Quiz — 63e94a20afa00_exercice 9.pdf

Question 1 sur 5 10:00

[{"id":355,"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'une fonction polynomiale s'obtient en appliquant la formule (x^n)' = n*x^(n-1) à chaque terme. Ici, on obtient : (x³)' = 3x², (-2x²)' = -4x, (5x)' = 5 et (-1)' = 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\": \"3x² - 4x + 5\", \"b\": \"3x² - 4x + 5x\", \"c\": \"x² - 4x + 5\", \"d\": ","_debug_options_count":4},{"id":356,"question":"Soit f(x) = (2x + 1)\/(x - 3). Quelle est la dérivée f'(x) ?","option_a":"(2x - 7)\/(x - 3)²","option_b":"(2x + 7)\/(x - 3)²","option_c":"(2x - 7)\/(x + 3)²","option_d":"(2x + 7)\/(x + 3)²","option_e":"","option_f":"","bonne_reponse":"a","explication":"On utilise la formule de dérivation d'un quotient : (u\/v)' = (u'v - uv')\/v². Ici, u = 2x + 1 (u' = 2) et v = x - 3 (v' = 1). Le calcul donne : (2*(x-3) - (2x+1)*1)\/(x-3)² = (2x - 6 - 2x - 1)\/(x-3)² = -7\/(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\": \"(2x - 7)\/(x - 3)²\", \"b\": \"(2x + 7)\/(x - 3)²\", \"c\": \"(2x - 7)\/(x","_debug_options_count":4},{"id":357,"question":"Quelle est la limite de la fonction f(x) = (3x² + 2x - 5)\/(x² - 4) quand x tend vers l'infini ?","option_a":"3","option_b":"2","option_c":"0","option_d":"1","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour les fonctions rationnelles, la limite en l'infini est égale au rapport des coefficients dominants. Ici, 3x²\/x² = 3. Les termes de degré inférieur 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\": \"3\", \"b\": \"2\", \"c\": \"0\", \"d\": \"1\"}}","_debug_options_count":4},{"id":358,"question":"Soit f(x) = e^(2x). Quelle est la valeur de f'(0) ?","option_a":"0","option_b":"1","option_c":"2","option_d":"e","option_e":"","option_f":"","bonne_reponse":"c","explication":"La dérivée de e^(kx) est k*e^(kx). Ici, f'(x) = 2*e^(2x). En x=0, f'(0) = 2*e^0 = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"0\", \"b\": \"1\", \"c\": \"2\", \"d\": \"e\"}}","_debug_options_count":4},{"id":359,"question":"Quelle est la solution de l'équation ln(x) = 2 ?","option_a":"x = e²","option_b":"x = 2","option_c":"x = 1\/e²","option_d":"x = √e","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction logarithme népérien ln(x) est bijective. L'équation ln(x) = 2 équivaut à x = e², car ln(e²) = 2 par 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\": \"x = e²\", \"b\": \"x = 2\", \"c\": \"x = 1\/e²\", \"d\": \"x = √e\"}}","_debug_options_count":4}]