Quiz — TdCalculDiff.pdf

Question 1 sur 10 20:00

[{"id":125593,"question":"Quelle est la dérivée de la fonction f(x) = x³ + 2x² - 5x + 1 ?","option_a":"3x² + 4x - 5","option_b":"x² + 4x - 5","option_c":"3x² + 4x + 1","option_d":"x³ + 4x - 5","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'un polynôme s'obtient en appliquant la formule (x^n)' = n*x^(n-1) à chaque terme.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"3x² + 4x - 5\", \"b\": \"x² + 4x - 5\", \"c\": \"3x² + 4x + 1\", \"d\": \"","_debug_options_count":4},{"id":125594,"question":"Si f'(x) = 0 pour tout x dans un intervalle, alors f est constante sur cet intervalle.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est le théorème de Rolle : si la dérivée est nulle sur un intervalle, la fonction est constante.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":125595,"question":"Quelle est la dérivée de la fonction g(x) = e^(2x) ?","option_a":"2e^(2x)","option_b":"e^(2x)","option_c":"2xe^(2x)","option_d":"e^(x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de e^(u(x)) est u'(x)*e^(u(x)). Ici, u(x) = 2x, donc u'(x) = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"2e^(2x)\", \"b\": \"e^(2x)\", \"c\": \"2xe^(2x)\", \"d\": \"e^(x)\"}}","_debug_options_count":4},{"id":125596,"question":"La fonction h(x) = x² - 4x + 3 admet un minimum en x = 2.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"h'(x) = 2x - 4. h'(2) = 0, et h''(x) = 2 \u003E 0, donc x=2 est un minimum.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":125597,"question":"Quelle est la dérivée de la fonction k(x) = ln(x² + 1) ?","option_a":"2x\/(x² + 1)","option_b":"1\/(x² + 1)","option_c":"2x","option_d":"ln(2x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de ln(u(x)) est u'(x)\/u(x). Ici, u(x) = x² + 1, donc u'(x) = 2x.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"2x\/(x² + 1)\", \"b\": \"1\/(x² + 1)\", \"c\": \"2x\", \"d\": \"ln(2x)\"}}","_debug_options_count":4},{"id":125598,"question":"Le théorème des accroissements finis s'applique à toute fonction dérivable sur un intervalle fermé.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le théorème nécessite que la fonction soit continue sur [a,b] et dérivable sur ]a,b[.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":125599,"question":"Quelle est la dérivée de la fonction m(x) = sin(3x) ?","option_a":"3cos(3x)","option_b":"cos(3x)","option_c":"3sin(3x)","option_d":"-3cos(3x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de sin(u(x)) est u'(x)*cos(u(x)). Ici, u(x) = 3x, donc u'(x) = 3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"3cos(3x)\", \"b\": \"cos(3x)\", \"c\": \"3sin(3x)\", \"d\": \"-3cos(3x)\"}}","_debug_options_count":4},{"id":125600,"question":"La fonction n(x) = x³ - 3x² + 2x est croissante sur l'intervalle ]-∞, 1[.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"n'(x) = 3x² - 6x + 2. Le discriminant est négatif, donc n'(x) \u003E 0 pour tout x.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":125601,"question":"Quelle est la dérivée de la fonction p(x) = (2x + 1)\/(x - 3) ?","option_a":"(2x - 7)\/(x - 3)²","option_b":"2\/(x - 3)","option_c":"(2x + 1)\/(x - 3)²","option_d":"(2x - 7)\/(x - 3)","option_e":"","option_f":"","bonne_reponse":"a","explication":"Utilisez la formule de dérivation d'un quotient : (u\/v)' = (u'v - uv')\/v².","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"(2x - 7)\/(x - 3)²\", \"b\": \"2\/(x - 3)\", \"c\": \"(2x + 1)\/(x - 3)²\",","_debug_options_count":4},{"id":125602,"question":"Si f''(x) \u003E 0 pour tout x, alors la fonction f est convexe sur son domaine de définition.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La convexité est définie par f''(x) \u003E 0. La concavité correspond à f''(x) \u003C 0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4}]