大學(xué)微分題目及答案解析

大學(xué)微分題目及答案解析

一、單項(xiàng)選擇題（每題2分，共10題）1.函數(shù)\(y=x^2\)的導(dǎo)數(shù)是（）A.\(2x\)B.\(x^3\)C.\(2\)D.\(1\)2.若\(y=\sinx\)，則\(y'\)為（）A.\(\cosx\)B.\(-\cosx\)C.\(\sinx\)D.\(-\sinx\)3.函數(shù)\(y=e^x\)的導(dǎo)數(shù)是（）A.\(e^x\)B.\(xe^{x-1}\)C.\(e\)D.\(1\)4.曲線\(y=x^3\)在點(diǎn)\((1,1)\)處的切線斜率是（）A.\(1\)B.\(2\)C.\(3\)D.\(4\)5.設(shè)\(y=\lnx\)，則\(y'\)等于（）A.\(\frac{1}{x}\)B.\(x\)C.\(-\frac{1}{x}\)D.\(x^2\)6.函數(shù)\(y=\cos(2x)\)的導(dǎo)數(shù)是（）A.\(-2\sin(2x)\)B.\(2\sin(2x)\)C.\(-\sin(2x)\)D.\(\sin(2x)\)7.若\(y=x^n\)（\(n\)為常數(shù)），則\(y'=\)（）A.\(nx^{n-1}\)B.\(nx^n\)C.\(x^{n-1}\)D.\(n\)8.函數(shù)\(y=\tanx\)的導(dǎo)數(shù)是（）A.\(\sec^2x\)B.\(-\sec^2x\)C.\(\csc^2x\)D.\(-\csc^2x\)9.曲線\(y=\frac{1}{x}\)在點(diǎn)\((2,\frac{1}{2})\)處的切線方程的斜率為（）A.\(-\frac{1}{4}\)B.\(\frac{1}{4}\)C.\(-\frac{1}{2}\)D.\(\frac{1}{2}\)10.函數(shù)\(y=\arcsinx\)的導(dǎo)數(shù)是（）A.\(\frac{1}{\sqrt{1-x^2}}\)B.\(-\frac{1}{\sqrt{1-x^2}}\)C.\(\frac{1}{1+x^2}\)D.\(-\frac{1}{1+x^2}\)二、多項(xiàng)選擇題（每題2分，共10題）1.下列函數(shù)中，導(dǎo)數(shù)為\(0\)的有（）A.\(y=5\)B.\(y=\pi\)C.\(y=0\)D.\(y=x^0\)2.以下求導(dǎo)正確的是（）A.\((x^3)'=3x^2\)B.\((\cosx)'=-\sinx\)C.\((e^{2x})'=2e^{2x}\)D.\((\ln2x)'=\frac{1}{x}\)3.函數(shù)\(y=f(x)\)在某點(diǎn)可導(dǎo)的等價(jià)條件有（）A.左導(dǎo)數(shù)等于右導(dǎo)數(shù)B.函數(shù)在該點(diǎn)連續(xù)C.函數(shù)在該點(diǎn)有極限D(zhuǎn).函數(shù)在該點(diǎn)的切線存在4.若\(y=u(x)v(x)\)，則\(y'\)可能是（）A.\(u'(x)v(x)+u(x)v'(x)\)B.\(u(x)v'(x)\)C.\(u'(x)v(x)\)D.\(u'(x)v(x)-u(x)v'(x)\)5.下列函數(shù)中，是復(fù)合函數(shù)的有（）A.\(y=\sin(2x+1)\)B.\(y=e^{x^2}\)C.\(y=\ln(x^2+1)\)D.\(y=\sqrt{x}\)6.曲線\(y=f(x)\)在某點(diǎn)處的切線方程相關(guān)的量有（）A.切點(diǎn)坐標(biāo)B.函數(shù)在該點(diǎn)的導(dǎo)數(shù)C.函數(shù)的定義域D.函數(shù)的值域7.以下關(guān)于導(dǎo)數(shù)的說(shuō)法正確的是（）A.導(dǎo)數(shù)表示函數(shù)的變化率B.導(dǎo)數(shù)為正函數(shù)單調(diào)遞增C.導(dǎo)數(shù)為負(fù)函數(shù)單調(diào)遞減D.導(dǎo)數(shù)為\(0\)函數(shù)一定是常數(shù)函數(shù)8.函數(shù)\(y=\frac{u(x)}{v(x)}\)（\(v(x)\neq0\)）的導(dǎo)數(shù)\(y'\)是（）A.\(\frac{u'(x)v(x)-u(x)v'(x)}{v^2(x)}\)B.\(\frac{u'(x)v(x)+u(x)v'(x)}{v^2(x)}\)C.\(\frac{u(x)v'(x)-u'(x)v(x)}{v^2(x)}\)D.\(\frac{u'(x)}{v'(x)}\)9.若\(f(x)\)可導(dǎo)，\(c\)為常數(shù)，則\((cf(x))'\)等于（）A.\(cf'(x)\)B.\(c+f'(x)\)C.\(f'(x)\)D.\(cf(x)\)10.下列函數(shù)中，導(dǎo)數(shù)是偶函數(shù)的有（）A.\(y=x^2\)B.\(y=\cosx\)C.\(y=\sinx\)D.\(y=e^x\)三、判斷題（每題2分，共10題）1.常數(shù)的導(dǎo)數(shù)是\(0\)。（）2.函數(shù)\(y=x\)的導(dǎo)數(shù)是\(0\)。（）3.若函數(shù)在某點(diǎn)不可導(dǎo)，則函數(shù)在該點(diǎn)一定不連續(xù)。（）4.曲線\(y=f(x)\)在某點(diǎn)的切線斜率就是函數(shù)在該點(diǎn)的導(dǎo)數(shù)。（）5.復(fù)合函數(shù)求導(dǎo)需要使用鏈?zhǔn)椒▌t。（）6.函數(shù)\(y=\lnx\)在\((0,+\infty)\)上可導(dǎo)。（）7.若\(y=f(x)\)的導(dǎo)數(shù)\(y'>0\)，則\(y=f(x)\)的圖像是上升的。（）8.兩個(gè)函數(shù)乘積的導(dǎo)數(shù)等于兩個(gè)函數(shù)導(dǎo)數(shù)的乘積。（）9.函數(shù)\(y=\sqrt{x}\)的導(dǎo)數(shù)是\(\frac{1}{2\sqrt{x}}\)。（）10.函數(shù)\(y=\arccosx\)的導(dǎo)數(shù)是\(\frac{1}{\sqrt{1-x^2}}\)。（）四、簡(jiǎn)答題（每題5分，共4題）1.求函數(shù)\(y=x^3+2x^2-3x+1\)的導(dǎo)數(shù)。答案：根據(jù)求導(dǎo)公式\((x^n)^\prime=nx^{n-1}\)，\(y^\prime=3x^2+4x-3\)。2.簡(jiǎn)述函數(shù)可導(dǎo)與連續(xù)的關(guān)系。答案：函數(shù)可導(dǎo)一定連續(xù)，但連續(xù)不一定可導(dǎo)?？蓪?dǎo)是連續(xù)的充分不必要條件，即若函數(shù)在某點(diǎn)可導(dǎo)，則在該點(diǎn)一定連續(xù)；但在某點(diǎn)連續(xù)的函數(shù)，在該點(diǎn)不一定可導(dǎo)。3.求\(y=\sin(3x)\)的導(dǎo)數(shù)。答案：令\(u=3x\)，\(y=\sinu\)，根據(jù)鏈?zhǔn)椒▌t\(y^\prime=(\sinu)^\prime\cdotu^\prime=\cosu\cdot3=3\cos(3x)\)。4.曲線\(y=x^2\)在點(diǎn)\((1,1)\)處的切線方程是什么？答案：先求導(dǎo)數(shù)\(y^\prime=2x\)，在點(diǎn)\((1,1)\)處切線斜率\(k=2\times1=2\)，由點(diǎn)斜式得切線方程\(y-1=2(x-1)\)，即\(y=2x-1\)。五、討論題（每題5分，共4題）1.討論導(dǎo)數(shù)在優(yōu)化問(wèn)題中的應(yīng)用。答案：在優(yōu)化問(wèn)題中，導(dǎo)數(shù)可用來(lái)找函數(shù)的極值點(diǎn)。通過(guò)求導(dǎo)令導(dǎo)數(shù)為\(0\)找到駐點(diǎn)，再結(jié)合實(shí)際問(wèn)題判斷駐點(diǎn)是否為最值點(diǎn)，從而解決如成本最小、利潤(rùn)最大等問(wèn)題。2.如何根據(jù)函數(shù)導(dǎo)數(shù)的正負(fù)判斷函數(shù)的單調(diào)性？答案：若函數(shù)\(y=f(x)\)在某區(qū)間內(nèi)\(f^\prime(x)>0\)，則函數(shù)在該區(qū)間單調(diào)遞增；若\(f^\prime(x)<0\)，則函數(shù)在該區(qū)間單調(diào)遞減。3.舉例說(shuō)明復(fù)合函數(shù)求導(dǎo)法則的重要性。答案：如求\(y=e^{\sinx}\)的導(dǎo)數(shù)，它是復(fù)合函數(shù)。利用