Quiz — EpreuveII_2000.pdf

Question 1 sur 10 20:00

[{"id":8100,"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 + 2","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'une fonction polynomiale se calcule en appliquant la règle (ax^n)' = n*ax^(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 : 3x","_debug_options_count":4},{"id":8101,"question":"Si f'(x) = 4x - 3, alors la fonction f est croissante sur l'intervalle où x \u003E 3\/4.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction est croissante lorsque sa dérivée est positive, soit 4x - 3 \u003E 0 ⇒ x \u003E 3\/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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":8102,"question":"Quelle est la dérivée de la fonction g(x) = (2x + 1)(x - 3) ?","option_a":"A : 2x - 5","option_b":"B : 4x - 5","option_c":"C : 2x + 1","option_d":"D : 4x + 5","option_e":"","option_f":"","bonne_reponse":"b","explication":"On utilise la règle du produit : (uv)' = u'v + uv'. Ici, g'(x) = 2(x - 3) + (2x + 1)*1 = 4x - 5.","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 : 2x - 5\", \"b\": \"B : 4x - 5\", \"c\": \"C : 2x + 1\", \"d\": \"D : 4x +","_debug_options_count":4},{"id":8103,"question":"La dérivée de la fonction h(x) = sin(2x) est h'(x) = 2cos(2x).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"On applique la règle de dérivation des fonctions composées : (sin(u))' = u'cos(u). Ici, h'(x) = 2cos(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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":8104,"question":"Quelle est la dérivée de la fonction k(x) = e^(3x) ?","option_a":"A : 3e^(3x)","option_b":"B : e^(3x)","option_c":"C : 3x e^(3x)","option_d":"D : e^(x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de e^(u(x)) est u'(x)e^(u(x)). Ici, k'(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\": \"A : 3e^(3x)\", \"b\": \"B : e^(3x)\", \"c\": \"C : 3x e^(3x)\", \"d\": \"D : ","_debug_options_count":4},{"id":8105,"question":"Si f'(x) = x² - 4, alors la fonction f admet un extremum en x = 2.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Un extremum se produit lorsque f'(x) = 0. Ici, x² - 4 = 0 ⇒ x = ±2. Il faut vérifier le signe de f' autour de ces points.","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":8106,"question":"Quelle est la dérivée de la fonction m(x) = ln(5x) ?","option_a":"A : 1\/x","option_b":"B : 5\/x","option_c":"C : 1\/(5x)","option_d":"D : 5\/(5x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de ln(u(x)) est u'(x)\/u(x). Ici, m'(x) = 5\/(5x) = 1\/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\": \"A : 1\/x\", \"b\": \"B : 5\/x\", \"c\": \"C : 1\/(5x)\", \"d\": \"D : 5\/(5x)\"}}","_debug_options_count":4},{"id":8107,"question":"La dérivée d'une constante est toujours égale à 1.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée d'une constante est toujours égale à 0, car la pente de la tangente à une droite horizontale est nulle.","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":8108,"question":"Quelle est la dérivée de la fonction n(x) = (x³ + 2x) \/ (x² + 1) ?","option_a":"A : (3x² + 2)(x² + 1) - (x³ + 2x)(2x) \/ (x² + 1)²","option_b":"B : (3x² + 2) \/ (2x)","option_c":"C : (x³ + 2x) \/ (2x)","option_d":"D : (3x² + 2)(x² + 1) + (x³ + 2x)(2x) \/ (x² + 1)²","option_e":"","option_f":"","bonne_reponse":"a","explication":"On utilise la règle du quotient : (u\/v)' = (u'v - uv') \/ v². Ici, n'(x) = [(3x² + 2)(x² + 1) - (x³ + 2x)(2x)] \/ (x² + 1)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 : (3x² + 2)(x² + 1) - (x³ + 2x)(2x) \/ (x² + 1)²\", \"b\": \"B ","_debug_options_count":4},{"id":8109,"question":"Si f'(x) = 0 pour tout x, alors la fonction f est une fonction constante.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Si la dérivée est nulle sur un intervalle, alors la fonction est constante sur cet intervalle (théorème des accroissements finis).","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}]