Quiz — 2. Etude de fonction.pdf

Question 1 sur 10 20:00

[{"id":41505,"question":"Quelle est la dérivée de la fonction f(x) = x³ - 2x² + 5 ?","option_a":"3x² - 4x","option_b":"3x² - 4x + 5","option_c":"x² - 4x","option_d":"3x² - 2x","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'un polynôme s'obtient en appliquant la règle de dérivation terme à terme : (x^n)' = n*x^(n-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\": \"3x² - 4x\", \"b\": \"3x² - 4x + 5\", \"c\": \"x² - 4x\", \"d\": \"3x² - 2","_debug_options_count":4},{"id":41506,"question":"Si f'(x) \u003E 0 sur un intervalle, alors la fonction f est :","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une dérivée positive sur un intervalle indique que la fonction est strictement croissante sur cet intervalle.","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":41507,"question":"Quelle est la limite de f(x) = (2x² + 3)\/(x² - 1) quand x tend vers +∞ ?","option_a":"2","option_b":"0","option_c":"1","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour les fonctions rationnelles, la limite à l'infini est le rapport des coefficients dominants (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":41508,"question":"Une fonction peut-elle avoir une asymptote verticale et une asymptote horizontale en même temps ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Oui, par exemple la fonction f(x) = (x² + 1)\/x a une asymptote verticale en x=0 et une asymptote oblique en y=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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":41509,"question":"Quelle est la dérivée seconde de f(x) = x⁴ - 3x³ ?","option_a":"12x² - 18x","option_b":"4x³ - 9x²","option_c":"12x² - 9x","option_d":"4x³ - 18x","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée première est f'(x) = 4x³ - 9x². La dérivée seconde s'obtient en dérivant f'(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\": \"12x² - 18x\", \"b\": \"4x³ - 9x²\", \"c\": \"12x² - 9x\", \"d\": \"4x³ -","_debug_options_count":4},{"id":41510,"question":"Si f'(a) = 0 et f''(a) \u003E 0, alors le point d'abscisse a est :","option_a":"Un minimum local","option_b":"Un maximum local","option_c":"Un point d'inflexion","option_d":"Un point de discontinuité","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le signe de la dérivée seconde en un point critique indique la nature de l'extremum : positif pour un minimum, négatif pour un maximum.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Un minimum local\", \"b\": \"Un maximum local\", \"c\": \"Un point d'infl","_debug_options_count":4},{"id":41511,"question":"La fonction f(x) = e^x a pour asymptote horizontale :","option_a":"y = 0","option_b":"y = 1","option_c":"Aucune","option_d":"x = 0","option_e":"","option_f":"","bonne_reponse":"a","explication":"Quand x tend vers -∞, e^x tend vers 0, donc la droite y=0 est une asymptote horizontale.","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 = 0\", \"b\": \"y = 1\", \"c\": \"Aucune\", \"d\": \"x = 0\"}}","_debug_options_count":4},{"id":41512,"question":"Vrai ou Faux ? Une fonction continue sur un intervalle est toujours dérivable sur cet intervalle.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La continuité n'implique pas la dérivabilité. Par exemple, la fonction valeur absolue est continue mais non dérivable en x=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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":41513,"question":"Quelle est l'équation de la tangente à la courbe de f(x) = ln(x) au point d'abscisse 1 ?","option_a":"y = x - 1","option_b":"y = x + 1","option_c":"y = 1","option_d":"y = x","option_e":"","option_f":"","bonne_reponse":"a","explication":"La tangente a pour équation y = f'(a)(x - a) + f(a). Ici, f(1)=0 et f'(1)=1, donc y = 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\": \"y = x - 1\", \"b\": \"y = x + 1\", \"c\": \"y = 1\", \"d\": \"y = x\"}}","_debug_options_count":4},{"id":41514,"question":"Si f est dérivable en a, alors la fonction f est nécessairement continue en a.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivabilité implique la continuité. Si une fonction est dérivable en un point, elle y est automatiquement continue.","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}]