Quiz — تمارين الدوال العددية في البكالوريا

Question 1 sur 10 20:00

[{"id":94621,"question":"Soit f une fonction définie sur ℝ par f(x) = x³ - 3x + 1. Quelle est sa dérivée f'(x) ?","option_a":"3x² - 3","option_b":"x² - 3","option_c":"3x² + 1","option_d":"x³ - 3","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'un polynôme se calcule terme à terme : f'(x) = 3x² - 3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"3x² - 3\", \"b\": \"x² - 3\", \"c\": \"3x² + 1\", \"d\": \"x³ - 3\"}}","_debug_options_count":4},{"id":94622,"question":"La fonction f(x) = 1\/x est-elle définie 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 car la division par zéro est impossible.","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":94623,"question":"Quelle est la limite de f(x) = (2x² + 3x - 1)\/(x² - 4) quand x tend vers +∞ ?","option_a":"2","option_b":"1","option_c":"0","option_d":"+∞","option_e":"","option_f":"","bonne_reponse":"a","explication":"En divisant numérateur et dénominateur par x², on obtient lim (2 + 3\/x - 1\/x²)\/(1 - 4\/x²) = 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\": \"1\", \"c\": \"0\", \"d\": \"+∞\"}}","_debug_options_count":4},{"id":94624,"question":"Soit f une fonction continue sur [a, b]. Le théorème des valeurs intermédiaires garantit-il qu'il existe c ∈ [a, b] tel que f(c) = (f(a) + f(b))\/2 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le théorème des valeurs intermédiaires garantit l'existence de c tel que f(c) = k pour tout k entre f(a) et f(b), mais pas nécessairement la moyenne.","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":94625,"question":"Quelle est la solution de l'équation ln(x) = 2 ?","option_a":"x = e²","option_b":"x = 2e","option_c":"x = ln(2)","option_d":"x = 2","option_e":"","option_f":"","bonne_reponse":"a","explication":"En appliquant l'exponentielle des deux côtés, on obtient 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\": \"x = e²\", \"b\": \"x = 2e\", \"c\": \"x = ln(2)\", \"d\": \"x = 2\"}}","_debug_options_count":4},{"id":94626,"question":"La fonction f(x) = x² - 4x + 3 admet-elle un minimum sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est une fonction polynôme du second degré de coefficient dominant positif, donc elle admet un minimum en x = -b\/(2a) = 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},{"id":94627,"question":"Soit f une fonction dérivable sur ℝ. Si f'(x) \u003E 0 pour tout x ∈ ℝ, alors f est-elle strictement croissante sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Si la dérivée est strictement positive sur un intervalle, la fonction y est 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":94628,"question":"Quelle est l'asymptote de la fonction f(x) = (3x² + 2x - 1)\/(x - 1) quand x tend vers +∞ ?","option_a":"y = 3x + 5","option_b":"y = x + 1","option_c":"y = 3x","option_d":"y = 2x + 3","option_e":"","option_f":"","bonne_reponse":"a","explication":"En effectuant la division euclidienne, on trouve f(x) = 3x + 5 + 4\/(x-1), donc l'asymptote est y = 3x + 5.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"y = 3x + 5\", \"b\": \"y = x + 1\", \"c\": \"y = 3x\", \"d\": \"y = 2x + 3\"}}","_debug_options_count":4},{"id":94629,"question":"Soit f une fonction paire. Que peut-on dire de f(-x) ?","option_a":"f(-x) = f(x)","option_b":"f(-x) = -f(x)","option_c":"f(-x) = 0","option_d":"f(-x) = 1\/f(x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"Par définition, une fonction paire vérifie f(-x) = f(x) pour tout x de son domaine de définition.","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) = f(x)\", \"b\": \"f(-x) = -f(x)\", \"c\": \"f(-x) = 0\", \"d\": \"f(-x","_debug_options_count":4},{"id":94630,"question":"La fonction f(x) = e^x - x admet-elle un point critique ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée f'(x) = e^x - 1 s'annule en x = 0, donc f admet un point critique en x = 0.","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}]