01总习题一

总习题一 1( 在“充分”、“必要”和“充分必要”三者中选择一个正确的填入下列空格内( (1)数列{xn}有界是数列{xn}收敛的________条件( 数列{xn}收敛是数列{xn}有界的________的条件( (2)f(x)在x0的某一去心邻域内有界是 存在的________条件( 存在是f(x)在x0的某一去心邻域内有界的________条件( (3) f(x)在x0的某一去心邻域内无界是 的________条件( 是f(x)在x0的某一去心邻域内无界的________条件( (4)f(x)当x(x0时的右极限f(x0()及左极限f(x0()都存在且相等是 存在的________条件( 解 (1) 必要( 充分( (2) 必要( 充分( (3) 必要( 充分( (4) 充分必要( 2( 选择以下题中给出的四个结论中一个正确的结论( 设f(x)(2x(3x(2( 则当x(0时( 有( )( (A)f(x)与x是等价无穷小( (B)f(x)与x同阶但非等价无穷小( (C)f(x)是比x高阶的无穷小( (D)f(x)是比x低阶的无穷小( 解 因为 (令2x(1(t( 3x(1(u) ( 所以f(x)与x同阶但非等价无穷小( 故应选B( 3( 设f(x)的定义域是[0( 1]( 求下列函数的定义域( (1) f(ex)( (2) f(ln x)( (3) f(arctan x)( (4) f(cos x)( 解 (1)由0(ex(1得x(0( 即函数f(ex)的定义域为(((( 0]( (2) 由0( ln x(1得1(x(e ( 即函数f(ln x)的定义域为[1( e]( (3) 由0( arctan x (1得0(x(tan 1( 即函数f(arctan x)的定义域为[0( tan 1]( (4) 由0( cos x(1得 (n(0( (1( (2( ( ( ()( 即函数f(cos x)的定义域为[ ]( (n(0( (1( (2( ( ( ()( 4( 设 ( ( 求f[f(x)]( g[g(x)]( f[g(x)]( g[f(x)]( 解 因为f(x)(0( 所以f[f(x)](f(x) ( 因为g(x)(0( 所以g[g(x)](0( 因为g(x)(0( 所以f[g(x)](0( 因为f(x)(0( 所以g[f(x)]((f 2(x) ( 5( 利用y(sin x的图形作出下列函数的图形( (1)y(|sin x|( (2)y(sin|x|( (3) ( 6( 把半径为R的一圆形铁片( 自中心处剪去中心角为(的一扇形后围成一无底圆锥( 试将这圆锥的体积 表 关于同志近三年现实表现材料材料类招标技术评分表图表与交易pdf视力表打印pdf用图表说话 pdf 为(的函数( 解 设围成的圆锥的底半径为r( 高为h( 依题意有 R(2((()(2(r ( ( ( 圆锥的体积为 (0(((2()( 7( 根据函数极限的定义证明 ( 证明 对于任意给定的((0( 要使 ( 只需|x(3|((( 取(((( 当0(|x(3|((时( 就有|x(3|((( 即 ( 所以 ( 8( 求下列极限( (1) ( (2) ( (3) ( (4) ( (5) (a(0( b(0( c(0)( (6) ( 解 (1)因为 ( 所以 ( (2) ( (3) ( (4) (提示( 用等价无穷小换)( (5) ( 因为 ( ( 所以 ( 提示( 求极限过程中作了变换ax(1(t( bx(1(u( cx(1(v( (6) ( 因为 ( ( 所以 ( 9( 设 ( 要使f(x)在(((( (()内连续( 应怎样选择数a? 解 要使函数连续( 必须使函数在x(0处连续( 因为 f(0)(a( ( ( 所以当a(0时( f(x)在x(0处连续( 因此选取a(0时( f(x)在(((( (()内连续( 10( 设 ( 求f(x)的间断点( 并说明间断点所属类形( 解 因为函数f(x)在x(1处无定义( 所以x(1是函数的一个间断点( 因为 (提示 )( (提示 )( 所以x(1是函数的第二类间断点( 又因为 ( ( 所以x(0也是函数的间断点( 且为第一类间断点( 11( 证明 ( 证明 因为 ( 且 ( ( 所以 ( 12( 证明方程sin x(x(1(0在开区间 内至少有一个根( 证明 设f(x)(sin x(x(1( 则函数f(x)在 上连续( 因为 ( ( ( 所以由零点定理( 在区间 内至少存在一点(( 使f(()(0( 这说明方程sin x(x(1(0在开区间 内至少有一个根( 13( 如果存在直线L( y(kx(b( 使得当x(((或x(((( x((()时( 曲线y(f(x)上的动点M(x( y)到直线L的距离d(M( L)(0( 则称L为曲线y(f(x)的渐近线( 当直线L的斜率k(0时( 称L为斜渐近线( (1)证明( 直线L( y(kx(b为曲线y(f(x)的渐近线的充分必要条件是 ( ( (2)求曲线 的斜渐近线( 证明 (1) 仅就x((的情况进行证明( 按渐近线的定义( y(kx(b是曲线y(f(x)的渐近线的充要条件是 ( 必要性( 设y(kx(b是曲线y(f(x)的渐近线( 则 ( 于是有 ( ( ( 同时有 ( ( 充分性( 如果 ( ( 则 ( 因此y(kx(b是曲线y(f(x)的渐近线( (2)因为 ( ( 所以曲线 的斜渐近线为y(2x(1( _1106115009.unknown _1118469459.unknown _1118469517.unknown _1118469564.unknown _1118469589.unknown _1118469615.unknown _1118469627.unknown _1118469639.unknown _1118469645.unknown _1118469654.unknown _1118469657.unknown _1118469660.unknown _1118469650.unknown _1118469642.unknown _1118469633.unknown _1118469636.unknown _1118469630.unknown _1118469621.unknown _1118469624.unknown _1118469618.unknown _1118469601.unknown _1118469608.unknown _1118469612.unknown _1118469604.unknown _1118469595.unknown _1118469598.unknown _1118469591.unknown _1118469577.unknown _1118469583.unknown _1118469585.unknown _1118469580.unknown _1118469570.unknown _1118469573.unknown _1118469567.unknown _1118469541.unknown _1118469553.unknown _1118469559.unknown _1118469562.unknown _1118469556.unknown _1118469547.unknown _1118469550.unknown _1118469544.unknown _1118469529.unknown _1118469535.unknown _1118469538.unknown _1118469532.unknown _1118469523.unknown _1118469526.unknown _1118469520.unknown _1118469484.unknown _1118469504.unknown _1118469511.unknown _1118469514.unknown _1118469508.unknown _1118469490.unknown _1118469495.unknown _1118469487.unknown _1118469471.unknown _1118469477.unknown _1118469481.unknown _1118469475.unknown _1118469465.unknown _1118469468.unknown _1118469462.unknown _1118469412.unknown _1118469435.unknown _1118469448.unknown _1118469453.unknown _1118469456.unknown _1118469450.unknown _1118469441.unknown _1118469445.unknown _1118469438.unknown _1118469423.unknown _1118469429.unknown _1118469432.unknown _1118469426.unknown _1118469417.unknown _1118469420.unknown _1118469415.unknown _1117263716.unknown _1118469398.unknown _1118469404.unknown _1118469407.unknown _1118469401.unknown _1118469392.unknown _1118469395.unknown _1118469388.unknown _1106116854.unknown _1117263612.unknown _1117263710.unknown _1106117183.unknown _1106117384.unknown _1106116228.unknown _1106116289.unknown _1106115037.unknown _1106057229.unknown _1106057397.unknown _1106057405.unknown _1106057409.unknown _1106114786.unknown _1106114861.unknown _1106057410.unknown _1106057407.unknown _1106057408.unknown _1106057406.unknown _1106057401.unknown _1106057403.unknown _1106057404.unknown _1106057402.unknown _1106057399.unknown _1106057400.unknown _1106057398.unknown _1106057259.unknown _1106057388.unknown _1106057392.unknown _1106057395.unknown _1106057396.unknown _1106057394.unknown _1106057390.unknown _1106057391.unknown _1106057389.unknown _1106057287.unknown _1106057297.unknown _1106057387.unknown _1106057386.unknown _1106057295.unknown _1106057267.unknown _1106057276.unknown _1106057263.unknown _1106057246.unknown _1106057253.unknown _1106057256.unknown _1106057249.unknown _1106057235.unknown _1106057239.unknown _1106057233.unknown _1106057181.unknown _1106057205.unknown _1106057217.unknown _1106057223.unknown _1106057226.unknown _1106057220.unknown _1106057209.unknown _1106057212.unknown _1106057208.unknown _1106057193.unknown _1106057197.unknown _1106057201.unknown _1106057195.unknown _1106057186.unknown _1106057189.unknown _1106057183.unknown _1106057145.unknown _1106057160.unknown _1106057167.unknown _1106057170.unknown _1106057164.unknown _1106057153.unknown _1106057156.unknown _1106057150.unknown _1106057065.unknown _1106057088.unknown _1106057091.unknown _1106057072.unknown _1106057028.unknown _1106057060.unknown _1106057024.unknown