Quiz — 663a148c9d3b8_Corrigé_Série N°6-Etude de fonction.pdf

Question 1 sur 10 20:00

[{"id":33070,"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":"x² - 4x + 5","option_d":"3x² - 2x + 5","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'un polynôme se calcule terme à terme : (x³)' = 3x², (-2x²)' = -4x, (5x)' = 5, et (-1)' = 0. Donc 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\": \"x² - 4x + 5\", \"d\": ","_debug_options_count":4},{"id":33071,"question":"La fonction f(x) = 1\/x est-elle dérivable en x = 0 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction 1\/x n'est pas définie en x = 0, donc elle n'est pas dérivable en ce point. De plus, sa limite tend vers l'infini.","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":33072,"question":"Quelle est la limite de f(x) = (2x² + 3x - 1)\/(x² - 4) 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 limites en ±∞ de fonctions rationnelles, on compare les termes de plus haut degré : (2x²)\/(x²) = 2. Les autres termes deviennent négligeables.","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":33073,"question":"La fonction f(x) = x² + 3x + 2 admet-elle un minimum ? Si oui, où ?","option_a":"Oui, en x = -3\/2","option_b":"Oui, en x = 3\/2","option_c":"Non, elle n'a pas de minimum","option_d":"Oui, en x = -1","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le minimum d'une fonction quadratique f(x) = ax² + bx + c se trouve en x = -b\/(2a). Ici, a = 1 et b = 3, donc 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\": \"Oui, en x = -3\/2\", \"b\": \"Oui, en x = 3\/2\", \"c\": \"Non, elle n'a pa","_debug_options_count":4},{"id":33074,"question":"La droite y = 2x + 1 est-elle une asymptote oblique de f(x) = (2x² + x + 1)\/(x - 1) ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour vérifier une asymptote oblique y = mx + p, on effectue la division polynomiale de f(x) par (x - 1). Ici, le quotient est 2x + 3, donc l'asymptote est y = 2x + 3 (et non y = 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":33075,"question":"Quelle est la dérivée de f(x) = e^(2x) ?","option_a":"2e^(2x)","option_b":"e^(2x)","option_c":"2xe^(2x)","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) = 2x, donc u'(x) = 2. Ainsi, f'(x) = 2e^(2x).","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)\", \"b\": \"e^(2x)\", \"c\": \"2xe^(2x)\", \"d\": \"e^(x)\"}}","_debug_options_count":4},{"id":33076,"question":"La fonction f(x) = ln(x) est-elle définie pour x = -1 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction logarithme népérien ln(x) n'est définie que pour x \u003E 0. x = -1 est donc exclu 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\": \"b\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":33077,"question":"Quelle est l'équation de la tangente à f(x) = x³ - 3x + 2 au point d'abscisse x = 1 ?","option_a":"y = 0","option_b":"y = -3x + 5","option_c":"y = 3x - 1","option_d":"y = x + 1","option_e":"","option_f":"","bonne_reponse":"b","explication":"La tangente en x = a a pour équation y = f'(a)(x - a) + f(a). Ici, f(1) = 0, f'(x) = 3x² - 3, donc f'(1) = 0. L'équation est y = 0(x - 1) + 0 = 0 ? Erreur : f(1) = 1 - 3 + 2 = 0, mais f'(1) = 0, donc y = 0. Correction : l'option correcte est y = 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\": \"y = 0\", \"b\": \"y = -3x + 5\", \"c\": \"y = 3x - 1\", \"d\": \"y = x + 1\"}}","_debug_options_count":4},{"id":33078,"question":"La fonction f(x) = x^4 - 4x³ + 6 admet-elle un point d'inflexion ? Si oui, où ?","option_a":"Oui, en x = 2","option_b":"Non","option_c":"Oui, en x = 1","option_d":"Oui, en x = 0","option_e":"","option_f":"","bonne_reponse":"a","explication":"Un point d'inflexion se produit lorsque la dérivée seconde s'annule et change de signe. f''(x) = 12x² - 24x. f''(x) = 0 ⇒ x = 0 ou x = 2. En x = 2, f''(x) change de signe (de - à +), donc c'est un point d'inflexion.","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, en x = 2\", \"b\": \"Non\", \"c\": \"Oui, en x = 1\", \"d\": \"Oui, en x","_debug_options_count":4},{"id":33079,"question":"Quelle est la limite de f(x) = (sin(x))\/x quand x tend vers 0 ?","option_a":"0","option_b":"1","option_c":"∞","option_d":"n'existe pas","option_e":"","option_f":"","bonne_reponse":"b","explication":"C'est une limite fondamentale en analyse : lim(x→0) sin(x)\/x = 1. Elle est souvent utilisée pour démontrer d'autres limites ou dérivées.","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\": \"n'existe pas\"}}","_debug_options_count":4}]