Quiz — td_derives_primitives.pdf

Question 1 sur 10 20:00

[{"id":27050,"question":"Quelle est la dérivée de la fonction f(x) = 3x² + 2x - 5 ?","option_a":"6x + 2","option_b":"3x + 2","option_c":"6x² + 2x","option_d":"3x² + 2","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'un polynôme se calcule terme par terme : (3x²)' = 6x, (2x)' = 2, et (-5)' = 0. Donc f'(x) = 6x + 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\": \"6x + 2\", \"b\": \"3x + 2\", \"c\": \"6x² + 2x\", \"d\": \"3x² + 2\"}}","_debug_options_count":4},{"id":27051,"question":"Si F est une primitive de f, alors f est la dérivée de F.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Par définition, si F est une primitive de f, alors F' = f. Donc f est bien la dérivée de F.","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":27052,"question":"Quelle est la primitive de la fonction f(x) = 4x³ ?","option_a":"x⁴ + C","option_b":"12x² + C","option_c":"x⁴","option_d":"16x⁴ + C","option_e":"","option_f":"","bonne_reponse":"a","explication":"La primitive d'un monôme xⁿ est (xⁿ⁺¹)\/(n+1) + C. Ici, n=3, donc la primitive est (4x⁴)\/4 + C = x⁴ + C.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"x⁴ + C\", \"b\": \"12x² + C\", \"c\": \"x⁴\", \"d\": \"16x⁴ + C\"}}","_debug_options_count":4},{"id":27053,"question":"La fonction f(x) = 1\/x admet-elle une primitive sur ℝ ?","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 continue sur tout ℝ. Elle admet des primitives sur ]-∞,0[ et ]0,+∞[ séparément.","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":27054,"question":"Quelle est la dérivée de la fonction f(x) = e^(2x) ?","option_a":"e^(2x)","option_b":"2e^(2x)","option_c":"e^(x)","option_d":"2e^x","option_e":"","option_f":"","bonne_reponse":"b","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\": \"b\", \"options\": {\"a\": \"e^(2x)\", \"b\": \"2e^(2x)\", \"c\": \"e^(x)\", \"d\": \"2e^x\"}}","_debug_options_count":4},{"id":27055,"question":"La primitive de sin(x) est -cos(x) + C.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de -cos(x) est sin(x), donc -cos(x) + C est bien une primitive de sin(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":27056,"question":"Quelle est la primitive de la fonction f(x) = 1\/(1+x²) ?","option_a":"arctan(x) + C","option_b":"ln(1+x²) + C","option_c":"1\/(1+x²) + C","option_d":"arcsin(x) + C","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de arctan(x) est 1\/(1+x²), donc arctan(x) + C est une primitive de 1\/(1+x²).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"arctan(x) + C\", \"b\": \"ln(1+x²) + C\", \"c\": \"1\/(1+x²) + C\", \"d\": ","_debug_options_count":4},{"id":27057,"question":"La fonction f(x) = x² + 3x admet-elle une primitive ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction f(x) = x² + 3x est un polynôme, donc continue sur ℝ. D'après le théorème fondamental de l'analyse, elle admet une primitive sur ℝ.","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":27058,"question":"Quelle est la dérivée de la fonction f(x) = ln(3x) ?","option_a":"1\/x","option_b":"3\/x","option_c":"1\/(3x)","option_d":"3\/(3x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de ln(u(x)) est u'(x)\/u(x). Ici, u(x)=3x, donc u'(x)=3. Ainsi, f'(x)=3\/(3x)=1\/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\": \"1\/x\", \"b\": \"3\/x\", \"c\": \"1\/(3x)\", \"d\": \"3\/(3x)\"}}","_debug_options_count":4},{"id":27059,"question":"Si F est une primitive de f, alors ∫ₐᵇ f(x) dx = F(b) - F(a).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est l'énoncé du théorème fondamental de l'analyse : l'intégrale d'une fonction sur [a,b] est égale à la différence des valeurs de sa primitive aux bornes.","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}]