Quiz — 1586626405_3-1 Exercice d'application_Question1.pdf

Question 1 sur 5 10:00

[{"id":1110,"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 - 5","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'une fonction polynôme se calcule en appliquant la règle : si f(x) = ax^n, alors f'(x) = nax^(n-1). Ici, 3x² donne 6x, -5x donne -5, et la constante 2 disparaît.","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":1111,"question":"Si f'(x) = 4x - 3, quel est le sens de variation de la fonction f ?","option_a":"Croissante sur ℝ","option_b":"Décroissante sur ℝ","option_c":"Croissante sur ]-∞; 3\/4[ et décroissante sur ]3\/4; +∞[","option_d":"Décroissante sur ]-∞; 3\/4[ et croissante sur ]3\/4; +∞[","option_e":"","option_f":"","bonne_reponse":"d","explication":"Le signe de la dérivée f'(x) = 4x - 3 dépend de x. f'(x) \u003E 0 quand x \u003E 3\/4 (fonction croissante) et f'(x) \u003C 0 quand x \u003C 3\/4 (fonction décroissante).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"d\", \"options\": {\"a\": \"Croissante sur ℝ\", \"b\": \"Décroissante sur ℝ\", \"c\": \"Croissan","_debug_options_count":4},{"id":1112,"question":"Quelle est la dérivée de la fonction g(x) = (2x + 1)\/(x - 3) ?","option_a":"g'(x) = (2x - 7)\/(x - 3)²","option_b":"g'(x) = (2x + 7)\/(x - 3)²","option_c":"g'(x) = 2\/(x - 3)","option_d":"g'(x) = (x - 3)\/(2x + 1)","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour une fonction rationnelle u\/v, la dérivée est (u'v - uv')\/v². Ici, u = 2x + 1 (u' = 2), v = x - 3 (v' = 1), donc g'(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\": \"g'(x) = (2x - 7)\/(x - 3)²\", \"b\": \"g'(x) = (2x + 7)\/(x - 3)²\", \"","_debug_options_count":4},{"id":1113,"question":"Soit f(x) = x³ - 6x² + 9x. À quel point la fonction f admet-elle un extremum local ?","option_a":"x = 0","option_b":"x = 1","option_c":"x = 3","option_d":"x = 6","option_e":"","option_f":"","bonne_reponse":"b","explication":"Un extremum local se produit lorsque f'(x) = 0. f'(x) = 3x² - 12x + 9 = 3(x² - 4x + 3) = 3(x - 1)(x - 3). Les solutions sont x = 1 et x = 3. Pour vérifier l'extremum, on étudie le signe de f'(x) 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\": \"x = 0\", \"b\": \"x = 1\", \"c\": \"x = 3\", \"d\": \"x = 6\"}}","_debug_options_count":4},{"id":1114,"question":"Quelle est la valeur de la dérivée de h(x) = √(x² + 1) en x = 2 ?","option_a":"1\/√5","option_b":"2\/√5","option_c":"√5\/2","option_d":"1\/2","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée de h(x) = √(x² + 1) est h'(x) = (2x)\/(2√(x² + 1)) = x\/√(x² + 1). En x = 2, h'(2) = 2\/√(4 + 1) = 2\/√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\": \"1\/√5\", \"b\": \"2\/√5\", \"c\": \"√5\/2\", \"d\": \"1\/2\"}}","_debug_options_count":4}]