Quiz — تمارين الدوال - المعادلات - الاشتقاقية.pdf

Question 1 sur 10 20:00

[{"id":6869,"question":"Quelle est la dérivée de la fonction f(x) = 3x² + 2x - 5 ?","option_a":"A. 6x + 2","option_b":"B. 3x + 2","option_c":"C. 6x² + 2","option_d":"D. 3x² + 2x","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'une fonction polynôme s'obtient en appliquant la règle : (ax^n)' = n·a·x^(n-1). Ici, f'(x) = 6x + 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\": \"A. 6x + 2\", \"b\": \"B. 3x + 2\", \"c\": \"C. 6x² + 2\", \"d\": \"D. 3x² +","_debug_options_count":4},{"id":6870,"question":"L'équation e^x = 5 admet-elle une solution réelle ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction exponentielle est strictement croissante et tend vers +∞, donc elle prend toutes les valeurs de ℝ+. Ainsi, e^x = 5 admet une solution unique x = ln(5).","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":6871,"question":"Quelle est la dérivée de la fonction f(x) = ln(2x + 1) ?","option_a":"A. 1\/(2x + 1)","option_b":"B. 2\/(2x + 1)","option_c":"C. 1\/x","option_d":"D. 2x\/(2x + 1)","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée de ln(u(x)) est u'(x)\/u(x). Ici, u(x) = 2x + 1, donc u'(x) = 2. Ainsi, f'(x) = 2\/(2x + 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\": \"A. 1\/(2x + 1)\", \"b\": \"B. 2\/(2x + 1)\", \"c\": \"C. 1\/x\", \"d\": \"D. 2x\/","_debug_options_count":4},{"id":6872,"question":"L'équation x³ - 3x + 2 = 0 admet-elle trois solutions réelles distinctes ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"En étudiant la fonction f(x) = x³ - 3x + 2, on trouve un maximum local en x = -1 (f(-1) = 4) et un minimum local en x = 1 (f(1) = 0). L'équation n'a donc que deux solutions réelles (x = 1 est une racine double).","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":6873,"question":"Quelle est la dérivée de la fonction f(x) = (x² + 1)\/(x - 1) ?","option_a":"A. (2x)\/(x - 1)","option_b":"B. (x² - 2x - 1)\/(x - 1)²","option_c":"C. (2x(x - 1) - (x² + 1))\/(x - 1)²","option_d":"D. (x² - 1)\/(x - 1)²","option_e":"","option_f":"","bonne_reponse":"c","explication":"On utilise la formule de dérivation d'un quotient : (u\/v)' = (u'v - uv')\/v². Ici, u = x² + 1 (u' = 2x) et v = x - 1 (v' = 1). Ainsi, f'(x) = (2x(x - 1) - (x² + 1))\/(x - 1)².","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"A. (2x)\/(x - 1)\", \"b\": \"B. (x² - 2x - 1)\/(x - 1)²\", \"c\": \"C. (2","_debug_options_count":4},{"id":6874,"question":"La fonction f(x) = x^4 - 4x³ admet-elle un point d'inflexion en x = 2 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Un point d'inflexion est un point où la dérivée seconde s'annule en changeant de signe. Ici, f''(x) = 12x² - 24x. f''(2) = 0 et f''(x) change de signe en x = 2 (f''(1) = -12 \u003C 0 et f''(3) = 36 \u003E 0), donc x = 2 est bien un point d'inflexion.","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":6875,"question":"Quelle est la solution de l'équation 2^(x+1) = 8 ?","option_a":"A. x = 1","option_b":"B. x = 2","option_c":"C. x = 3","option_d":"D. x = 0","option_e":"","option_f":"","bonne_reponse":"b","explication":"On réécrit l'équation sous la forme 2^(x+1) = 2³. Comme la fonction exponentielle est injective, x + 1 = 3, donc 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. x = 1\", \"b\": \"B. x = 2\", \"c\": \"C. x = 3\", \"d\": \"D. x = 0\"}}","_debug_options_count":4},{"id":6876,"question":"La dérivée de la fonction f(x) = cos(3x) est-elle f'(x) = -3sin(3x) ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de cos(u(x)) est -u'(x)sin(u(x)). Ici, u(x) = 3x, donc u'(x) = 3. Ainsi, f'(x) = -3sin(3x).","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":6877,"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. Ainsi, f'(x) = 2e^(2x).","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. e^(x","_debug_options_count":4},{"id":6878,"question":"L'équation ln(x) = 0 admet-elle une solution réelle ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction logarithme est définie pour x \u003E 0 et ln(1) = 0. Ainsi, x = 1 est la solution unique de l'équation ln(x) = 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}]