Quiz — Series d_exercices Etude Fonction.pdf

Question 1 sur 10 20:00

[{"id":53817,"question":"Quelle est la dérivée de la fonction f(x) = x³ - 2x² + 5x - 1 ?","option_a":"3x² - 4x + 5","option_b":"3x² - 4x + 5x","option_c":"x³ - 4x + 5","option_d":"3x² - 2x + 5","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'un polynôme s'obtient en appliquant la règle : (x^n)' = n*x^(n-1). Ainsi, f'(x) = 3x² - 4x + 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\": \"3x² - 4x + 5\", \"b\": \"3x² - 4x + 5x\", \"c\": \"x³ - 4x + 5\", \"d\": ","_debug_options_count":4},{"id":53818,"question":"Si une fonction f est dérivable en a, alors f est continue en a.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est un théorème fondamental : toute fonction dérivable en un point est nécessairement continue en ce point.","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":53819,"question":"Quelle est la limite de f(x) = (2x² + 3x - 1)\/(x² - 4) quand x tend vers +∞ ?","option_a":"2","option_b":"3\/2","option_c":"1","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour les limites à l'infini d'une fonction rationnelle, on compare les degrés du numérateur et du dénominateur. Ici, les deux sont de degré 2, donc la limite 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\": \"3\/2\", \"c\": \"1\", \"d\": \"∞\"}}","_debug_options_count":4},{"id":53820,"question":"La fonction f(x) = 1\/x admet une asymptote verticale en x = 0.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction f(x) = 1\/x n'est pas définie en x = 0 et sa limite tend vers ±∞ quand x tend vers 0, donc elle admet bien une asymptote verticale 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},{"id":53821,"question":"Quelle est la dérivée de f(x) = e^(3x) ?","option_a":"3e^(3x)","option_b":"e^(3x)","option_c":"3x*e^(3x)","option_d":"e^(x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de e^(u(x)) est u'(x)*e^(u(x)). Ici, u(x) = 3x, donc u'(x) = 3. Ainsi, f'(x) = 3e^(3x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"3e^(3x)\", \"b\": \"e^(3x)\", \"c\": \"3x*e^(3x)\", \"d\": \"e^(x)\"}}","_debug_options_count":4},{"id":53822,"question":"Une fonction peut avoir une asymptote oblique sans avoir de dérivée.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une asymptote oblique existe si la limite de f(x)\/x quand x tend vers ±∞ est finie et non nulle. Cela implique généralement que la fonction est dérivable, mais ce n'est pas une condition nécessaire.","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":53823,"question":"Quelle est la valeur de f'(2) pour f(x) = ln(x² + 1) ?","option_a":"4\/5","option_b":"2\/5","option_c":"1\/2","option_d":"0","option_e":"","option_f":"","bonne_reponse":"a","explication":"f'(x) = (2x)\/(x² + 1). En x = 2, f'(2) = 4\/(4 + 1) = 4\/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\": \"4\/5\", \"b\": \"2\/5\", \"c\": \"1\/2\", \"d\": \"0\"}}","_debug_options_count":4},{"id":53824,"question":"La fonction f(x) = x² - 4x + 3 est croissante sur l'intervalle [2, +∞[.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"f'(x) = 2x - 4. f'(x) ≥ 0 quand x ≥ 2, donc la fonction est bien croissante sur [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":53825,"question":"Quelle est l'équation de l'asymptote oblique de f(x) = (x² + 1)\/x ?","option_a":"y = x","option_b":"y = x + 1","option_c":"y = x - 1","option_d":"y = 1\/x","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour trouver l'asymptote oblique, on effectue la division euclidienne : (x² + 1)\/x = x + 1\/x. Quand x tend vers ±∞, 1\/x tend vers 0, donc l'asymptote oblique est 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\": \"y = x\", \"b\": \"y = x + 1\", \"c\": \"y = x - 1\", \"d\": \"y = 1\/x\"}}","_debug_options_count":4},{"id":53826,"question":"La fonction f(x) = (x - 1)\/(x + 2) admet une asymptote verticale en x = -2.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction n'est pas définie en x = -2 et sa limite tend vers ±∞ quand x tend vers -2, donc elle admet bien une asymptote verticale en x = -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}]