4.8 KiB
4.8 KiB
| 1 | id | question | A | B | C | D | answer |
|---|---|---|---|---|---|---|---|
| 2 | 0 | 求极限：$\lim_{x\rightarrow0}\frac{\int_{x^2}^x{\frac{\sin\left(xt\right)}{t}}\mathrm{d}t}{x^2}=$____ | $\frac{5}{6}$ | 1 | $\frac{7}{6}$ | $\frac{4}{3}$ | B |
| 3 | 1 | 设$n$为正整数，求极限：$\lim_{x\rightarrow+\infty}\left[\frac{x^n}{\left(x-1\right)\left(x-2\right)\cdots\left(x-n\right)}\right]^x=$____ | $e^{\frac{(n-1)(n+1)}{2}}$ | $e^{\frac{(n-1)n}{2}}$ | $e^{\frac{n(n+1)}{2}}$ | $e^{\frac{n^{2}}{2}}$ | C |
| 4 | 2 | 设平面区域$D$由直线$y=\frac{1}{2}x-\frac{1}{2\sqrt{5}}$、$y=2x-\frac{2}{\sqrt{5}}$和$y=x$围成，函数$z=3xy+3$在$D$上的最大值和最小值分别是M和m，则____ | $M=6,m=3$ | $M=\dfrac{27}{5},m=3$ | $M=\dfrac{18}{5},m=3$ | $M={\frac{27}{5}},m={\frac{117}{40}}$ | D |
| 5 | 3 | 设函数$f\left(x\right)$连续，且$f\left(x\right)>0$，求积分：$int_0^1{\ln f\left(x+t\right)}\mathrm{d}t=$____ | $\int_0^x{\ln \frac{f\left( t+2 \right)}{f\left( t \right)}}\mathrm{d}t+\int_0^1{\ln f\left( t \right)}\mathrm{d}t$ | $\int_0^1{\ln \frac{f\left( t+2 \right)}{f\left( t \right)}}\mathrm{d}t+\int_0^1{\ln f\left( t \right)}\mathrm{d}t$ | $\int_0^2x{\ln \frac{f\left( t+1 \right)}{f\left( t \right)}}\mathrm{d}t+\int_0^1{\ln f\left( t \right)}\mathrm{d}t$ | $\int_0^x{\ln \frac{f\left( t+2 \right)}{f\left( t \right)}}\mathrm{d}t+\int_0^1{\ln f\left( t \right)}\mathrm{d}t$ | A |
| 6 | 4 | 设有界区域$\Omega$由平面$2x+y+2z=2$与三个坐标平面围成，$\Sigma$为整个表面的外侧；\\计算曲面积分：$I=\iint_{\Sigma}{\left(x^2+1\right)\mathrm{d}y\mathrm{d}z-2y\mathrm{d}z\mathrm{d}x+3z\mathrm{d}x\mathrm{d}y}=$____ | $\frac{1}{2}$ | 1 | $\frac{3}{2}$ | $\frac{5}{2}$ | A |
| 7 | 5 | 已知$\int_0^{+\infty}{\frac{\sin x}{x}\mathrm{d}x=\frac{\pi}{2}}$，则$\int_0^{+\infty}{\int_0^{+\infty}{\frac{\sin x\sin\left(x+y\right)}{x\left(x+y\right)}}}\mathrm{d}x\mathrm{d}y$=____ | $\frac{\pi ^2}{12}$ | $\frac{\pi ^2}{8}$ | $\frac{\pi ^2}{4}$ | $\frac{\pi ^2}{2}$ | B |
| 8 | 6 | 设曲线$C=\left\{(x,y,z):x={\sqrt{3}}\cos(t),y={\sqrt{3}}\sin(t),z={\frac{2}{3}}t^{\frac{3}{2}},0\leq t\leq5\right\}$，则曲线积分$\int_C(x^2+y^2)\mathrm{d}s=$____ | $\frac{3}{4}\left(16\sqrt{2}-3\sqrt{3}\right)$ | $2\bigl(16\sqrt{2}-3\sqrt{3}\bigr)$ | $\frac{9}{4}\left(16\sqrt{2}-3\sqrt{3}\right)$ | $\frac{3}{2}\left(16\sqrt{2}-3\sqrt{3}\right)$ | B |
| 9 | 7 | 计算二重积分:$\iint_D{x\mathrm{d}x\mathrm{d}y}=$.其中$D$为由直线$y=-x+2,x$轴以及曲线$y=\sqrt{2x-x^2}$所围成的平面区域.____ | $\frac{\pi}{4}+\frac{1}{3}$ | $\frac{\pi}{2}+\frac{1}{3}$ | $\frac{\pi}{2}+\frac{1}{4}$ | $\frac{\pi}{2}+\frac{2}{3}$ | A |
| 10 | 8 | 求极限：$L=\lim_{n\rightarrow\infty}\sqrt{n}\left(1-\sum_{k=1}^n{\frac{1}{n+\sqrt{k}}}\right)$=____ | $\frac{1}{3}$ | $\frac{2}{3}$ | 1 | $\frac{4}{3}$ | B |
| 11 | 9 | $x=1$是函数$f\left(x\right)=\frac{bx^2+x+1}{ax+1}$的可去间断点，求$a,b$的值？____ | $a=-1,b=-2$ | $a=-2,b=-1$ | $a=-2,b=-2$ | $a=-1,b=0$ | A |
| 12 | 10 | 求$\sum_{n=1}^{\infty}{\frac{\left(-1\right)^{n+1}-2^n}{n}x^n}$的和函数.____ | $\ln\left(1-2x-x^2\right),x\in\left[-\dfrac{1}{2},\dfrac{1}{2}\right)$ | $\ln\left(1-x-x^2\right),x\in\left(-\dfrac{1}{2},\dfrac{1}{2}\right)$ | $\ln\left(1-2x-2x^2\right),x\in\left[-\dfrac{1}{2},\dfrac{1}{2}\right)$ | $\ln\left(1-x-2x^2\right),x\in\left[-\dfrac{1}{2},\dfrac{1}{2}\right)$ | D |
| 13 | 11 | 求极限：$\lim_{x\rightarrow0}\frac{\sqrt{1+x\cos x}-\sqrt{1+\sin x}}{x^3}=$____ | $-\dfrac{1}{3}$ | $-\dfrac{1}{4}$ | $-\dfrac{1}{5}$ | $-\dfrac{1}{6}$ | D |
| 14 | 12 | 求极限：$\lim_{n\rightarrow\infty}\sum_{k=1}^n{\frac{k}{\left(k+1\right)!}}=$____ | 1 | 0 | -1 | 2 | A |
| 15 | 13 | 已知曲线C是圆$(x-1)^{2}+(y-6)^{2}=25$上从点$A(1,1)$沿逆时针方向到$B(4,2)$的一段弧，则$\oint_{C}(3\ln(1+y)+5x^{2})\mathrm{d}x+\Bigl({\frac{3x}{1+y}}-2y\Bigr)\mathrm{d}y=$____ | $108+3\ln\Bigl(\frac{27}{2}\Bigr)$ | $3\ln\left(\dfrac{81}{2}\right)-102$ | $102+3\ln\Bigl(\frac{81}{2}\Bigr)$ | $3\ln\left(\dfrac{27}{2}\right)-97$ | C |
| 16 | 14 | 设$D$是全平面，$f\left(x\right)=\begin{cases} x\text{，}-1\leq x\leq2\\ 0\text{，}\text{其他}\\ \end{cases}$；计算$\iint_D{f\left(x\right)f\left(x^2-y\right)}\mathrm{d}\sigma=$____ | $\frac{9}{4}$ | $\frac{5}{2}$ | $\frac{11}{4}$ | 3 | A |
| 17 | 15 | 计算广义积分：$\int_0^{+\infty}{\frac{\mathrm{d}x}{\left(1+x^2\right)\left(1+x^4\right)}}$____ | $\frac{\pi}{8}$ | $\frac{\pi}{4}$ | $\frac{\pi}{2}$ | $\frac{\pi}{3}$ | B |
| 18 | 16 | 已知数列$\left\{a_n\right\}$，其中$a_n=\cos n\alpha$，其前$n$$项和为$S_n$.求：$S_n=$____ | $\frac{\cos \frac{n}{2}\alpha \cdot \sin \frac{n\alpha}{2}}{\sin \frac{\alpha}{2}}$ | $\frac{\cos \frac{n+1}{2}\alpha \cdot \sin \frac{n\alpha}{2}}{\sin \frac{\alpha}{2}}$ | $\frac{\cos \frac{n+1}{2}\alpha \cdot \sin \frac{n\alpha}{2}}{\sin \frac{\alpha}{3}}$ | $\frac{\cos \frac{n-1}{2}\alpha \cdot \sin \frac{n\alpha}{2}}{\sin \frac{\alpha}{2}}$ | B |
| 19 | 17 | 计算定积分：$\int_{-1}^1{\frac{\mathrm{d}x}{\left(1+\mathrm{e}^x\right)\left(1+x^2\right)}}$____ | $\frac{\pi}{8}$ | $\frac{\pi}{4}$ | $\frac{\pi}{2}$ | $\pi$ | B |
| 20 | 18 | 求极限：$\lim_{x\rightarrow0}\frac{\tan2x^2-x^2}{\sin x^2+3x^2}=$____ | $\frac{1}{2}$ | $\frac{1}{4}$ | $\frac{1}{8}$ | $\frac{3}{5}$ | B |