Quiz — devoir-de-contrôle-n°6--2008-2009[sayeda-monastir].pdf

Question 1 sur 10 20:00

[{"id":26770,"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 polynomiale se calcule terme par terme : 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":26771,"question":"L'équation 2x² - 5x + 2 = 0 admet-elle deux solutions réelles distinctes ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le discriminant Δ = (-5)² - 4*2*2 = 25 - 16 = 9 \u003E 0, donc deux solutions réelles distinctes.","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":26772,"question":"Quelle est la dérivée de la fonction f(x) = (2x + 1)\/(x - 3) ?","option_a":"A. (2x - 7)\/(x - 3)²","option_b":"B. (2x + 7)\/(x - 3)²","option_c":"C. (2x - 1)\/(x - 3)²","option_d":"D. (2x + 1)\/(x - 3)²","option_e":"","option_f":"","bonne_reponse":"a","explication":"En utilisant la formule de dérivation d'un quotient : f'(x) = [2(x - 3) - (2x + 1)*1]\/(x - 3)² = (2x - 6 - 2x - 1)\/(x - 3)² = -7\/(x - 3)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. (2x - 7)\/(x - 3)²\", \"b\": \"B. (2x + 7)\/(x - 3)²\", \"c\": \"C. (2","_debug_options_count":4},{"id":26773,"question":"La fonction f(x) = x³ - 3x² + 4 admet-elle un extremum local en x = 2 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"f'(x) = 3x² - 6x. En x = 2, f'(2) = 12 - 12 = 0. f''(x) = 6x - 6, donc f''(2) = 6 \u003E 0 : minimum local.","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":26774,"question":"Quelle est la solution de l'équation ln(x) = 2 ?","option_a":"A. x = e²","option_b":"B. x = 2","option_c":"C. x = 1\/e²","option_d":"D. x = ln(2)","option_e":"","option_f":"","bonne_reponse":"a","explication":"Par définition du logarithme, ln(x) = 2 ⇒ x = e².","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. x = e²\", \"b\": \"B. x = 2\", \"c\": \"C. x = 1\/e²\", \"d\": \"D. x = l","_debug_options_count":4},{"id":26775,"question":"La dérivée de la fonction f(x) = e^(3x) est-elle f'(x) = 3e^(3x) ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de e^(u(x)) est u'(x)e^(u(x)). Ici, u(x) = 3x ⇒ u'(x) = 3 ⇒ f'(x) = 3e^(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":26776,"question":"Quelle est la valeur de la dérivée de f(x) = √(x² + 1) en x = 0 ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. -1","option_d":"D. 2","option_e":"","option_f":"","bonne_reponse":"b","explication":"f'(x) = x\/√(x² + 1). En x = 0, f'(0) = 0\/1 = 0. Mais attention, la dérivée en 0 est bien 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\": \"A. 0\", \"b\": \"B. 1\", \"c\": \"C. -1\", \"d\": \"D. 2\"}}","_debug_options_count":4},{"id":26777,"question":"L'équation différentielle y' + 2y = 0 admet-elle pour solution y = Ce^(-2x) ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La solution générale de y' + 2y = 0 est y = Ce^(-2x), où C est une 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":26778,"question":"Quelle est la dérivée seconde de f(x) = x³ - 2x² + x ?","option_a":"A. 6x - 4","option_b":"B. 6x² - 4x","option_c":"C. 3x² - 4x + 1","option_d":"D. 6x - 4x","option_e":"","option_f":"","bonne_reponse":"a","explication":"f'(x) = 3x² - 4x + 1 ⇒ f''(x) = 6x - 4.","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 - 4\", \"b\": \"B. 6x² - 4x\", \"c\": \"C. 3x² - 4x + 1\", \"d\": \"D","_debug_options_count":4},{"id":26779,"question":"La fonction f(x) = x² - 4x + 3 est-elle croissante sur l'intervalle [2, +∞[ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"f'(x) = 2x - 4. Pour x ≥ 2, f'(x) ≥ 0 ⇒ f est croissante sur [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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4}]