Quiz — DMD.pdf

Question 1 sur 10 20:00

[{"id":46623,"question":"Quelle est la dérivée de la fonction f(x) = 3x² + 2x - 5 ?","option_a":"6x + 2","option_b":"3x + 2","option_c":"6x² + 2","option_d":"3x² + 2x","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'une fonction polynomiale se calcule terme par terme : (3x²)' = 6x, (2x)' = 2, et (-5)' = 0. Donc 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\": \"6x + 2\", \"b\": \"3x + 2\", \"c\": \"6x² + 2\", \"d\": \"3x² + 2x\"}}","_debug_options_count":4},{"id":46624,"question":"La limite de la fonction f(x) = (2x² + 3x - 1)\/(x² - 1) lorsque x tend vers l'infini est égale à :","option_a":"2","option_b":"0","option_c":"1","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour calculer cette limite, on divise le numérateur et le dénominateur par x² (terme de plus haut degré) : f(x) = (2 + 3\/x - 1\/x²)\/(1 - 1\/x²). Lorsque x tend vers l'infini, les termes en 1\/x et 1\/x² tendent vers 0, donc la limite est 2\/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\": \"2\", \"b\": \"0\", \"c\": \"1\", \"d\": \"∞\"}}","_debug_options_count":4},{"id":46625,"question":"Vrai ou Faux ? La fonction f(x) = x³ - 3x² + 2 est décroissante sur l'intervalle [0, 2].","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Pour étudier le sens de variation, on calcule la dérivée : f'(x) = 3x² - 6x. Sur [0, 2], f'(x) ≤ 0 (car 3x² - 6x = 3x(x - 2) ≤ 0), donc la fonction est décroissante sur cet intervalle. L'affirmation est donc fausse.","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":46626,"question":"Quelle est la limite de la fonction f(x) = (sin x)\/x lorsque x tend vers 0 ?","option_a":"0","option_b":"1","option_c":"∞","option_d":"La limite n'existe pas","option_e":"","option_f":"","bonne_reponse":"b","explication":"C'est une limite classique en analyse : lim(x→0) (sin x)\/x = 1. Cette limite est souvent utilisée pour définir la dérivée de la fonction sinus en 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\": \"0\", \"b\": \"1\", \"c\": \"∞\", \"d\": \"La limite n'existe pas\"}}","_debug_options_count":4},{"id":46627,"question":"Vrai ou Faux ? Si f'(x) = 0 en un point x₀, alors f admet un extremum en x₀.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Un point où f'(x₀) = 0 est un point critique, mais ce n'est pas forcément un extremum. Il faut étudier le signe de f' autour de x₀ pour conclure (théorème de la dérivée seconde ou changement de signe).","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":46628,"question":"Quelle est la dérivée de la fonction f(x) = e^(2x + 1) ?","option_a":"2e^(2x + 1)","option_b":"e^(2x + 1)","option_c":"(2x + 1)e^(2x)","option_d":"2xe^(2x + 1)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'une fonction exponentielle e^u est u' * e^u. Ici, u = 2x + 1, donc u' = 2. Ainsi, f'(x) = 2 * e^(2x + 1).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"2e^(2x + 1)\", \"b\": \"e^(2x + 1)\", \"c\": \"(2x + 1)e^(2x)\", \"d\": \"2xe","_debug_options_count":4},{"id":46629,"question":"La fonction f(x) = x³ - 6x² + 9x + 1 admet-elle un maximum local en x = 1 ?","option_a":"Oui","option_b":"Non","option_c":"On ne peut pas savoir","option_d":"Seulement un minimum","option_e":"","option_f":"","bonne_reponse":"a","explication":"Calculons f'(x) = 3x² - 12x + 9. En x = 1, f'(1) = 3 - 12 + 9 = 0. La dérivée seconde f''(x) = 6x - 12 donne f''(1) = -6 \u003C 0, donc f admet un maximum local en x = 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Oui\", \"b\": \"Non\", \"c\": \"On ne peut pas savoir\", \"d\": \"Seulement u","_debug_options_count":4},{"id":46630,"question":"Vrai ou Faux ? La fonction f(x) = 1\/x est dérivable en x = 0.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction f(x) = 1\/x n'est pas définie en x = 0, donc elle ne peut pas y être dérivable. De plus, sa limite en 0 tend vers l'infini, ce qui confirme son absence de dérivée en ce point.","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":46631,"question":"Quelle est la limite de la fonction f(x) = (x² - 4)\/(x - 2) lorsque x tend vers 2 ?","option_a":"0","option_b":"4","option_c":"2","option_d":"La limite n'existe pas","option_e":"","option_f":"","bonne_reponse":"b","explication":"On peut simplifier l'expression : (x² - 4)\/(x - 2) = (x - 2)(x + 2)\/(x - 2) = x + 2 (pour x ≠ 2). Donc lim(x→2) f(x) = 2 + 2 = 4.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"0\", \"b\": \"4\", \"c\": \"2\", \"d\": \"La limite n'existe pas\"}}","_debug_options_count":4},{"id":46632,"question":"Quelle est la dérivée de la fonction f(x) = ln(3x² + 1) ?","option_a":"6x\/(3x² + 1)","option_b":"3x\/(3x² + 1)","option_c":"(3x² + 1)\/6x","option_d":"6x","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de ln(u) est u'\/u. Ici, u = 3x² + 1, donc u' = 6x. Ainsi, f'(x) = 6x\/(3x² + 1).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"6x\/(3x² + 1)\", \"b\": \"3x\/(3x² + 1)\", \"c\": \"(3x² + 1)\/6x\", \"d\": ","_debug_options_count":4}]