Quiz — تمارين شاملة حول المناقشة البيانية بكل انواعها - الأستاذ مويات

Question 1 sur 10 20:00

[{"id":39380,"question":"Quelle est la dérivée de la fonction f(x) = x² + 3x - 5 ?","option_a":"A) 2x + 3","option_b":"B) x² + 3","option_c":"C) 2x - 5","option_d":"D) x + 3","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de x² est 2x, celle de 3x est 3, et la dérivée de -5 est 0. Donc f'(x) = 2x + 3.","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) 2x + 3\", \"b\": \"B) x² + 3\", \"c\": \"C) 2x - 5\", \"d\": \"D) x + 3\"}","_debug_options_count":4},{"id":39381,"question":"Si f'(x) = 4x - 8, pour quelles valeurs de x la fonction f est-elle décroissante ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction est décroissante lorsque f'(x) \u003C 0, soit 4x - 8 \u003C 0 → x \u003C 2.","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":39382,"question":"Quelle est la nature du point critique de la fonction f(x) = x³ - 3x² à x = 1 ?","option_a":"A) Minimum local","option_b":"B) Maximum local","option_c":"C) Point d'inflexion","option_d":"D) Aucun extremum","option_e":"","option_f":"","bonne_reponse":"b","explication":"f'(x) = 3x² - 6x. En x=1, f'(1)=0. f''(x)=6x-6, f''(1)=0. Mais f'''(x)=6≠0, donc c'est un point d'inflexion. Cependant, en analysant le signe de f' autour de x=1, on voit un changement de signe : f'(0.9)\u003E0 et f'(1.1)\u003C0, donc c'est un maximum local.","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) Minimum local\", \"b\": \"B) Maximum local\", \"c\": \"C) Point d'infl","_debug_options_count":4},{"id":39383,"question":"La fonction f(x) = x⁴ - 4x³ a-t-elle un point d'inflexion en x = 2 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"f''(x) = 12x² - 24x. En x=2, f''(2)=0. Mais f'''(x)=24x-24, f'''(2)=24≠0, donc c'est bien 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":39384,"question":"Quelle est l'équation de la tangente à la courbe de f(x) = x² + 1 au point d'abscisse x = 1 ?","option_a":"A) y = 2x + 1","option_b":"B) y = 2x","option_c":"C) y = 2x - 1","option_d":"D) y = x + 1","option_e":"","option_f":"","bonne_reponse":"c","explication":"f(1)=2, f'(1)=2. L'équation de la tangente est y = f'(1)(x-1) + f(1) → y = 2(x-1) + 2 → y = 2x.","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) y = 2x + 1\", \"b\": \"B) y = 2x\", \"c\": \"C) y = 2x - 1\", \"d\": \"D) ","_debug_options_count":4},{"id":39385,"question":"La fonction f(x) = x³ est-elle strictement croissante sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"f'(x) = 3x² ≥ 0 pour tout x ∈ ℝ, et f'(x)=0 seulement en x=0. Donc f est strictement croissante 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":39386,"question":"Quelle est la dérivée seconde de la fonction f(x) = e^x ?","option_a":"A) e^x","option_b":"B) 0","option_c":"C) 1","option_d":"D) x·e^x","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de e^x est e^x, donc la dérivée seconde est aussi e^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) e^x\", \"b\": \"B) 0\", \"c\": \"C) 1\", \"d\": \"D) x·e^x\"}}","_debug_options_count":4},{"id":39387,"question":"Si f'(x) = 0 pour tout x ∈ ℝ, alors la fonction f est nécessairement une constante.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Si f'(x)=0, alors f est constante sur tout intervalle où elle est dérivable. Cependant, si f n'est pas dérivable en certains points, elle peut ne pas être constante partout.","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":39388,"question":"Quelle est la limite de f(x) = (x² + 1)\/x quand x tend vers +∞ ?","option_a":"A) 0","option_b":"B) 1","option_c":"C) +∞","option_d":"D) -∞","option_e":"","option_f":"","bonne_reponse":"c","explication":"En divisant numérateur et dénominateur par x, on obtient f(x) = x + 1\/x. Quand x→+∞, 1\/x→0, donc f(x)→+∞.","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) 0\", \"b\": \"B) 1\", \"c\": \"C) +∞\", \"d\": \"D) -∞\"}}","_debug_options_count":4},{"id":39389,"question":"La fonction f(x) = sin(x) a-t-elle des points critiques ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"f'(x) = cos(x). Les points critiques sont les solutions de cos(x)=0, soit x = π\/2 + kπ pour k ∈ ℤ.","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}]