Quiz — Série n°2(Fonctions composées).pdf

Question 1 sur 10 20:00

[{"id":128793,"question":"Soit f(x) = x² et g(x) = 3x + 1. Quelle est la fonction composée (f∘g)(x) ?","option_a":"A. (3x + 1)²","option_b":"B. 3x² + 1","option_c":"C. 9x² + 6x + 1","option_d":"D. 3x + 1²","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction composée (f∘g)(x) = f(g(x)) = f(3x + 1) = (3x + 1)². Il suffit d'appliquer la fonction f à l'expression de g(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\": \"A. (3x + 1)²\", \"b\": \"B. 3x² + 1\", \"c\": \"C. 9x² + 6x + 1\", \"d\":","_debug_options_count":4},{"id":128794,"question":"La dérivée de la fonction composée f(g(x)) est toujours égale à f'(g(x)) × g'(x).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est la formule de dérivation des fonctions composées, connue sous le nom de 'règle de la chaîne'. Cette affirmation est donc vraie.","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":128795,"question":"Soit h(x) = sin(2x). Quelle est la dérivée h'(x) ?","option_a":"A. cos(2x)","option_b":"B. 2cos(2x)","option_c":"C. 2sin(2x)","option_d":"D. cos(x)","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée de sin(u) est u' × cos(u). Ici, u = 2x, donc u' = 2. Ainsi, h'(x) = 2 × cos(2x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"A. cos(2x)\", \"b\": \"B. 2cos(2x)\", \"c\": \"C. 2sin(2x)\", \"d\": \"D. cos","_debug_options_count":4},{"id":128796,"question":"Soit k(x) = e^(x²). Quelle est la dérivée k'(x) ?","option_a":"A. e^(x²)","option_b":"B. 2x × e^(x²)","option_c":"C. x × e^(x²)","option_d":"D. e^(2x)","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée de e^(u) est u' × e^(u). Ici, u = x², donc u' = 2x. Ainsi, k'(x) = 2x × e^(x²).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"A. e^(x²)\", \"b\": \"B. 2x × e^(x²)\", \"c\": \"C. x × e^(x²)\", \"d\"","_debug_options_count":4},{"id":128797,"question":"La fonction composée (g∘f)(x) est toujours égale à (f∘g)(x).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La composition de fonctions n'est pas commutative. Par exemple, si f(x) = x + 1 et g(x) = x², alors (f∘g)(x) = x² + 1 et (g∘f)(x) = (x + 1)². Ces deux fonctions ne sont pas égales.","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":128798,"question":"Soit m(x) = ln(3x + 5). Quelle est la dérivée m'(x) ?","option_a":"A. 1\/(3x + 5)","option_b":"B. 3\/(3x + 5)","option_c":"C. 1\/(x + 5\/3)","option_d":"D. 3x\/(3x + 5)","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée de ln(u) est u'\/u. Ici, u = 3x + 5, donc u' = 3. Ainsi, m'(x) = 3\/(3x + 5).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"A. 1\/(3x + 5)\", \"b\": \"B. 3\/(3x + 5)\", \"c\": \"C. 1\/(x + 5\/3)\", \"d\":","_debug_options_count":4},{"id":128799,"question":"Soit n(x) = (2x + 1)^4. Quelle est la dérivée n'(x) ?","option_a":"A. 4(2x + 1)^3","option_b":"B. 8(2x + 1)^3","option_c":"C. 4(2x + 1)^4","option_d":"D. 8x(2x + 1)^3","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée de u^4 est 4u^3 × u'. Ici, u = 2x + 1, donc u' = 2. Ainsi, n'(x) = 4 × (2x + 1)^3 × 2 = 8(2x + 1)^3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"A. 4(2x + 1)^3\", \"b\": \"B. 8(2x + 1)^3\", \"c\": \"C. 4(2x + 1)^4\", \"d","_debug_options_count":4},{"id":128800,"question":"La dérivée de la fonction composée f(g(x)) peut s'écrire sans utiliser la règle de la chaîne.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La règle de la chaîne est indispensable pour dériver une fonction composée. Sans elle, il est impossible de calculer correctement la dérivée.","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":128801,"question":"Soit p(x) = cos(5x - 2). Quelle est la dérivée p'(x) ?","option_a":"A. -sin(5x - 2)","option_b":"B. 5sin(5x - 2)","option_c":"C. -5sin(5x - 2)","option_d":"D. sin(5x - 2)","option_e":"","option_f":"","bonne_reponse":"c","explication":"La dérivée de cos(u) est -u' × sin(u). Ici, u = 5x - 2, donc u' = 5. Ainsi, p'(x) = -5 × sin(5x - 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\": \"A. -sin(5x - 2)\", \"b\": \"B. 5sin(5x - 2)\", \"c\": \"C. -5sin(5x - 2)\"","_debug_options_count":4},{"id":128802,"question":"La fonction composée f(g(x)) est définie si et seulement si l'image de g(x) appartient au domaine de définition de f.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour que f(g(x)) soit définie, il faut que g(x) appartienne au domaine de définition de f. C'est une condition nécessaire et suffisante.","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}]