Quiz — Correction_série 6 Rep fémonine.pdf

Question 1 sur 10 20:00

[{"id":61888,"question":"Quelle est la dérivée de la fonction f(x) = 3x² + 5x - 2 ?","option_a":"f'(x) = 6x + 5","option_b":"f'(x) = 3x + 5","option_c":"f'(x) = 6x² + 5","option_d":"f'(x) = 3x² + 5x","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'une fonction polynôme se calcule terme par terme : (3x²)' = 6x et (5x)' = 5. La constante -2 a une dérivée nulle.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"f'(x) = 6x + 5\", \"b\": \"f'(x) = 3x + 5\", \"c\": \"f'(x) = 6x² + 5\", ","_debug_options_count":4},{"id":61889,"question":"Si f'(x) \u003E 0 pour tout x ∈ ℝ, alors la fonction f est strictement croissante sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est vrai uniquement si f est continue sur ℝ. Sans continuité, la fonction peut avoir des discontinuités et ne pas être strictement croissante.","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":61890,"question":"Quelle est la dérivée de la fonction f(x) = (2x + 1) * e^x ?","option_a":"f'(x) = 2e^x + (2x + 1)e^x","option_b":"f'(x) = 2e^x","option_c":"f'(x) = (2x + 3)e^x","option_d":"f'(x) = 2(2x + 1)e^x","option_e":"","option_f":"","bonne_reponse":"c","explication":"On utilise la règle du produit : (u*v)' = u'v + uv'. Ici, u = 2x + 1 (u' = 2) et v = e^x (v' = e^x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"f'(x) = 2e^x + (2x + 1)e^x\", \"b\": \"f'(x) = 2e^x\", \"c\": \"f'(x) = (","_debug_options_count":4},{"id":61891,"question":"La tangente à la courbe de f en x = a a pour équation y = f'(a)(x - a) + f(a).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est vrai : l'équation de la tangente utilise le coefficient directeur f'(a) et passe par le point (a, f(a)).","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":61892,"question":"Quelle est la dérivée de la fonction f(x) = ln(3x² + 1) ?","option_a":"f'(x) = 6x \/ (3x² + 1)","option_b":"f'(x) = 3x \/ (3x² + 1)","option_c":"f'(x) = 6x \/ (x² + 1)","option_d":"f'(x) = 3 \/ (3x² + 1)","option_e":"","option_f":"","bonne_reponse":"a","explication":"On utilise la dérivée de ln(u) : (ln(u))' = u'\/u. Ici, u = 3x² + 1, donc u' = 6x.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"f'(x) = 6x \/ (3x² + 1)\", \"b\": \"f'(x) = 3x \/ (3x² + 1)\", \"c\": \"f","_debug_options_count":4},{"id":61893,"question":"Si f'(x) = 0 en un point, alors ce point est toujours un extremum de la fonction.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Faux : un point où f'(x) = 0 peut être un extremum, mais aussi un point d'inflexion ou un point où la fonction n'est pas définie.","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":61894,"question":"Quelle est la dérivée de la fonction f(x) = sin(2x + π\/3) ?","option_a":"f'(x) = 2cos(2x + π\/3)","option_b":"f'(x) = cos(2x + π\/3)","option_c":"f'(x) = 2sin(2x + π\/3)","option_d":"f'(x) = -2cos(2x + π\/3)","option_e":"","option_f":"","bonne_reponse":"a","explication":"On utilise la dérivée de sin(u) : (sin(u))' = u'cos(u). Ici, u = 2x + π\/3, donc u' = 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\": \"f'(x) = 2cos(2x + π\/3)\", \"b\": \"f'(x) = cos(2x + π\/3)\", \"c\": \"f'","_debug_options_count":4},{"id":61895,"question":"La fonction f(x) = x³ - 3x² + 4 a-t-elle un extremum en x = 1 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai : f'(x) = 3x² - 6x. En x = 1, f'(1) = 3 - 6 = -3 ≠ 0, donc ce n'est pas un extremum. Correction : f'(1) = 0, donc c'est un point critique. Il faut vérifier le changement de signe de f' autour de x=1 pour confirmer l'extremum. Réponse corrigée : Vrai (sous réserve de vérification).","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":61896,"question":"Quelle est la dérivée de la fonction f(x) = e^(3x + 2) ?","option_a":"f'(x) = e^(3x + 2)","option_b":"f'(x) = 3e^(3x + 2)","option_c":"f'(x) = (3x + 2)e^(3x + 2)","option_d":"f'(x) = 3e^x","option_e":"","option_f":"","bonne_reponse":"b","explication":"On utilise la dérivée de e^u : (e^u)' = u'e^u. Ici, u = 3x + 2, donc u' = 3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"f'(x) = e^(3x + 2)\", \"b\": \"f'(x) = 3e^(3x + 2)\", \"c\": \"f'(x) = (3","_debug_options_count":4},{"id":61897,"question":"Si f'(x) change de signe en x = a, alors f admet un extremum en x = a.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai : le changement de signe de la dérivée indique un extremum (minimum ou maximum) pour la fonction f.","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}]