Quiz — Serie 12- Etude de fonction.pdf

Question 1 sur 10 20:00

[{"id":7315,"question":"Quelle est la dérivée de la fonction f(x) = x³ - 2x² + 5x - 1 ?","option_a":"A : 3x² - 4x + 5","option_b":"B : 3x² - 4x + 5x","option_c":"C : x² - 2x + 5","option_d":"D : 3x² - 4x","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'un polynôme s'obtient en appliquant la règle de dérivation terme à terme : (xⁿ)' = n·xⁿ⁻¹.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"A : 3x² - 4x + 5\", \"b\": \"B : 3x² - 4x + 5x\", \"c\": \"C : x² - 2x","_debug_options_count":4},{"id":7316,"question":"La fonction f(x) = (x² - 1)\/(x - 1) est-elle continue en x = 1 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction n'est pas définie en x = 1 (dénominateur nul), donc elle est discontinue en ce point.","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":7317,"question":"Quelle est la limite de f(x) = (2x² + 3x - 5)\/(x² - 4) lorsque x tend vers +∞ ?","option_a":"A : 0","option_b":"B : 2","option_c":"C : +∞","option_d":"D : -∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"Pour les limites en ±∞ de fonctions rationnelles, on compare les degrés des polynômes au numérateur et au dénominateur.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"A : 0\", \"b\": \"B : 2\", \"c\": \"C : +∞\", \"d\": \"D : -∞\"}}","_debug_options_count":4},{"id":7318,"question":"Une fonction dérivable sur un intervalle est-elle toujours continue sur cet intervalle ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Toute fonction dérivable sur un intervalle est nécessairement continue sur cet intervalle (théorème fondamental).","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":7319,"question":"Quelle est la dérivée de la fonction f(x) = e^(2x) ?","option_a":"A : e^(2x)","option_b":"B : 2e^(2x)","option_c":"C : 2x·e^(2x)","option_d":"D : e^(x)","option_e":"","option_f":"","bonne_reponse":"b","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\": \"b\", \"options\": {\"a\": \"A : e^(2x)\", \"b\": \"B : 2e^(2x)\", \"c\": \"C : 2x·e^(2x)\", \"d\": \"D :","_debug_options_count":4},{"id":7320,"question":"La fonction f(x) = x³ + 2x² - 5x + 1 admet-elle un extremum local ?","option_a":"A : Oui, en x = -2","option_b":"B : Oui, en x = 1","option_c":"C : Non","option_d":"D : Oui, en x = -1","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'extremum local est trouvé en résolvant f'(x) = 0. Ici, f'(x) = 3x² + 4x - 5, et x = -2 est une solution.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"A : Oui, en x = -2\", \"b\": \"B : Oui, en x = 1\", \"c\": \"C : Non\", \"d","_debug_options_count":4},{"id":7321,"question":"Quelle est l'asymptote oblique de la fonction f(x) = (x² + 3x - 2)\/(x - 1) ?","option_a":"A : y = x + 4","option_b":"B : y = x + 2","option_c":"C : y = 2x + 1","option_d":"D : y = x - 1","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour trouver une asymptote oblique, on effectue la division euclidienne du numérateur par le dénominateur.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"A : y = x + 4\", \"b\": \"B : y = x + 2\", \"c\": \"C : y = 2x + 1\", \"d\":","_debug_options_count":4},{"id":7322,"question":"La fonction f(x) = |x| est-elle dérivable en x = 0 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction valeur absolue n'est pas dérivable en x = 0 car sa courbe présente un point anguleux (dérivée à gauche ≠ dérivée à droite).","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":7323,"question":"Quelle est la dérivée de la fonction f(x) = ln(x² + 1) ?","option_a":"A : 2x\/(x² + 1)","option_b":"B : 2x","option_c":"C : 1\/(x² + 1)","option_d":"D : x\/(x² + 1)","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\": \"A : 2x\/(x² + 1)\", \"b\": \"B : 2x\", \"c\": \"C : 1\/(x² + 1)\", \"d\": \"D","_debug_options_count":4},{"id":7324,"question":"Si f'(x) \u003E 0 sur un intervalle, que peut-on dire de la fonction f sur cet intervalle ?","option_a":"A : f est décroissante","option_b":"B : f est croissante","option_c":"C : f est constante","option_d":"D : f est discontinue","option_e":"","option_f":"","bonne_reponse":"b","explication":"Si la dérivée est positive sur un intervalle, la fonction est strictement croissante sur cet intervalle.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"A : f est décroissante\", \"b\": \"B : f est croissante\", \"c\": \"C : ","_debug_options_count":4}]