矢量導(dǎo)數(shù)題目及答案

一、單項(xiàng)選擇題（每題2分，共10題）1.矢量函數(shù)\(\vec{A}(t)=t\vec{i}+t^{2}\vec{j}\)的導(dǎo)數(shù)\(\vec{A}'(t)\)是（）A.\(\vec{i}+2t\vec{j}\)B.\(2t\vec{i}+\vec{j}\)C.\(\vec{i}+t\vec{j}\)D.\(t\vec{i}+2\vec{j}\)2.若\(\vec{r}(t)=3\cost\vec{i}+3\sint\vec{j}\)，則\(\vec{r}'(t)\)為（）A.\(3\sint\vec{i}-3\cost\vec{j}\)B.\(-3\sint\vec{i}+3\cost\vec{j}\)C.\(-3\sint\vec{i}-3\cost\vec{j}\)D.\(3\sint\vec{i}+3\cost\vec{j}\)3.矢量\(\vec{F}(x,y,z)=x\vec{i}+y\vec{j}+z\vec{k}\)關(guān)于\(x\)的偏導(dǎo)數(shù)\(\frac{\partial\vec{F}}{\partialx}\)是（）A.\(\vec{i}\)B.\(\vec{j}\)C.\(\vec{k}\)D.\(\vec{i}+\vec{j}+\vec{k}\)4.已知\(\vec{A}(t)=(t^{2}+1)\vec{i}+t\vec{j}\)，\(\vec{A}'(1)\)的值是（）A.\(2\vec{i}+\vec{j}\)B.\(\vec{i}+2\vec{j}\)C.\(3\vec{i}+\vec{j}\)D.\(\vec{i}+3\vec{j}\)5.對(duì)于矢量函數(shù)\(\vec{B}(t)=\sint\vec{i}+\cost\vec{j}\)，\(\vec{B}''(t)\)是（）A.\(-\sint\vec{i}-\cost\vec{j}\)B.\(\sint\vec{i}-\cost\vec{j}\)C.\(-\sint\vec{i}+\cost\vec{j}\)D.\(\sint\vec{i}+\cost\vec{j}\)6.若\(\vec{r}(u,v)=u\vec{i}+v\vec{j}+uv\vec{k}\)，\(\frac{\partial\vec{r}}{\partialu}\)是（）A.\(\vec{i}+v\vec{k}\)B.\(\vec{j}+u\vec{k}\)C.\(\vec{i}+\vec{j}+\vec{k}\)D.\(u\vec{i}+v\vec{j}\)7.矢量\(\vec{F}(t)=e^{t}\vec{i}+e^{-t}\vec{j}\)的導(dǎo)數(shù)是（）A.\(e^{t}\vec{i}-e^{-t}\vec{j}\)B.\(e^{t}\vec{i}+e^{-t}\vec{j}\)C.\(-e^{t}\vec{i}+e^{-t}\vec{j}\)D.\(-e^{t}\vec{i}-e^{-t}\vec{j}\)8.設(shè)\(\vec{A}(x)=x^{3}\vec{i}+x^{2}\vec{j}\)，則\(\frac{d\vec{A}}{dx}\)為（）A.\(3x^{2}\vec{i}+2x\vec{j}\)B.\(2x^{2}\vec{i}+3x\vec{j}\)C.\(x^{2}\vec{i}+x\vec{j}\)D.\(3x\vec{i}+2\vec{j}\)9.矢量函數(shù)\(\vec{V}(t)=\lnt\vec{i}+t^{3}\vec{j}\)的導(dǎo)數(shù)\(\vec{V}'(t)\)是（）A.\(\frac{1}{t}\vec{i}+3t^{2}\vec{j}\)B.\(t\vec{i}+3t^{2}\vec{j}\)C.\(\frac{1}{t}\vec{i}-3t^{2}\vec{j}\)D.\(t\vec{i}-3t^{2}\vec{j}\)10.已知\(\vec{r}(t)=(t+1)\vec{i}+t^{2}\vec{j}+t^{3}\vec{k}\)，\(\vec{r}'(0)\)等于（）A.\(\vec{i}\)B.\(\vec{i}+2\vec{j}\)C.\(\vec{i}+3\vec{k}\)D.\(\vec{i}+2\vec{j}+3\vec{k}\)二、多項(xiàng)選擇題（每題2分，共10題）1.以下關(guān)于矢量導(dǎo)數(shù)性質(zhì)正確的有（）A.\((\vec{A}+\vec{B})'=\vec{A}'+\vec{B}'\)B.\((k\vec{A})'=k\vec{A}'\)（\(k\)為常數(shù)）C.\((\vec{A}\cdot\vec{B})'=\vec{A}'\cdot\vec{B}+\vec{A}\cdot\vec{B}'\)D.\((\vec{A}\times\vec{B})'=\vec{A}'\times\vec{B}+\vec{A}\times\vec{B}'\)2.設(shè)\(\vec{A}(t)=f(t)\vec{i}+g(t)\vec{j}\)，\(\vec{B}(t)=m(t)\vec{i}+n(t)\vec{j}\)，則（）A.\((\vec{A}(t)+\vec{B}(t))'=f'(t)\vec{i}+g'(t)\vec{j}+m'(t)\vec{i}+n'(t)\vec{j}\)B.\((\vec{A}(t)\cdot\vec{B}(t))'=f'(t)m(t)+f(t)m'(t)+g'(t)n(t)+g(t)n'(t)\)C.\(\vec{A}'(t)=f'(t)\vec{i}+g'(t)\vec{j}\)D.\((\vec{A}(t)\times\vec{B}(t))'=(g'(t)m(t)-g(t)m'(t))\vec{k}\)3.對(duì)于矢量函數(shù)\(\vec{r}(t)=x(t)\vec{i}+y(t)\vec{j}+z(t)\vec{k}\)，下列說(shuō)法正確的是（）A.其導(dǎo)數(shù)\(\vec{r}'(t)=x'(t)\vec{i}+y'(t)\vec{j}+z'(t)\vec{k}\)B.若\(\vec{r}(t)\)表示位置矢量，\(\vec{r}'(t)\)表示速度矢量C.\(\vec{r}''(t)\)表示加速度矢量D.求導(dǎo)滿(mǎn)足與標(biāo)量函數(shù)求導(dǎo)類(lèi)似的四則運(yùn)算規(guī)則4.已知\(\vec{F}(x,y,z)=P(x,y,z)\vec{i}+Q(x,y,z)\vec{j}+R(x,y,z)\vec{k}\)，則（）A.\(\frac{\partial\vec{F}}{\partialx}=\frac{\partialP}{\partialx}\vec{i}+\frac{\partialQ}{\partialx}\vec{j}+\frac{\partialR}{\partialx}\vec{k}\)B.\(\frac{\partial\vec{F}}{\partialy}=\frac{\partialP}{\partialy}\vec{i}+\frac{\partialQ}{\partialy}\vec{j}+\frac{\partialR}{\partialy}\vec{k}\)C.\(\frac{\partial\vec{F}}{\partialz}=\frac{\partialP}{\partialz}\vec{i}+\frac{\partialQ}{\partialz}\vec{j}+\frac{\partialR}{\partialz}\vec{k}\)D.\(\vec{F}\)關(guān)于某一變量的偏導(dǎo)數(shù)是矢量5.以下矢量函數(shù)求導(dǎo)正確的有（）A.若\(\vec{A}(t)=t^{n}\vec{i}\)，則\(\vec{A}'(t)=nt^{n-1}\vec{i}\)B.若\(\vec{B}(t)=\cost\vec{j}\)，則\(\vec{B}'(t)=-\sint\vec{j}\)C.若\(\vec{C}(t)=e^{at}\vec{k}\)，則\(\vec{C}'(t)=ae^{at}\vec{k}\)D.若\(\vec{D}(t)=\ln(bt)\vec{i}\)（\(b\neq0\)），則\(\vec{D}'(t)=\frac{1}{t}\vec{i}\)6.關(guān)于矢量函數(shù)\(\vec{A}(t)\)和\(\vec{B}(t)\)，有（）A.\(\frackseiwg4{dt}(\vec{A}(t)\cdot\vec{B}(t))=\vec{A}'(t)\cdot\vec{B}(t)+\vec{A}(t)\cdot\vec{B}'(t)\)B.\(\fraciwqu64m{dt}(\vec{A}(t)\times\vec{B}(t))=\vec{A}'(t)\times\vec{B}(t)+\vec{A}(t)\times\vec{B}'(t)\)C.若\(\vec{A}(t)\)和\(\vec{B}(t)\)是常矢量，則\(\fraccq6i6q6{dt}(\vec{A}(t)\cdot\vec{B}(t))=0\)D.若\(\vec{A}(t)\)和\(\vec{B}(t)\)是常矢量，則\(\fracuo66uiy{dt}(\vec{A}(t)\times\vec{B}(t))=0\)7.設(shè)\(\vec{r}(t)\)是矢量函數(shù)，\(k\)為常數(shù)，則（）A.\((k\vec{r}(t))'=k\vec{r}'(t)\)B.\(\frac6y6yeyo{dt}(\vec{r}(t)+k\vec{i})=\vec{r}'(t)\)C.\(\fracuokicye{dt}(\vec{r}(t)\cdotk\vec{i})=k\vec{r}'(t)\cdot\vec{i}\)D.\(\fracmq66e6e{dt}(\vec{r}(t)\timesk\vec{i})=k\vec{r}'(t)\times\vec{i}\)8.對(duì)于二維矢量函數(shù)\(\vec{A}(x,y)=u(x,y)\vec{i}+v(x,y)\vec{j}\)，偏導(dǎo)數(shù)相關(guān)正確的是（）A.\(\frac{\partial\vec{A}}{\partialx}=\frac{\partialu}{\partialx}\vec{i}+\frac{\partialv}{\partialx}\vec{j}\)B.\(\frac{\partial\vec{A}}{\partialy}=\frac{\partialu}{\partialy}\vec{i}+\frac{\partialv}{\partialy}\vec{j}\)C.混合偏導(dǎo)數(shù)\(\frac{\partial^{2}\vec{A}}{\partialx\partialy}=\frac{\partial^{2}u}{\partialx\partialy}\vec{i}+\frac{\partial^{2}v}{\partialx\partialy}\vec{j}\)D.混合偏導(dǎo)數(shù)\(\frac{\partial^{2}\vec{A}}{\partialy\partialx}=\frac{\partial^{2}u}{\partialy\partialx}\vec{i}+\frac{\partial^{2}v}{\partialy\partialx}\vec{j}\)9.已知矢量函數(shù)\(\vec{M}(t)=t^{2}\vec{i}-t\vec{j}+e^{t}\vec{k}\)，則（）A.\(\vec{M}'(t)=2t\vec{i}-\vec{j}+e^{t}\vec{k}\)B.\(\vec{M}''(t)=2\vec{i}+e^{t}\vec{k}\)C.\(\vec{M}'(0)=-\vec{j}+\vec{k}\)D.\(\vec{M}''(0)=2\vec{i}+\vec{k}\)10.關(guān)于矢量導(dǎo)數(shù)與標(biāo)量函數(shù)導(dǎo)數(shù)關(guān)系，正確的有（）A.矢量函數(shù)的導(dǎo)數(shù)運(yùn)算基于標(biāo)量函數(shù)導(dǎo)數(shù)運(yùn)算B.矢量函數(shù)對(duì)某變量求導(dǎo)時(shí)，各分量按標(biāo)量函數(shù)求導(dǎo)規(guī)則求導(dǎo)C.兩者求導(dǎo)的基本法則類(lèi)似但矢量導(dǎo)數(shù)結(jié)果是矢量D.標(biāo)量函數(shù)導(dǎo)數(shù)可視為矢量函數(shù)導(dǎo)數(shù)的特殊情況三、判斷題（每題2分，共10題）1.矢量函數(shù)\(\vec{A}(t)\)的導(dǎo)數(shù)\(\vec{A}'(t)\)方向一定與\(\vec{A}(t)\)方向相同。（）2.若\(\vec{A}(t)\)是常矢量函數(shù)，則\(\vec{A}'(t)=\vec{0}\)。（）3.對(duì)于矢量\(\vec{F}(x,y,z)\)，\(\frac{\partial\vec{F}}{\partialx}\)和\(\frac{\partial\vec{F}}{\partialy}\)的方向一定不同。（）4.\((\vec{A}\cdot\vec{B})'=\vec{A}'\cdot\vec{B}'\)。（）5.矢量函數(shù)\(\vec{r}(t)\)的二階導(dǎo)數(shù)\(\vec{r}''(t)\)表示其加速度。（）6.若\(\vec{A}(t)\)和\(\vec{B}(t)\)的導(dǎo)數(shù)都為\(\vec{0}\)，則\((\vec{A}(t)\times\vec{B}(t))'=\vec{0}\)。（）7.對(duì)矢量函數(shù)\(\vec{V}(t)\)求導(dǎo)時(shí)，與求標(biāo)量函數(shù)導(dǎo)數(shù)方法完全一樣。（）8.設(shè)\(\vec{F}(x)=x\vec{i}+x^{2}\vec{j}\)，則\(\frac{d\vec{F}}{dx}=\vec{i}+2x\vec{j}\)。（）9.矢量函數(shù)\(\vec{M}(t)\)在某點(diǎn)導(dǎo)數(shù)為\(\vec{0}\)，則\(\vec{M}(t)\)在該點(diǎn)是常矢量。（）10.若\(\vec{A}(t)\)是矢量函數(shù)，\(k\)為非零常數(shù)，則\((k\vec{A}(t))'=k\vec{A}'(t)\)。（）四、簡(jiǎn)答題（每題5分，共4題）1.簡(jiǎn)述矢量函數(shù)求導(dǎo)的基本規(guī)則。答：矢量函數(shù)求導(dǎo)與標(biāo)量函數(shù)求導(dǎo)類(lèi)似，和差求導(dǎo)等于求導(dǎo)的和差，數(shù)乘求導(dǎo)是常數(shù)乘導(dǎo)數(shù)。點(diǎn)積求導(dǎo)為\((\vec{A}\cdot\vec{B})'=\vec{A}'\cdot\vec{B}+\vec{A}\cdot\vec{B}'\)，叉積求導(dǎo)為\((\vec{A}\times\vec{B})'=\vec{A}'\times\vec{B}+\vec{A}\times\vec{B}'\)，各分量按標(biāo)量函數(shù)求導(dǎo)規(guī)則求導(dǎo)。2.已知矢量函數(shù)\(\vec{A}(t)=t\vec{i}+t^{2}\vec{j}\)，求\(\vec{A}(t)\)從\(t=1\)到\(t=2\)的平均變化