Quiz — td_courb.pdf

Question 1 sur 10 20:00

[{"id":10513,"question":"Soit la fonction f(x) = x² - 4x + 3. Quelle est la dérivée f'(x) ?","option_a":"2x - 4","option_b":"2x + 4","option_c":"x² - 4","option_d":"x - 4","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de x² est 2x, celle de -4x est -4, et celle de 3 est 0. Donc f'(x) = 2x - 4.","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 - 4\", \"b\": \"2x + 4\", \"c\": \"x² - 4\", \"d\": \"x - 4\"}}","_debug_options_count":4},{"id":10514,"question":"La fonction f(x) = x³ - 3x² a-t-elle un extremum en x = 2 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée f'(x) = 3x² - 6x s'annule en x = 2. En étudiant le signe de f'(x), on voit qu'elle change de signe autour de x = 2, donc il y a un extremum 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":10515,"question":"Quelle est l'équation de la tangente à la courbe de f(x) = x² en x = 1 ?","option_a":"y = 2x - 1","option_b":"y = x + 1","option_c":"y = 2x + 1","option_d":"y = x - 1","option_e":"","option_f":"","bonne_reponse":"a","explication":"La pente de la tangente est f'(1) = 2*1 = 2. L'équation est y = f'(1)(x - 1) + f(1) = 2(x - 1) + 1 = 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\": \"y = 2x - 1\", \"b\": \"y = x + 1\", \"c\": \"y = 2x + 1\", \"d\": \"y = x - 1","_debug_options_count":4},{"id":10516,"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 ne peut pas y être dérivable.","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":10517,"question":"Quelle est la dérivée de la fonction f(x) = e^(2x) ?","option_a":"2e^(2x)","option_b":"e^(2x)","option_c":"2x e^(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\": \"2x e^(2x)\", \"d\": \"e^x\"}}","_debug_options_count":4},{"id":10518,"question":"Si f'(x) \u003E 0 pour tout x dans un intervalle I, que peut-on dire de la fonction f sur I ?","option_a":"f est décroissante sur I","option_b":"f est constante sur I","option_c":"f est croissante sur I","option_d":"f est périodique sur I","option_e":"","option_f":"","bonne_reponse":"c","explication":"Si la dérivée est positive sur un intervalle, la fonction est strictement croissante sur cet intervalle.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"f est décroissante sur I\", \"b\": \"f est constante sur I\", \"c\": \"f","_debug_options_count":4},{"id":10519,"question":"La tangente à une courbe en un point peut-elle être verticale ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Oui, si la dérivée tend vers l'infini en ce point, la tangente est verticale (exemple : f(x) = √x 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":10520,"question":"Quelle est la dérivée de la fonction f(x) = ln(x² + 1) ?","option_a":"2x \/ (x² + 1)","option_b":"2x","option_c":"1 \/ (x² + 1)","option_d":"2 \/ (x² + 1)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de ln(u(x)) est u'(x)\/u(x). Ici u(x) = x² + 1, donc u'(x) = 2x. Ainsi f'(x) = 2x \/ (x² + 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\": \"2x \/ (x² + 1)\", \"b\": \"2x\", \"c\": \"1 \/ (x² + 1)\", \"d\": \"2 \/ (x² ","_debug_options_count":4},{"id":10521,"question":"Si f'(a) = 0 et f''(a) \u003E 0, que peut-on conclure sur la fonction f en x = a ?","option_a":"f a un maximum en a","option_b":"f a un minimum en a","option_c":"f est constante en a","option_d":"f est décroissante en a","option_e":"","option_f":"","bonne_reponse":"b","explication":"Si la dérivée s'annule et que la dérivée seconde est positive, la fonction a un minimum local en ce point.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"f a un maximum en a\", \"b\": \"f a un minimum en a\", \"c\": \"f est con","_debug_options_count":4},{"id":10522,"question":"La fonction f(x) = sin(x) a-t-elle une tangente horizontale en x = π\/2 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de sin(x) est cos(x). En x = π\/2, cos(π\/2) = 0, donc la tangente est horizontale.","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}]