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18C
22D
26B
Giải thích thêm:
ta có: v=s'(t)=3t²-6t+6
a=s"(t)=6t-6
Thời điểm gia tốc bị triệt tiêu khi a=0
⇔6t-6=0
⇔t=1
Vậy v=3.1²-6.1+6=3 (m/s)
32A
34C
35A
cho mình hỏi là tại sao ở câu 26 lại phải đạo hàm thêm lần nữa vậy?
1.a
\(\lim\limits_{x\rightarrow2}\dfrac{x^3+3x^2-9x-2}{x^3-x-6}=\lim\limits_{x\rightarrow2}\dfrac{\left(x-2\right)\left(x^2+5x+1\right)}{\left(x-2\right)\left(x^2+2x+3\right)}\)
\(=\lim\limits_{x\rightarrow2}\dfrac{x^2+5x+1}{x^2+2x+3}=\dfrac{15}{11}\)
b.
\(\lim\limits_{x\rightarrow-\infty}\left(\sqrt{x^2-x+3}+x\right)=\lim\limits_{x\rightarrow-\infty}\dfrac{-x+3}{\sqrt{x^2-x+3}-x}\)
\(=\lim\limits_{x\rightarrow-\infty}\dfrac{-1+\dfrac{3}{x}}{-\sqrt{1-\dfrac{1}{x}+\dfrac{3}{x^2}}-1}=\dfrac{-1}{-2}=\dfrac{1}{2}\)
4.
\(\lim\limits_{x\rightarrow8}f\left(x\right)=\lim\limits_{x\rightarrow8}\dfrac{\sqrt[3]{x}-2}{x-8}=\lim\limits_{x\rightarrow8}\dfrac{x-8}{\left(x-8\right)\left(\sqrt[3]{x^2}+2\sqrt[3]{x}+4\right)}=\lim\limits_{x\rightarrow8}\dfrac{1}{\sqrt[3]{x^2}+2\sqrt[3]{x}+4}\)
\(=\dfrac{1}{4+4+4}=\dfrac{1}{12}\)
\(f\left(8\right)=3.8-20=4\)
\(\Rightarrow\lim\limits_{x\rightarrow8}f\left(x\right)\ne f\left(8\right)\)
\(\Rightarrow\) Hàm gián đoạn tại \(x=8\)
5.
\(\lim\limits_{x\rightarrow0^+}f\left(x\right)=\lim\limits_{x\rightarrow0^+}\dfrac{\sqrt[]{1+2x}-1+1-\sqrt[3]{1+3x}}{x}=\lim\limits_{x\rightarrow0^+}\dfrac{\dfrac{2x}{\sqrt[]{1+2x}+1}-\dfrac{3x}{1+\sqrt[3]{1+3x}+\sqrt[3]{\left(1+3x\right)^2}}}{x}\)
\(=\lim\limits_{x\rightarrow0^+}\left(\dfrac{2}{\sqrt[]{1+2x}+1}-\dfrac{3}{1+\sqrt[3]{1+3x}+\sqrt[3]{\left(1+3x\right)^2}}\right)=\dfrac{2}{1+1}-\dfrac{3}{1+1+1}=0\)
\(f\left(0\right)=\lim\limits_{x\rightarrow0^-}f\left(x\right)=\lim\limits_{x\rightarrow0^-}\left(3x^2-2x\right)=0\)
\(\Rightarrow\lim\limits_{x\rightarrow0^+}f\left(x\right)=\lim\limits_{x\rightarrow0^-}f\left(x\right)=f\left(0\right)\)
\(\Rightarrow\) Hàm liên tục tại \(x=0\)
6.
\(\lim\limits_{x\rightarrow0^+}f\left(x\right)=\lim\limits_{x\rightarrow0^+}\dfrac{\sqrt[]{4x+1}-\sqrt[3]{6x+1}}{x^2}\)
\(=\lim\limits_{x\rightarrow0^+}\dfrac{\sqrt[]{4x+1}-\left(2x+1\right)+\left(2x+1-\sqrt[3]{6x+1}\right)}{x^2}\)
\(=\lim\limits_{x\rightarrow0^+}\dfrac{\dfrac{-x^2}{\sqrt[]{4x+1}+2x+1}+\dfrac{x^2\left(8x+12\right)}{\left(2x+1\right)^2+\left(2x+1\right)\sqrt[3]{6x+1}+\sqrt[3]{\left(6x+1\right)^2}}}{x^2}\)
\(=\lim\limits_{x\rightarrow0^+}\left(\dfrac{-1}{\sqrt[]{4x+1}+2x+1}+\dfrac{8x+12}{\left(2x+1\right)^2+\left(2x+1\right)\sqrt[3]{6x+1}+\sqrt[3]{\left(6x+1\right)^2}}\right)\)
\(=\dfrac{-1}{1+1}+\dfrac{12}{1+1+1}=\dfrac{7}{2}\)
\(f\left(0\right)=\lim\limits_{x\rightarrow0^-}f\left(x\right)=\lim\limits_{x\rightarrow0^-}\left(2-3x\right)=2\)
\(\Rightarrow\lim\limits_{x\rightarrow0^+}f\left(x\right)\ne\lim\limits_{x\rightarrow0^-}f\left(x\right)\)
\(\Rightarrow\) Hàm gián đoạn tại \(x=0\)
Đáy là bát giác đều (8 cạnh) nên chóp có 8 mặt bên
Cộng thêm mặt đáy nữa nên ta sẽ có tổng cộng 9 mặt
\(L=\lim\limits_{x\rightarrow+\infty}\left(2x^2-\sqrt{x^2-x}.\sqrt[3]{8x^3+12x^2-3x}\right)\)
Đặt \(f\left(x\right)=2x^2-\sqrt{x^2-x}.\sqrt[3]{8x^3+12x^2-3x}\)
Ta có:
\(2.f\left(x\right)=4x^2-\sqrt{4x^2-4x}.\sqrt[3]{8x^3+12x^2-3x}\)
\(=1+\left(4x^2-1\right)-\sqrt{4x^2-4x}.\sqrt[3]{8x^3+12x^2-3x}\)
\(=1+\left(2x-1\right)\left(2x+1-\sqrt[3]{8x^3+12x^2-3x}\right)+\left(2x-1-\sqrt{4x^2-4x}\right).\sqrt[3]{8x^3+12x^2-3x}\)
Đặt \(A\left(x\right)=\left(2x-1\right)\left(2x+1-\sqrt[3]{8x^3+12x^2-3x}\right)\)
\(B\left(x\right)=\left(2x-1-\sqrt{4x^2-4x}\right).\sqrt[3]{8x^3+12x^2-3x}\)
\(A\left(x\right)=\left(2x-1\right)\left(2x+1-\sqrt[3]{8x^3+12x^2-3x}\right)\)
\(=\dfrac{\left(2x-1\right)\left(8x^3+12x^2+6x+1-8x^3-12x^2+3x\right)}{\left(2x+1\right)^2+\sqrt[3]{\left(8x^3+12x^2-3x\right)^2}+\left(2x+1\right)\sqrt[3]{8x^3+12x^2-3x}}\)
\(=\dfrac{\left(2x-1\right)\left(9x+1\right)}{\left(2x+1\right)^2+\sqrt[3]{\left(8x^3+12x^2-3x\right)^2}+\left(2x+1\right)\sqrt[3]{8x^3+12x^2-3x}}\)
\(\Rightarrow\lim\limits_{x\rightarrow+\infty}A\left(x\right)=\lim\limits_{x\rightarrow+\infty}\dfrac{\left(2-\dfrac{1}{x}\right)\left(9+\dfrac{1}{x}\right)}{\left(2+\dfrac{1}{x}\right)^2+\sqrt[3]{\left(8+\dfrac{12}{x}-\dfrac{3}{x^2}\right)^2}+\left(2+\dfrac{1}{x}\right)\sqrt[3]{8+\dfrac{12}{x}-\dfrac{3}{x^2}}}\)
\(=\dfrac{2.9}{2^2+4+2.2}\)
\(=\dfrac{3}{2}\)
\(B\left(x\right)=\left(2x-1-\sqrt{4x^2-4x}\right).\sqrt[3]{8x^3+12x^2-3x}\)
\(=\dfrac{\left(4x^2-4x+1-4x^2+4x\right).\sqrt[3]{8x^3+12x^2-3x}}{2x-1+\sqrt{4x^2-4x}}\)
\(=\dfrac{\sqrt[3]{8x^3+12x^2-3x}}{2x-1+\sqrt{4x^2-4x}}\)
\(\Rightarrow\lim\limits_{x\rightarrow+\infty}B\left(x\right)=\lim\limits_{x\rightarrow+\infty}\dfrac{\sqrt[3]{8+\dfrac{12}{x}-\dfrac{3}{x^2}}}{2-\dfrac{1}{x}+\sqrt{4-\dfrac{4}{x}}}\)
\(=\dfrac{2}{2+2}\)
\(=\dfrac{1}{2}\)
\(\Rightarrow2L=\lim\limits_{x\rightarrow+\infty}\left[2f\left(x\right)\right]\)
\(=\lim\limits_{x\rightarrow+\infty}\left[1+A\left(x\right)+B\left(x\right)\right]\)
\(=1+\lim\limits_{x\rightarrow+\infty}A\left(x\right)+\lim\limits_{x\rightarrow+\infty}B\left(x\right)\)
\(=1+\dfrac{3}{2}+\dfrac{1}{2}\)
\(=3\)
\(\Rightarrow L=\dfrac{3}{2}\)