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{"common":{"save":0,"post_id":"7851","level":3,"total":10,"point":10,"point_extra":0},"segment":[{"id":"5871","post_id":"7851","mon_id":"1159285","chapter_id":"1159380","question":"<p>Cho hàm s\u1ed1 <span class=\"math-tex\">$y=f(x)$<\/span> có \u0111\u1ea1o hàm là <span class=\"math-tex\">$f^\\prime(x)=8x^3+\\sin x,\\forall x\\in R$<\/span> và <span class=\"math-tex\">$f(0)=3$<\/span>. Bi\u1ebft <span class=\"math-tex\">$F(x)$<\/span> là nguyên hàm c\u1ee7a <span class=\"math-tex\">$f(x)$<\/span> th\u1ecfa mãn <span class=\"math-tex\">$F(0)=2$<\/span>, khi \u0111ó <span class=\"math-tex\">$F(1)$<\/span> b\u1eb1ng<\/p>","options":["<strong>A.<\/strong> <span class=\"math-tex\">$\\dfrac{32}{5}+\\cos1$<\/span>","<strong>B.<\/strong> <span class=\"math-tex\">$\\dfrac{32}{5}-\\cos1$<\/span>","<strong>C.<\/strong> <span class=\"math-tex\">$\\dfrac{32}{5}-\\sin1$<\/span>","<strong>D.<\/strong> <span class=\"math-tex\">$\\dfrac{32}{5}+\\sin1$<\/span>"],"correct":"3","level":"3","hint":"","answer":"<p>Ch\u1ecdn <span style=\"color:#16a085;\"><strong>C.<\/strong> <span class=\"math-tex\">$\\dfrac{32}{5}-\\sin1$<\/span>.<\/span><\/p><p>Ta có <span class=\"math-tex\">$f(x)=\\int(8x^3+\\sin x)dx$<\/span> <span class=\"math-tex\">$=2x^4-\\cos x+C_1$<\/span>.<\/p><p>Mà <span class=\"math-tex\">$f(0)=3$<\/span> nên <span class=\"math-tex\">$2.0^4-\\cos 0+C_1=3$<\/span>, suy ra <span class=\"math-tex\">$C_1=4$<\/span>.<\/p><p>V\u1eady <span class=\"math-tex\">$f(x)=2x^4-\\cos x+4$<\/span>.<\/p><p>Khi \u0111ó <span class=\"math-tex\">$F(x)=\\int(2x^4-\\cos x+4)dx=\\dfrac{2}{5}x^5-\\sin x+4x+C_2$<\/span>.<\/p><p>Mà <span class=\"math-tex\">$F(0)=2$<\/span> nên <span class=\"math-tex\">$0+C_2=2$<\/span>, suy ra <span class=\"math-tex\">$C_2=2$<\/span>.<\/p><p>V\u1eady <span class=\"math-tex\">$F(x)=\\dfrac{2}{5}x^5-\\sin x+4x+2$<\/span>.<\/p><p>Do \u0111ó <span class=\"math-tex\">$F(1)=\\dfrac{2}{5}.1^5-\\sin1+4.1+2=\\dfrac{32}{5}-\\sin1$<\/span>.<\/p>","type":"choose","extra_type":"classic","user_id":"131","test":"0","date":"2024-10-20 08:23:24","option_type":"math","len":0},{"id":"5874","post_id":"7851","mon_id":"1159285","chapter_id":"1159380","question":"<p>Cho hàm s\u1ed1 <span class=\"math-tex\">$f(x)$<\/span> tho\u1ea3 mãn <span class=\"math-tex\">$f(2)=-\\dfrac{1}{25}$<\/span> và <span class=\"math-tex\">$f^\\prime(x)=4x^3[f(x)]^2$<\/span> v\u1edbi m\u1ecdi x ∈ R. Giá tr\u1ecb c\u1ee7a <span class=\"math-tex\">$f(1)$<\/span> b\u1eb1ng<\/p>","options":["<strong>A.<\/strong> <span class=\"math-tex\">$-\\dfrac{391}{400}$<\/span>","<strong>B.<\/strong> <span class=\"math-tex\">$-\\dfrac{1}{40}$<\/span>","<strong>C.<\/strong> <span class=\"math-tex\">$-\\dfrac{41}{400}$<\/span>","<strong>D.<\/strong> <span class=\"math-tex\">$-\\dfrac{1}{10}$<\/span>"],"correct":"4","level":"3","hint":"","answer":"<p>Ch\u1ecdn <span style=\"color:#16a085;\"><strong>D.<\/strong> <span class=\"math-tex\">$-\\dfrac{1}{10}$<\/span>.<\/span><\/p><p><span class=\"math-tex\">$f^\\prime(x)=4x^3[f(x)]^2$<\/span> ⇒ <span class=\"math-tex\">$-\\dfrac{f^\\prime(x)}{[f(x)]^2}=-4x^2$<\/span> ⇒ <span class=\"math-tex\">$\\bigg[\\dfrac{1}{f(x)}\\bigg]^\\prime=-4x^3$<\/span> ⇒ <span class=\"math-tex\">$\\dfrac{1}{f(x)}=-x^4+C$<\/span>.<\/p><p>Vì <span class=\"math-tex\">$f(2)=-\\dfrac{1}{25}$<\/span> nên <span class=\"math-tex\">$-25=-2^4+C$<\/span>, suy ra <span class=\"math-tex\">$C=-9$<\/span>.<\/p><p>V\u1eady <span class=\"math-tex\">$f(x)=-\\dfrac{1}{x^4+9}$<\/span> ⇒ <span class=\"math-tex\">$f(1)=-\\dfrac{1}{10}$<\/span>.<\/p>","type":"choose","extra_type":"classic","user_id":"131","test":"0","date":"2024-10-20 08:30:22","option_type":"math","len":0},{"id":"5879","post_id":"7851","mon_id":"1159285","chapter_id":"1159380","question":"<p>Cho hàm s\u1ed1 <span class=\"math-tex\">$y=f(x)$<\/span> \u0111\u1ed3ng bi\u1ebfn và có \u0111\u1ea1o hàm liên t\u1ee5c trên \u211d th\u1ecfa mãn <span class=\"math-tex\">$(f^\\prime(x))^2=f(x)e^x,\\forall x\\in R$<\/span> và <span class=\"math-tex\">$f(0)=2$<\/span>. Khi \u0111ó <span class=\"math-tex\">$f(2)$<\/span> thu\u1ed9c kho\u1ea3ng nào sau \u0111ây?<\/p>","options":["<strong>A.<\/strong> (12 ; 13)","<strong>B.<\/strong> (9 ; 10)","<strong>C.<\/strong> (11 ; 12)","<strong>D.<\/strong> (13 ; 14)"],"correct":"2","level":"3","hint":"","answer":"<p>Ch\u1ecdn <span style=\"color:#16a085;\"><strong>B.<\/strong> (9 ; 10).<\/span><\/p><p>Vì hàm s\u1ed1 <span class=\"math-tex\">$y=f(x)$<\/span> \u0111\u1ed3ng bi\u1ebfn và có \u0111\u1ea1o hàm liên t\u1ee5c trên \u211d và <span class=\"math-tex\">$f(0)=2$<\/span> nên <span class=\"math-tex\">$f^\\prime(x)\\ge0$<\/span> và <span class=\"math-tex\">$f(x)>0$<\/span> v\u1edbi m\u1ecdi <span class=\"math-tex\">$x\\in [0;+\\infty)$<\/span>.<\/p><p>Vì <span class=\"math-tex\">$(f^\\prime(x))^2=f(x)e^x,\\forall x\\in R$<\/span> nên <span class=\"math-tex\">$f^\\prime(x)=\\sqrt{f(x)}.e^\\frac{x}{2},\\forall x\\in[0;+\\infty)$<\/span>.<\/p><p>Suy ra <span class=\"math-tex\">$\\dfrac{f^\\prime(x)}{2\\sqrt{f(x)}}=\\dfrac{1}{2}e^\\frac{x}{2},\\forall x\\in[0;+\\infty)$<\/span>.<\/p><p>L\u1ea5y nguyên hàm hai v\u1ebf ta \u0111\u01b0\u1ee3c <span class=\"math-tex\">$\\sqrt{f(x)}=e^\\frac{x}{2}+C,\\forall x\\in[0;+\\infty)$<\/span>.<\/p><p>Mà <span class=\"math-tex\">$f(0)=2$<\/span>, ta \u0111\u01b0\u1ee3c <span class=\"math-tex\">$C=\\sqrt{2}-1$<\/span>. <\/p><p>T\u1eeb \u0111ó tính \u0111\u01b0\u1ee3c <span class=\"math-tex\">$f(2)=(e+\\sqrt{2}-1)^2\\approx9,81$<\/span> ∈ (9 ; 10).<\/p>","type":"choose","extra_type":"classic","user_id":"131","test":"0","date":"2024-10-20 08:51:54","option_type":"txt","len":1}]}