00e4lj minst en underkategori innan du startar.\");\r\n return;\r\n }\r\n\r\n adaptivStatus.underkategorier = valda;\r\n adaptivStatus.nuvarandeNiva = \"E\";\r\n adaptivStatus.rattIRad = 0;\r\n adaptivStatus.felIRad = 0;\r\n adaptivStatus.avsluta = false;\r\n\r\n document.getElementById(\"adaptivPopup\").style.display = \"none\";\r\n visaN\u00e4staAdaptivUppgift();\r\n });\r\n});\r\n\r\n\r\n<\/script>\r\n\r\n<\/body>\r\n<\/html>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>Uppgiftsbibliotek – Ma4 (Under konstruktion) H\u00e4r kan du s\u00f6ka p\u00e5 uppgifter fr\u00e5n Mahifis uppgiftsbibliotek. V\u00e4lj vilket omr\u00e5de du vill jobba med samt vilken sv\u00e5righetsgrad. D\u00e4refter f\u00e5r du ett antal slumpm\u00e4ssigt utvalda uppgifter. Alla uppgifter har facit och vissa har \u00e4ven l\u00f6sningsf\u00f6rslag. Du kan byta uppgift genom att trycka p\u00e5 uppgiftsnumret till v\u00e4nster n\u00e4r du s\u00f6ker […]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"om_disable_all_campaigns":false,"_uag_custom_page_level_css":"","_monsterinsights_skip_tracking":false,"site-sidebar-layout":"no-sidebar","site-content-layout":"page-builder","ast-site-content-layout":"full-width-container","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"disabled","ast-breadcrumbs-content":"","ast-featured-img":"disabled","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"set","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"class_list":["post-39933","page","type-page","status-publish","hentry"],"aioseo_notices":[],"aioseo_head":"\n\t\t<!-- All in One SEO 4.9.10 - aioseo.com -->\n\t<meta name=\"description\" content=\"Uppgiftsbibliotek - Ma4 (Under konstruktion) H\u00e4r kan du s\u00f6ka p\u00e5 uppgifter fr\u00e5n Mahifis uppgiftsbibliotek. V\u00e4lj vilket omr\u00e5de du vill jobba med samt vilken sv\u00e5righetsgrad. D\u00e4refter f\u00e5r du ett antal slumpm\u00e4ssigt utvalda uppgifter. Alla uppgifter har facit och vissa har \u00e4ven l\u00f6sningsf\u00f6rslag. 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