Quiz — Chapitre 4 -Les fonctions réelles-.pdf

Question 1 sur 10 20:00

[{"id":50833,"question":"Quelle est la dérivée de la fonction f(x) = x² + 3x - 5 ?","option_a":"2x + 3","option_b":"x² + 3","option_c":"2x² + 3","option_d":"x + 3","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'une fonction polynôme s'obtient en appliquant la règle de dérivation terme à terme : f'(x) = 2x + 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\": \"2x + 3\", \"b\": \"x² + 3\", \"c\": \"2x² + 3\", \"d\": \"x + 3\"}}","_debug_options_count":4},{"id":50834,"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 elle présente un 'point anguleux' à cet endroit.","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":50835,"question":"Quelle est la limite de (sin x)\/x quand x tend vers 0 ?","option_a":"0","option_b":"1","option_c":"∞","option_d":"Indéterminée","option_e":"","option_f":"","bonne_reponse":"b","explication":"D'après le théorème des gendarmes, la limite de (sin x)\/x quand x tend vers 0 est égale à 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"0\", \"b\": \"1\", \"c\": \"∞\", \"d\": \"Indéterminée\"}}","_debug_options_count":4},{"id":50836,"question":"Une fonction continue sur un intervalle fermé est-elle toujours dérivable sur cet intervalle ?","option_a":"Oui","option_b":"Non","option_c":"Parfois","option_d":"Jamais","option_e":"","option_f":"","bonne_reponse":"b","explication":"La continuité n'implique pas la dérivabilité. Par exemple, la fonction f(x) = |x| est continue mais pas dérivable en x = 0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Oui\", \"b\": \"Non\", \"c\": \"Parfois\", \"d\": \"Jamais\"}}","_debug_options_count":4},{"id":50837,"question":"Quelle est la dérivée de la fonction f(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":50838,"question":"La fonction f(x) = x³ - 3x + 2 est-elle croissante sur l'intervalle [-2, 2] ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée f'(x) = 3x² - 3 est positive sur [-2, -1] et [1, 2], et négative sur [-1, 1]. La fonction n'est donc pas croissante sur tout l'intervalle.","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":50839,"question":"Quelle est la limite de (1 + 1\/n)^n quand n tend vers l'infini ?","option_a":"0","option_b":"1","option_c":"e","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"c","explication":"Cette limite définit le nombre e, base des logarithmes naturels, et vaut environ 2,718.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"0\", \"b\": \"1\", \"c\": \"e\", \"d\": \"∞\"}}","_debug_options_count":4},{"id":50840,"question":"Une fonction dérivable en un point est-elle toujours continue en ce point ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivabilité implique la continuité. Si une fonction est dérivable en un point, elle y est nécessairement continue.","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":50841,"question":"Quelle est la dérivée de la fonction f(x) = ln(x² + 1) ?","option_a":"2x\/(x² + 1)","option_b":"2x","option_c":"1\/(x² + 1)","option_d":"x\/(x² + 1)","option_e":"","option_f":"","bonne_reponse":"a","explication":"En utilisant la règle de la chaîne, 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\": \"2x\", \"c\": \"1\/(x² + 1)\", \"d\": \"x\/(x² + 1)\"}","_debug_options_count":4},{"id":50842,"question":"La fonction f(x) = x^4 - 4x³ + 6x² est-elle convexe sur ℝ ?","option_a":"Oui","option_b":"Non","option_c":"Parfois","option_d":"Jamais","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée seconde f''(x) = 12x² - 24x + 12 = 12(x² - 2x + 1) = 12(x - 1)² est toujours positive ou nulle. La fonction est donc convexe sur ℝ.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Oui\", \"b\": \"Non\", \"c\": \"Parfois\", \"d\": \"Jamais\"}}","_debug_options_count":4}]