实用主义-公共通识类高等数学

y3x2y1xy1yx35xx2x2y 3 222121y(x2)(x3 x x y1 11x11x3 )(x2 y(1)(x2)2x3

1616116y(x3x)(x5)5x 5x5 111 y(x23x2)(x6)

1x668已知物体的运动规律为st3(m)求这物体在t2秒(s)时的速度解v(s)3t2v|t212(米/秒)9f(x)为偶函数f(0)存在f(0)limf(x)f(0)limf(x)f(0)limf(x)f(0)f(0) x x x10ysinx在具有下列横坐标的各点处切线的斜率解因为ycosx所以斜率分别为k1cos21k2cos1

x23 11ycosx上点

3解ysinx sin3x 故在点(,1处y13(x3 法线方y12(x) 12求曲线yex在点(01)处的切线方程解yexy|x01故在(01)处的切线方程y11(x0)13在抛物线yx2上取横坐标为x11及x23的两点作过这两点的割线问该抛物线上哪解y2x割线斜率为ky(3)y(1)914 2x414讨论下列函数在x0处的连续性与可导性y

x

x0 (1)

xy(0)0limylim|sinx|lim(sinx)0limylim|sinx|limsinx0

x x y(0)limy(x)y(0)lim|sinx||sin0| x x x x y(0)limy(x)y(0)lim|sinx| x x 而y(0)y(0)所以函数在x0处不可导解因为limy(x)

x2sin1 limy(x)y(0) limxsin 15设函数f(x) x1为了使函数f(x)在x1处连续且可导ab应取什么值 x

f(x)limx2

f(x)lim(axb)abf(1)limx212f(1)limaxb1lima(x1)ab1lima(x1)a x10x x x x10x16已知f(x) x0求f(0)及f(0)又f(0)是否存在 x f(0)limf(x)f(0)limx01f(0)limf(x)f(0)limx200 17已知 求f(x) xx<0时f(x)sinxf(x)cosxx>0时f(x)xff(0)limf(x)f(0)limsinx01 f(0)limf(x)f(0limx01所以f(0)1 x0xf(x)cosx x0 xxya2ya2kya2 yya2(xx)x x02

y令y0并注意x0y0ax00x02x0为切线在x轴上的距令x0并注意xya2y

y2y为切线在y轴上的距x0 x0S1|2x0||2y0|2|x0y0|2a22习题1推导余切函数及余割函数的导数公式(cotx)csc2x (cscx)cscxcotx解(cotx)(cosx)sinxsinxcosxcosxsin2xcos2x csc2x sin2 sin2 sin2(cscx)(1)cosxcscxcotx sin22求下列函数的导数y47212 y5x32x3exy2tanxsecysinxcosxyx2lnxy3excosxylnxxyexln3yx2lnxcosxs1sint(1)y(47212)(4x57x42x1 20x628x52x220282 y(5x32x3ex)15x22xy(sinxcosx)(sinx)cosxsinx(cosx)cosxcosxsinx(sinx)cos2xy(x2lnx)2xlnxx21x(2lnx1)xy(3excosx)3excosx3ex(sinx)3ex(cosxsin1xlny(lnx) 1lnx y(exln3)exx2ex2xex(x2) y(x2lnxcosx)2xlnxcosxx21cosxx2lnx(sinx2xlnxcosxxcosxx2lnxsinxs(1sint)cost(1cost)(1sint)(sint)1sintcost 3求下列函数在给定点处的导数ysinxcosx

x

xsin1cos,求 df(x3x2f(0)f(2)5 (1)ycosxsin cossin3131x 2 cossin22 2x dsincos1sin1sincos 1sincos1222(1)d

2 f(x) 2xf(0)3f(2)17(5 2该物体的速度解(1)v(t)s(t)v0gt(2)令v(t)0即v0gt0得tv0这就是物体达到最高点的时刻g5求曲线y2sinxx2上横坐标为x0的点处的切线方程和法线方程y2cosx2xy|x02又当x0时y0所以所求的切线方程为y1x26求下列函数的导数ye3x2ysin2xa2a2y(arcsinylncosx(1)ye3x2(3x2)e3x2(6x)6xe3x2y1(1x2)12x2x1 1 1 1 a2 (a2 2(a2 (a2 2(a2x2)2a2 (ex 1(ex 11y2arcsinx(arcsinx)2arcsin1y1(cosx) cos cos7求下列函数的导数y ye2cos3xyarccos1xy1lnxysin2xxxy xyln(xa2x2)yln(secxtanyln(cscxcotx解(1)y (12x)x 11 1

1 3 (1x2)

2(1x2)

2(1x2)2(

(1x2)1 y

2)cos3xe

2 )cos3x2

2(sin1 12e2cos3x3e2sin3x2e2(cos3x6sin3x)1x 1x

(1)

| xx2x2x2x2 x (1ln ycos2x2xsin2x12xcos2xsin2x12121(y1 (x) 1(1(

2xa22xa2xa22a22a2x

a2x2 (a2xa22axa22a2a2 (secxtanx)secxtanxsec2xsecxsecxtan secxtan (cscxcotx)cscxcotxcsc2xcscx 8求下列函数的导数yarcsinx22ylntanx21ln21ln2yearctanxysinnxcosnxyarctanx1xyarcsinxy1x1x1xy

11x1解(1)y2arcsinxarcsinx2arcsinx (1(21(2 1 2 1(2

4 1(tanx 1sec2x(x 1sec2x1cscxtan2

tan2

2 tan 212121ln2y1ln2x (1ln2

2lnx(ln21ln2x1ln211x1ln211ln2 yearctanx

x)earctanx (

earctanx 1earctanx1(x)22 2x(1ynsinn1x(sinx)cosnxsinnx(sinnsinn1xcosxcosnxsinnx(sinnsinn1x(cosxcosnxsinxsinnx)nsinn1xcos(n1)x (x (x1)(x1)

1(xx

x 1(xx

(x 1 arccosx111 arccosx111arcsin(arccos 21x2(arccos

1

(arccos [ln(ln 1(ln 11 ln(ln ln(lnx)lnx xlnxln(ln 11121 21)(1x1x)(1x1(1x121 21)1x21x211(1x)2x(1

(1x) (1x)(1x) 1 111f2(x)111112f2(x)g9.设函数f(x)和g(x)可导且f2(x)g2(x)0试求函数y 解y 111f2(x)111112f2(x)g2f2(x)gf2(x)gf(x)f(xf2(x)g(1)(1)yf(x2)(x2)f(x2)2x2xf(2)yf(sin2x)(sin2x)ff(sin2x)2sinxcosxf(cos2x)2cosx(sinsin2x[f(sin2x)f11求下列函数的导数ych(shxyshxechyth(lnyarctan(thylnchx 2ch2ych2x1x(1)ysh(shx)(shx)sh(shx)chxychxechxshxechxshxechx(chxsh2x) (ln ch2(ln xch2(lny3sh2xchx2chxshxshxchx(3shx2) (1x2) ch2(1 ch2(1 (x21) 1(x2 x42x2e4xy (e2x)2e2e4x(e2x)2 (th 1 1th2xch2 sh2ch2

ch2 ch2xsh2x12sh2y1(ch (ch2x)shx 2chch 2ch4 chx2ch4shxshxshxch2xshxshx(ch2x1)sh3xth3xchx ch3x

x

xch3xx1y2chx1chx1

1

1 x x x x x sh2x1x (x (x x12求下列函数的导数yarctanx22ylnxyetetetylncos1xyy

sin2xxx4yxarcsinx42yarcsin2t(1)y2sinxcosy2arctanx 11 arctanx21x241xnln

x2 y 1nlnx y(etet)(etet)(etet)(etet) (etet (e2tysec1(cos1)sec1(sin1)(1)1tan1x x

x

x

1(1)

sin222x4xx22x4xxx x2

x

1) yarcsinx 1 (2x)arcsinx2x2x4 1x22

11(2t (2t 2(111(2t1(2t1(2t 2(1t2)2(1t2)

|1t2习题1求函数的二阶导数y2x2lnxye2x1yxcosxyetsinta2a2ytanxy x3y1x2)arctanxyexxyxex2yln(x1x2)解(1)y4x1x

y41ye2x122e2x1y2e2x1yxcosxycosxxsinysinxsinxxcosx2sinxxcosxyet(cosxsinx)et(sinxcosx)2etcosta2a2 (a2a2a22a2x2 y a2x2 a2y1(1x2)2x1 1

(a2x2)a2 ysec2y2secx(secx)2secxsecxtanx2sec2xtanx(x3

(x3y6x(x31)23x22(x31)3x6x(2x31)(x3 (x3y2xarctanx(1x2) 1y2arctanx2x1yexxex1ex(x1) y[ex(x1)ex]x2ex(x1)2xex(x22x2) yex2xex2(2x)ex2(12x2)121yex22x(12x2)ex24x2xex21211 1x

1x2 x1

) y1(1x2)1 1(1x)211(1x)212设f(x)(x10)6f解f(x)6(x10)5f(x)30(x10)4ff3f(x)存在,yd2yyf(x2)yln[f(x)](1)yf(x2)(x2)2xfy2f(x2)2x2xf(x2)2f(x2)4x2f(2)y f(x)fyf(x)f(x)f(x)f(x)f(x)f(x)[f(x)]2[f4dx1导出 d2xy d3x3(y)2yy

[f(1d2xddxd1d1dxy1y dy dy(y)2 (2)d3xdydydxy(y)3y3(y)2y13(y)2yy d2s2s0解dsAcostd2sA2sintd2s就是物体运动的加速度d2s2sA2sint2Asint0y2y0解yC1exC2ex(C1exC2ex)(C1exC2ex)07验证函数yexsinx满足关系式y2y2y0解yexsinxexcosxex(sinxcosyex(sinxcosx)ex(cosxsinx)2excosxy2y2y2excosx2ex(sinxcosx)2exsin2excosx2exsinx2excosx2exsinx0yxna1xn1a2xn2an1xan(a1a2an都是常数ysin2xyxlnxyxex(1)ynxn1(n1)a1xn2(n2)a2xn3an1yn(n1)xn2(n1)(n2)a1xn3(n2)(n3)a2xn4an2y(n)n(n1)(n2)21x0n!y2sinxcosxsin2xy2cos2x2sin(2x)

cos(2x

)

sin(2x2)2 2y(4)23cos(2x2)23sin(2x3) y(n)2n1sin[2x(n1)]2ylnx1y1x1y(n)(1)(2)(3)(n2)xn1(1)n2(n2)!(1)n(n2)! yexxexyexexxex2exxexy2exexxex3exxex9求下列函数所指定的阶的导数yexcosx,求y(4yxshx,求y(100yx2sin2x,求y(50(1)令uexvcosxvsinxvcosxvsinxv(4)cosx所 ex[cosx4(sinx)6(cosx)4sinxcosx]4excosxuxvshxu1vchxvshxv(99)chxv(100)sh所 y(100)u(100)vC1u(99)vC2u(98)vC98uv(98)C99uv(99)u100chxxshx令ux2vsin2xu2xu2

v(48)248sin(2x48)248sin2x2v(49)249cos2xv(50)250sin2x 所 y(50)u(50)vC1u(49)vC2u(48)vC48uv(48)C49uv(49) C48uv(48)C49uv(49)u 50492228sin2x502x249cos2xx2(250sin2x)2y22xxyexy(1)方程两边求导数得2yy2y2xy0于 y y(2)方程两边求导数于 (y2ax)yayx2yayx2y2于 (xexy)yeyexyyxexye于 (1xey)yey 1 2x3y3a3在点(4a,4a处的切线方程和法线方程解方程两边求导数得2x32

3y0于 yx3y在点

a,2a2 222y

a(x

xy2a2 2y2a(x2a (1)方程两边求导数得yxyyxy(x)yxy yy2x21 yb2xa2yb2yxyb2 a2

b2a2y2

a2 a2ysec2(xy) sin2(xy)cos2(xy)11 cos2(x sin2(x y2y2(11)2(1y2) yeyxe ey1 1(y 2yeyy(2y)ey(y)ey(3y)ye2y(3y)(2y)2 (2y)2 (2y)34用对数求导法求下列函数的导数y(x)x1y 5x2yx2(3x)4(xxsinxxsinx(1)lnyxln|x|x1ylnxx1ln(1x)x1 1于 y(x)x[lnx1]1 1x1lny1ln|x5|1ln(x22) 1y1112x x5x2 5 x5x2于 y

[112x]5

x 5x2lny1ln(x2)4ln(3x)5ln(x1)21 45 2(x2)3 x于 x2(3x)4 45(x 2(x x xlny1lnx1lnsinx1ln(1ex) y11cotx 4(1ex于 yxsinx1ex[11cotx

xsinx1ex[22cotxex] 4(1ex) ex15求下列参数方程所确定的函数的导数dy

yx(1sinyy 解 t (2)dyycossin x1sin6xetsint求当tdy的值yet dyytetcostetsintcostsint xtetsintet

1 当t3时

dx 2331331

37写出下列曲线在所给参数值相应的点处的切线方程和法线方程ycos

在t处4

y (1dyyt2sin2t 2sin(2 2当t4时2

dx

cos4

2

x02y00y22(x222xy202y (x2即2x4y102 y6at(1t2)3at22t x3a(1t2)3at2t3a3at2

2t xt3a t2时dy224x06ay012adx1 y12a4(x6a y12a3(x6a) x 2y

xax

f(t)存在且不为零ytft(t)f1tx(1)dyy1d2y(y)t21tx d2y(y csc t cott xt 4 xtasint asint a2sin3t d2y(y) 22e2t4t e2t xt e3t xt dyytf(t)tf(t)f(t)td2y(yx)t

f

f9d3y

ytytarctanxytarctan解(1)dy(tt3)13t2dx d2

(13t2

1

)3

4 1(1dy4 t3(1t2)

1(2)dy(tarctant)1t21t (1d2y 1t2

(1t2dy t41 10落在平静水面上的石头产生同心波纹6m/s,S当t2时r6212解水深为h时r1hS1h2 水的体积为V1hS1h1h2h3dV

3h2dhdh4dV h2已知h5(m)dV4m3/min)因此dh4dV4416 h2 12溶液自深18cm直径12cm的正漏斗中漏入一直径为10cm的圆柱形筒中开始时漏斗中盛满了溶液12cm时1cm/min问此162181r2y52h r

ry61816231y22 3(3)y5 162181y352h 1y2y52h当y12时yt11122 160.64

时的y及dyy|x2dy|x2x1(3x21)x|x2y|x2x0dy|x2x01(3x21)x|x2x0y|x2 dy|x2x001(3x21)x|x22设函数yf(x)的图形如图所示试在图(a)、(b)、(c)、(d)中分别标出在点x0的dy、yydy并说明其正负(a)y0dy0y0dy0y0dy0y0dy03求下列函数的微分xy1 xxyxsin2xx2yx2yx2e2xyarcsin1x2yarctan1x21(1)y11所以dy(11)dx 因为ysin2x2xcos2xdy(sin2x2x

x21 1(x21)x2x21 1(x21)x2x2x2 (xx2dyydx[ln2(1x)]dx[2ln(1x)1]dx2ln(1x)dx x dyydx(arcsin1x2)dx

1|x|11

dx2tan(12x2)sec2(12x2)4xdx8xtan(12x2)sec2(12x2)dxdydarctan1x2 d(1x21x21(1x21

1 2x(1x2)2x(1x2)dx4xdx1(1x21

1dyd[Asin(t)]Acos(t)d(t)Acos(t)dx4将适当的函数填入下列括号内,使等式成立 )2dx )costdt )sinxdx )1dxx )e2xdx )1dxx )sec23xdx(1)d2xC)2dxd(3x2C)3xdx2d(sintC)costdtd(1cosxC)sinxdxd(ln(1x)C)1dxxd(1e2xC)e2xdx2d(2xC)1dxxd(1tan3xC)sec23xdx3s2l(12f2)SdS2l(12f2)df8ff 改变了多少？又如果不变R增加1cm问扇形面积大约改变了多少？2SdS(1R2)d1R2将60

R10030

S11002()43.63 (2)SdS(1R2)dRRRR3S 7计算下列三角函数值的近似值(1)(2)cos29cos()cossin()310.874676 2(2)已知f(xx)f(x)f(x)x当f(x)tanx时有tan(xx)tanxsec2xx所tan136tan(3)tan3sec23120.9650948(1)(2)arccos(1)f(xx)f(x)f(x)xf(x)arcsinx时1arcsin(xx)arcsinx 111arcsin0.5002arcsin(0.50.0002)arcsin

2 (2)f(xx)f(x)f(x)xf(x)arccosx时1arccos(xx)arccosx 111arccos0.4995arccos(0.50.0005)

2 9x较小时证明下列近似公式11x1tan45ln1002的近似值已知当|x|较小时f(x0x)f(x0)f(x0)x取f(x)tanxx00xx则tanxtan(0x)tan0sec0xsec20xx已知当|x|较小时f(x0x)f(x0)f(x0)x取f(x)lnxx01xx则ln(1x)ln1(lnx)|x1xx已知当|x|较小时f(x0x)f(x0)f(x0)x取f(x1x01xx则x1 x1x1

)(1)310(1)3(2)6(1)f(xnx则当|x|较小时f(1xf(1)f(1)x11xn39963100041031 10(114)9.987 3n665664126112(111)2.0052 6V1D3dV1D2D因为计算球体体积时2%以内 1D2 dVV

116

2%D

2%3 过2%312某厂生产如图所示的扇形板半径R200mm要求中心角55产品检验时一般用测量弦长l的办法来间接测量中心角如果测量弦长l时的误差101mm问此而引起的中心角测量误差x是多少？解由lRsin得2arcsinl2arcsin 当55时l2Rsin2|l|l当l1847l01时

1111(l

l2 10.10.00056(弧度)

f(x)在点x0可导是f(x)在点x0连续 条件f(x)在点x0连续是f(x)在点x0可导 f(x)在点x0的左导数f(x0)及右导数f(x0)都存在且相等是f(x)在点x0可导 条件 解(1)充分必要充分必要充分必要2选择下述题中给出的四个结论中一个正确的结论 (A)limhf(a1)f(a存在Blimf(a2hf(ah)存在 (Climf(ahf(ah存在(Dlimf(af(ah)存在 提示limf(af(ahlimf(ahf(alimf(axf(a) 3设有一根细棒取棒的一端作为原点棒上任一点的做标x为于是分布在区间上细棒的质量m是x的函数mm(x)应怎样确定细棒在点x0处的线密度(对于均匀细棒来说解mm(x0x)m(x0)在区间[x0x0x]上的平均线密度mm(x0x)m(x0) limmlimm(x0x)m(x0)m(x)x0 4根据导数的定义f(x1的导数xx解ylimx1x1lim lim 1x x0x(x x0(x x01e

xx0 x(1)f(0)limf(x)f(0limsinx0 x f(x)f f(0)lim lim limln(1x)xlne1 x

而且f(0)f(0)所以f(0)存在且f (2)因为f(0)limf(x)f(0)lim1 x x 1e

1x1f(0)limf(x)f(0)lim1e

0 x x 1e6

f(x)xsin

xx