{"common":{"save":0,"post_id":"7450","level":2,"total":10,"point":10,"point_extra":0},"segment":[{"id":"5991","post_id":"7450","mon_id":"1158924","chapter_id":"1159187","question":"<p>Cho hàm s\u1ed1 <span class=\"math-tex\">$f(x)=(2022-x)(2021-2x)(2020-3x)...(1-2022x)$<\/span>. Tính <span class=\"math-tex\">$f^\\prime(-1)$<\/span>.<\/p>","options":["<b>A.<\/b> <span class=\"math-tex\">$1011.2023^{2022}$<\/span>","<b>B.<\/b> <span class=\"math-tex\">$-2022.2023^{1011}$<\/span>","<b>C.<\/b> <span class=\"math-tex\">$2022.1011^{2023}$<\/span>","<b>D.<\/b> <span class=\"math-tex\">$-1011.2023^{2022}$<\/span>"],"correct":"4","level":"2","hint":"","answer":"<p>Ch\u1ecdn <span style=\"color:#16a085;\"><b>D.<\/b> <span class=\"math-tex\">$-1011.2023^{2022}$<\/span><\/span><\/p><p>Ta có: <span class=\"math-tex\">$f(x)=(2022-x)(2021-2x)(2020-3x)...(1-2022x)$<\/span><\/p><p>⇒ <span class=\"math-tex\">$f^\\prime(x)=(-1).(2021-2x)(2020-3x)...(1-2022x) \\\\+(-2).(2021-x)(2020-3x)...(1-2022x) \\\\+...+(-2022).(2021-x)(2021-2x)...(2-2021x)$<\/span><\/p><p>⇒ <span class=\"math-tex\">$f^\\prime(-1)=(-1).2023^{2021}+(-2).2023^{2021}+...+(-2022).2023^{2021}$<\/span><\/p><p><span class=\"math-tex\">$=(-1).(1+2+...+2022).2023^{2021}$<\/span><\/p><p><span class=\"math-tex\">$=(-1).\\dfrac{2022.2023}{2}.2023^{2021}$<\/span><\/p><p><span class=\"math-tex\">$-1011.2023^{2021}$<\/span><\/p>","type":"choose","extra_type":"classic","user_id":"131","test":"0","date":"2024-05-30 05:26:44","option_type":"math","len":0},{"id":"5992","post_id":"7450","mon_id":"1158924","chapter_id":"1159187","question":"<p>Cho hàm s\u1ed1 <span class=\"math-tex\">$f(x)=e^{\\sin x}$<\/span>. T\u1eadp nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng trình <span class=\"math-tex\">$f^\\prime(x)+f(x)=0$<\/span> là<\/p>","options":["<b>A.<\/b> <span class=\"math-tex\">$S=\\{\\pi+k\\pi | k\\in Z\\}$<\/span>","<b>B.<\/b> <span class=\"math-tex\">$S=\\{k2\\pi | k\\in Z\\}$<\/span>","<b>C.<\/b> <span class=\"math-tex\">$S=\\{\\pi+k2\\pi | k\\in Z\\}$<\/span>","<b>D.<\/b> <span class=\"math-tex\">$S=\\{k\\pi | k\\in Z\\}$<\/span>"],"correct":"3","level":"2","hint":"","answer":"<p>Ch\u1ecdn <span style=\"color:#16a085;\"><b>C.<\/b> <span class=\"math-tex\">$S=\\{\\pi+k2\\pi | k\\in Z\\}$<\/span><\/span><\/p><p>Ta có: <span class=\"math-tex\">$f(x)=e^{\\sin x}$<\/span> ⇒ <span class=\"math-tex\">$f^\\prime(x)=\\cos xe^{\\sin x}$<\/span>.<\/p><p>Ph\u01b0\u01a1ng trình \u0111ã cho tr\u1edf thành<\/p><p><span class=\"math-tex\">$\\cos xe^{\\sin x}+e^{\\sin x}=0$<\/span><\/p><p>⇔ <span class=\"math-tex\">$e^{\\sin x}(\\cos x+1)=0$<\/span> <\/p><p>⇔ <span class=\"math-tex\">$\\cos x=-1$<\/span> (vì <span class=\"math-tex\">$e^{\\sin x} > 0$<\/span> v\u1edbi m\u1ecdi x∈R)<\/p><p>⇔ <span class=\"math-tex\">$x=\\pi+k2\\pi$<\/span> (k∈Z)<\/p><p>V\u1eady t\u1eadp nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng trình là <span class=\"math-tex\">$S=\\{\\pi+k2\\pi | k\\in Z\\}$<\/span><\/p>","type":"choose","extra_type":"classic","user_id":"131","test":"0","date":"2024-05-30 05:40:31","option_type":"math","len":0},{"id":"5994","post_id":"7450","mon_id":"1158924","chapter_id":"1159187","question":"<p>Xét hàm s\u1ed1 <span class=\"math-tex\">$f(x)=\\ln\\bigg(\\dfrac{x}{x+1}\\bigg)$<\/span> trên kho\u1ea3ng <span class=\"math-tex\">$(0;+\\infty)$<\/span>. Giá tr\u1ecb c\u1ee7a bi\u1ec3u th\u1ee9c <span class=\"math-tex\">$P=f^\\prime(1)+f^\\prime(2)+...+f^\\prime(99)$<\/span> b\u1eb1ng:<\/p>","options":["<b>A.<\/b> <span class=\"math-tex\">$P=\\dfrac{1}{100}$<\/span>","<b>B.<\/b> <span class=\"math-tex\">$P=\\dfrac{49}{50}$<\/span>","<b>C.<\/b> <span class=\"math-tex\">$P=\\dfrac{99}{100}$<\/span>","<b>D.<\/b> <span class=\"math-tex\">$P=\\dfrac{1}{50}$<\/span>"],"correct":"3","level":"2","hint":"","answer":"<p>Ch\u1ecdn <span style=\"color:#16a085;\"><b>C.<\/b> <span class=\"math-tex\">$P=\\dfrac{99}{100}$<\/span><\/span><\/p><p>Ta có: <span class=\"math-tex\">$f(x)=\\ln\\bigg(\\dfrac{x}{x+1}\\bigg)$<\/span> = <span class=\"math-tex\">$\\ln x-\\ln(x+1)$<\/span> (vì x > 0)<\/p><p>Suy ra <span class=\"math-tex\">$f^\\prime(x)=\\dfrac{1}{x}-\\dfrac{1}{x+1}$<\/span><\/p><p>Do \u0111ó <span class=\"math-tex\">$P=\\dfrac{1}{1}-\\dfrac{1}{2}+\\dfrac{1}{2}-\\dfrac{1}{3}+...+\\dfrac{1}{99}-\\dfrac{1}{100}=\\dfrac{1}{1}-\\dfrac{1}{100}=\\dfrac{99}{100}$<\/span><\/p>","type":"choose","extra_type":"classic","user_id":"131","test":"0","date":"2024-05-30 05:49:28","option_type":"math","len":0}]}