From 5c2864f77cfddd37fb0c0bf265c1a02399667dc3 Mon Sep 17 00:00:00 2001 From: wangweiye7840 Date: Fri, 7 Jul 2023 14:32:37 +0800 Subject: [PATCH] =?UTF-8?q?=E5=BD=95=E5=85=A5=E7=A9=BA=E4=B8=AD=E8=AF=BE?= =?UTF-8?q?=E5=A0=82=E9=80=89=E6=8B=A9=E6=80=A7=E5=BF=85=E4=BF=AE=E7=AC=AC?= =?UTF-8?q?=E4=BA=8C=E5=86=8C=E5=AF=BC=E6=95=B0=E5=8F=8A=E5=85=B6=E5=BA=94?= =?UTF-8?q?=E7=94=A8=E4=BE=8B=E9=A2=98=E4=B8=8E=E4=B9=A0=E9=A2=98=20?= =?UTF-8?q?=E5=B9=B6=E5=AF=B9=E5=BA=94=E5=8D=95=E5=85=83?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- 工具v2/批量收录题目.py | 2 +- 题库0.3/Problems.json | 1408 ++++++++++++++++++++++++++++++++++++++++ 2 files changed, 1409 insertions(+), 1 deletion(-) diff --git a/工具v2/批量收录题目.py b/工具v2/批量收录题目.py index ea57280c..c87f6bc3 100644 --- a/工具v2/批量收录题目.py +++ b/工具v2/批量收录题目.py @@ -1,5 +1,5 @@ #修改起始id,出处,文件名 -starting_id = 19006 #起始id设置, 来自"寻找空闲题号"功能 +starting_id = 19054 #起始id设置, 来自"寻找空闲题号"功能 raworigin = "" #题目来源的前缀(中缀在.tex文件中) filename = r"C:\Users\wangweiye\Documents\wwy sync\临时工作区\空中课堂必修第二册例题与习题.tex" #题目的来源.tex文件 editor = "王伟叶" #编辑者姓名 diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index a4daba5b..812c03cb 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -488865,6 +488865,1414 @@ "space": "4em", "unrelated": [] }, + "019054": { + "id": "019054", + "content": "已知在使用某种杀菌剂$t$小时后室内的细菌数量为$f(t)=10^5+10^4 t-10^3 t^2$.\\\\\n(1) 求$f'(10)$;\\\\\n(2) $f'(10)$的实际意义是什么?", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第二册导数及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019055": { + "id": "019055", + "content": "``复兴号''G19 次列车 16: 00 从北京南站驶向济南西站. 由于北京南站处于城区, 列车要通过限速路段后方能提速. 列车出发 0.1 小时后, 行驶路程$s$(单位: 千米) 与时间$x$(单位: 小时) 可以大致地用函数关系表示为$s=958 x^2-49 x+6.7$($0.1 \\leq x \\leq 0.21$). 求16: 12这一时刻``复兴号''$\\mathrm{G} 19$次列车的瞬时速度.", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第二册导数及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019056": { + "id": "019056", + "content": "将原油精炼为汽油、柴油、塑胶等各种不同产品, 需要对原油进行冷却和加热. 已知原油的温度$y$(单位: $^{\\circ} \\mathrm{C}$) 与时间$x$(单位: $\\mathrm{h}$) 满足函数关系式$y=x^2-7 x+15$($0 \\leq x \\leq 8$). 计算$x=2$与$x=6$时, 原油温度的瞬时变化率, 并说明它们的意义.", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第二册导数及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019057": { + "id": "019057", + "content": "在高台跳水运动过程中, 运动员的身体重心相对于水面的高度$h$(单位: $\\mathrm{m}$) 与起跳后的时间$t$(单位: $\\mathrm{s}$) 满足函数关系$h=-4.9 t^2+2.8 t+11$.\\\\\n(1) 求该运动员在时间段$[1,1.5]$内的平均速度;\\\\\n(2) 求该运动员在$t=1$时的瞬时速度;\\\\\n(3) 记$f(t)=-4.9 t^2+2.8 t+11$, 求$f'(\\dfrac{2}{7})$, 并说明$f'(\\dfrac{2}{7})$的实际意义, 及此刻运动员的位置.", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第二册导数及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019058": { + "id": "019058", + "content": "已知晩间蜥蜴的体温近似地满足$T(t)=\\dfrac{120}{t+5}+15$, 其中$T(t)$为蜥蜴的体温(单位: ${ }^{\\circ} \\mathrm{C}$) , $t$为太阳落山后的时间(单位: $\\min$).\\\\\n(1) 当$t=10$时, 蜥蜴体温的瞬时变化率是多少?\\\\\n(2) 蜥蜴体温的瞬时变化率为$-1^{\\circ} \\mathrm{C} / \\mathrm{min}$时的时刻$t$是多少? (结果精确到$0.01)$.", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第二册导数及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019059": { + "id": "019059", + "content": "已知$f(x)=x^2$, 求曲线$y=f(x)$在点$P(1,1)$处的切线方程.", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第二册导数及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019060": { + "id": "019060", + "content": "已知$f(x)=x^3$, 求曲线$y=f(x)$在点$P(0,0)$处的切线方程.", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第二册导数及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019061": { + "id": "019061", + "content": "如图, $A, B, C, D, E, F, G$为曲线$y=f(x)$上的点, 在这七点中分别指出满足下列条件的所有的点:\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-0.5,0) -- (5,0) node [below] {$x$};\n\\draw [->] (0,-0.5) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (-0.5,0.5) cos (0.25,1.5) sin (1,2.5) cos (1.5,2) sin (2,1.5) cos (2.25,1.6) sin (2.5,1.7) cos (2.9,1.2) sin (3.3,0.7) cos (3.6,1.1) sin (3.9,1.5) cos (4.3,1.2) sin (4.7,0.9);\n\\filldraw (-0.4,{(0.5-1.5)*cos((-0.4+0.5)/(0.25+0.5)*90)+1.5}) circle (0.03) node [above] {$A$};\n\\filldraw (0.3,{(0.5-1.5)*cos((0.3+0.5)/(0.25+0.5)*90)+1.5}) circle (0.03) node [below] {$B$};\n\\filldraw (2.25,1.6) circle (0.03) node [above] {$C$};\n\\filldraw (2.9,1.2) circle (0.03) node [below] {$D$};\n\\filldraw (3.3,0.7) circle (0.03) node [below] {$E$};\n\\filldraw (3.9,1.5) circle (0.03) node [above] {$F$};\n\\filldraw (4.6,{(1.5-1.2)*cos((4.6-3.9)/(4.3-3.9)*90)+1.2}) circle (0.03) node [below] {$G$};\n\\end{tikzpicture}\n\\end{center}\n(1) 在哪些点处, 曲线的切线斜率为$0$?\\\\\n(2) 在哪些点处, 曲线的切线斜率为正?\\\\\n(3) 在哪些点处, 曲线的切线斜率为负?\\\\\n(4) 在哪一点处, 曲线的切线斜率最大?\\\\\n(5) 在哪一点处, 曲线的切线斜率最小?", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第二册导数及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019062": { + "id": "019062", + "content": "已知曲线$y=x^2$的一条切线的斜率是$-4$, 求其切点的坐标.", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第二册导数及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019063": { + "id": "019063", + "content": "求函数$y=k x+b$的导数.", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第二册导数及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019064": { + "id": "019064", + "content": "已知函数$y=f(x)$, 其中$f(x)=\\dfrac{1}{x^2}$. 求$f'(2)$.", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第二册导数及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019065": { + "id": "019065", + "content": "求正弦函数$y=\\sin x$的驻点.", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第二册导数及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019066": { + "id": "019066", + "content": "设函数$y=\\ln x$的图像为曲线$C$.\\\\\n(1) 求曲线$C$在点$P(\\mathrm{e}, 1)$处的切线方程;\\\\\n(2) 求平行于直线$y=x$的曲线$C$的切线方程以及切点坐标.", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第二册导数及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019067": { + "id": "019067", + "content": "用导数的定义求下列函数的导数:\\\\\n(1) $y=x^2+x$;\\\\\n(2) $y=x^2-x$.", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第二册导数及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019068": { + "id": "019068", + "content": "已知函数$y=f(x)$的导函数为$y'=f'(x)$, 定义方程$f(x)=f'(x)$的实数根$x_0$叫做函数$y=f(x)$的``新驻点''. 设$f(x)=\\sin x$, 则$y=f(x)$在区间$(0, \\pi)$上的``新驻点''为\\blank{50}.", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第二册导数及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019069": { + "id": "019069", + "content": "若曲线$y=x^2$在点$P(a, a^2)$处的切线与两坐标轴围成的三角形的面积为$\\dfrac{1}{4}$, 求实数$a$的值.", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第二册导数及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019070": { + "id": "019070", + "content": "求下列函数$y=f(x)$的导数, 其中:\\\\\n(1) $f(x)=x^2 \\sin x$;\\\\\n(2) $f(x)=\\dfrac{x^2}{x+2}$;\\\\\n(3) $f(x)=(x-2)^2$.", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第二册导数及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019071": { + "id": "019071", + "content": "设实数$a>0$且$a \\neq 1$, 求证: $(\\log _a x)'=\\dfrac{1}{x \\ln a}$.", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第二册导数及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019072": { + "id": "019072", + "content": "当 J 物质中某一成分占比为$x \\%$时, 就称其是``纯净度为$x$度的 J 物质''. 已知将单位数量的 J 物质提纯到纯净度为$x$度时所需费用$c(x)=\\dfrac{5284}{100-x}$($800$且$a \\neq 1$, 求证: $(a^x)'=a^x \\ln a$.", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第二册导数及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019077": { + "id": "019077", + "content": "某个弹簧振子在振动过程中的位移$y$(单位: $\\mathrm{mm}$) 与时间$t$(单位: $\\mathrm{s}$) 满足关系$y=f(t)$, 其中$f(t)=18 \\sin (\\dfrac{2 \\pi}{3} t-\\dfrac{\\pi}{2})$. 求$f'(3)$, 并解释它的实际意义.", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第二册导数及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019078": { + "id": "019078", + "content": "已知函数$y=f(x)$与其导函数$y=f'(x)$的定义域均为$\\mathbf{R}$. 判断下列命题的真假, 并说明理由.\\\\\n(1) 若函数$y=f(x)$是奇函数, 则函数$y=f'(x)$为偶函数;\\\\\n(2) 若函数$y=f'(x)$是偶函数, 则函数$y=f(x)$是奇函数.", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第二册导数及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019079": { + "id": "019079", + "content": "定义在$\\mathbf{R}$上的偶函数$y=f(x)$满足$f(2-x)+f(x)=0$, 且在$x=1$处的导数$f'(1)=-2$, 则曲线$y=f(x)$在点$(-7, f(-7))$处的切线方程为\\blank{50}.", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第二册导数及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019080": { + "id": "019080", + "content": "已知$f(x)=x^2-4 x+2$, 求函数$y=f(x)$的单调区间.", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第二册导数及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019081": { + "id": "019081", + "content": "已知$f(x)=x^3+x^2-x-1$, 求函数$y=f(x)$的单调区间.", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第二册导数及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019082": { + "id": "019082", + "content": "已知$f(x)=x^3$, 求函数$y=f(x)$的单调区间.", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第二册导数及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019083": { + "id": "019083", + "content": "已知$f(x)=x^{-2}$, 求函数$y=f(x)$的单调区间.", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第二册导数及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019084": { + "id": "019084", + "content": "已知函数$y=f(x)$满足: 在区间$(0,1)$上$f'(x)>1$. 则下图所示的函数图像中, 有可能表示函数$y=f(x)$的图像的序号是\\blank{50}.\\\\\n\\textcircled{1}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-0.2,0) -- (1.5,0) node [below] {$x$};\n\\draw [->] (0,-0.2) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [dashed] (1,0) node [below] {$1$} -- (1,1) -- (0,1) node [left] {$1$} (0,0) -- (1,1);\n\\draw [domain = 0:1.4, thick] plot ({sin(\\x/pi*180)},\\x);\n\\end{tikzpicture} \\textcircled{2}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-0.2,0) -- (1.5,0) node [below] {$x$};\n\\draw [->] (0,-0.2) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [dashed] (1,0) node [below] {$1$} -- (1,1) -- (0,1) node [left] {$1$} (0,0) -- (1,1);\n\\draw [domain = 0:1.2, thick] plot (\\x,{sin(\\x*70)/sin(70)});\n\\end{tikzpicture} \\textcircled{3}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-0.2,0) -- (1.5,0) node [below] {$x$};\n\\draw [->] (0,-0.2) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [dashed] (1,0) node [below] {$1$} -- (1,1) -- (0,1) node [left] {$1$} (0,0) -- (1,1);\n\\draw [domain = 0:1.1, thick] plot (\\x,{\\x*\\x});\n\\end{tikzpicture} \\textcircled{4}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-0.2,0) -- (1.5,0) node [below] {$x$};\n\\draw [->] (0,-0.2) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [dashed] (1,0) node [below] {$1$} -- (1,1) -- (0,1) node [left] {$1$} (0,0) -- (1,1);\n\\draw [domain = 0:1.4, thick] plot ({\\x},{sqrt(\\x+0.3)-sqrt(0.3)});\n\\end{tikzpicture}", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第二册导数及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019085": { + "id": "019085", + "content": "如图为函数$y=f(x)$的导函数$y=f'(x)$的图像, 那么函数$y=f(x)$的图像可能为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.3]\n\\draw [->] (-3,0) -- (5,0) node [below] {$x$};\n\\draw [->] (0,-5) -- (0,5) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw [domain = -2:4] plot (\\x,{(\\x+1)*(\\x-3)});\n\\filldraw (-1,0) circle (0.1) node [below] {$-1$} (3,0) circle (0.1) node [below] {$3$};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{\\begin{tikzpicture}[>=latex,xscale = 0.2, yscale = 0.07]\n\\draw [->] (-4,0) -- (6,0) node [below] {$x$};\n\\draw [->] (0,-15) -- (0,15) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw [domain = -3.5:5.5, samples = 200] plot (\\x,{-3*\\x-\\x*\\x+\\x*\\x*\\x/3+1});\n\\draw [dashed] (-1,{8/3}) -- (-1,0) node [below] {$-1$} (3,-8)--(3,0) node [above] {$3$};\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex,xscale = 0.2, yscale = 0.07]\n\\draw [->] (-4,0) -- (6,0) node [below] {$x$};\n\\draw [->] (0,-15) -- (0,15) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw [domain = -3.5:5.5, samples = 200] plot (\\x,{3*\\x+\\x*\\x-\\x*\\x*\\x/3-1});\n\\draw [dashed] (-1,{-8/3}) -- (-1,0) node [above] {$-1$} (3,8)--(3,0) node [below] {$3$};\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex,xscale = 0.2, yscale = 0.07]\n\\draw [->] (-4,0) -- (6,0) node [below] {$x$};\n\\draw [->] (0,-15) -- (0,15) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw [domain = -2:4, samples = 200] plot (\\x,{(\\x+1)*(\\x-1)*(\\x-3)});\n\\filldraw (-1,0) circle (0.1) node [below] {$-1$} (3,0) circle (0.1) node [below] {$3$};\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex,xscale = 0.2, yscale = 0.07]\n\\draw [->] (-4,0) -- (6,0) node [below] {$x$};\n\\draw [->] (0,-15) -- (0,15) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw [domain = -2:4, samples = 200] plot (\\x,{-(\\x+1)*(\\x-1)*(\\x-3)});\n\\filldraw (-1,0) circle (0.1) node [below] {$-1$} (3,0) circle (0.1) node [below] {$3$};\n\\end{tikzpicture}}", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第二册导数及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019086": { + "id": "019086", + "content": "证明: 函数$y=\\cos x-2 x$在区间$(-\\infty,+\\infty)$上严格减.", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第二册导数及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019087": { + "id": "019087", + "content": "求函数$y=\\dfrac{\\mathrm{e}^x}{x-2}$的单调区间.", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第二册导数及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019088": { + "id": "019088", + "content": "若函数$y=x^3-a x+1$既有单调增区间, 又有单调减区间, 则实数$a$的取值范围是\\blank{50}.", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第二册导数及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019089": { + "id": "019089", + "content": "设$a \\in \\mathbf{R}$, 记$f(x)=\\dfrac{1}{2} x^2-a \\ln x$, 试根据$a$的不同取值, 讨论函数$y=f(x)$的单调性.", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第二册导数及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019090": { + "id": "019090", + "content": "已知$f(x)=-x^2+6 x-1$, 求函数$y=f(x)$的单调区间和极值.", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第二册导数及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019091": { + "id": "019091", + "content": "求正弦函数$y=\\sin x$的单调区间和极值.", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第二册导数及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019092": { + "id": "019092", + "content": "已知$f(x)=\\dfrac{1}{3} x^3-x+2$, 求函数$y=f(x)$的单调区间和极值.", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第二册导数及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019093": { + "id": "019093", + "content": "设$a$、$b$是实数, $a \\neq 0$, 若$x=a$为函数$y=a(x-a)^2(x-b)$的极大值点, 则\\bracket{20}.\n\\fourch{$ab$}{$a ba^2$}", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第二册导数及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019094": { + "id": "019094", + "content": "设函数$y=f(x)$的定义域为$\\mathbf{R}$, $x_0$($x_0 \\neq 0$)是$y=f(x)$的极大值点, 以下结论一定正确的是\\bracket{20}.\n\\twoch{对任意$x \\in \\mathbf{R}$, $f(x) \\leq f(x_0)$}{$-x_0$是$y=f(-x)$的极小值点}{$-x_0$是$y=-f(x)$的极小值点}{$-x_0$是$y=-f(-x)$的极小值点}", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第二册导数及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019095": { + "id": "019095", + "content": "设$a$、$b$是实数, $a \\neq b$, 若$f(x)=(x-a)(x-b)^2$, 函数$y=f(x)$的导数为$y'=f'(x)$, 且$y=f(x)$和$y'=f'(x)$的零点均属于集合$\\{-3,1,3\\}$, 求函数$y=f(x)$的极小值.", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第二册导数及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019096": { + "id": "019096", + "content": "已知$f(x)=-x^2+6 x-1$, 求函数$y=f(x)$在区间$[0,7]$上的最大值与最小值.", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第二册导数及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019097": { + "id": "019097", + "content": "已知$f(x)=\\dfrac{1}{3} x^3-x+2$, 求函数$y=f(x)$在区间$[0,3]$上的最大值与最小值.", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第二册导数及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019098": { + "id": "019098", + "content": "设$a \\in \\mathbf{R}$, 已知$f(x)=a x^3+2 x^2-4 x+5$, 当$x=\\dfrac{2}{3}$时, 函数$y=f(x)$有极值, 求函数$y=f(x)$在区间$[-3,3]$上的最小值.", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第二册导数及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019099": { + "id": "019099", + "content": "设$a$是实数, $f(x)=(x^2-4)(x-a)$, 若$f'(-1)=0$, 则函数$y=f(x)$在区间$[-2,2]$上的最大值是\\blank{50}.", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第二册导数及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019100": { + "id": "019100", + "content": "设$m$是实数, 若函数$y=-x^3+3 x^2-1$在区间$(m, m+5)$上存在最大值, 则$m$的取值范围是\\bracket{20}.\n\\fourch{$[-1,2)$}{$(-1,2)$}{$[-3,2)$}{$(-3,2)$}", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第二册导数及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019101": { + "id": "019101", + "content": "函数$y=\\cos x+(x+1) \\sin x+1$在区间$[0,2 \\pi]$上的最小值、最大值分别为\\bracket{20}.\n\\fourch{$-\\dfrac{\\pi}{2}$、$\\dfrac{\\pi}{2}$}{$-\\dfrac{3 \\pi}{2}$、$\\dfrac{\\pi}{2}$}{$-\\dfrac{\\pi}{2}$、$\\dfrac{\\pi}{2}+2$}{$-\\dfrac{3 \\pi}{2}$、$\\dfrac{\\pi}{2}+2$}", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第二册导数及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019102": { + "id": "019102", + "content": "求函数$y=x(\\ln x+1)$在区间$[\\dfrac{\\mathrm{e}}{2}, \\mathrm{e}]$上的最大值.", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第二册导数及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019103": { + "id": "019103", + "content": "下图是一张边长为$3$的正方形硬纸板, 先在它的四个角上裁去边长为$x$的四个小正方形, 再折叠成无盖纸盒. 当裁去的小正方形边长$x$发生变化时, 纸盒的容积$V$会随之发生变化. 当$x$在什么范围内变化时, 容积$V$随着$x$的增大而增大? 当$x$在什么范围内变化时, 容积$V$随着$x$的增大而减少? 当$x$取何值时, 容积$V$最大? 最大值是多少? (纸板厚度忽略不计)\n\\begin{center}\n \\begin{tikzpicture}[>=latex]\n \\draw (0,0) rectangle (2,2);\n \\draw [dashed] (0.3,0) --++ (0,0.3) --++ (-0.3,0);\n \\draw [dashed] (1.7,0) --++ (0,0.3) --++ (0.3,0);\n \\draw [dashed] (0.3,2) --++ (0,-0.3) --++ (-0.3,0);\n \\draw [dashed] (1.7,2) --++ (0,-0.3) --++ (0.3,0);\n \\draw (1,0) node [below] {$3$};\n \\draw (2,1.85) node [right] {$x$};\n \\end{tikzpicture}\n \\begin{tikzpicture}[>=latex]\n \\draw (0,0) coordinate (A1) --++ (1.4,0) coordinate (B1) --++ (45:0.7) coordinate (C1) --++ (-1.4,0) coordinate (D1) -- cycle;\n \\draw (A1) ++ (0,-0.3) coordinate (A);\n \\draw (B1) ++ (0,-0.3) coordinate (B);\n \\draw (C1) ++ (0,-0.3) coordinate (C);\n \\draw (D1) ++ (0,-0.3) coordinate (D);\n \\draw (A) ++ (0.3,0.3) coordinate (E);\n \\draw (C) ++ (-0.3,0) coordinate (F);\n \\draw (E)--(D)--(F) (D)--(D1);\n \\draw (A1)--(A) (B1)--(B) (C1)--(C) (D1)--(D) (A)--(B)--(C);\n \\draw (0,-0.15) node [left] {$x$};\n \\draw (0.7,-0.3) node [below] {$3-2x$};\n \\draw (0.5,-1.3);\n \\end{tikzpicture}\n \\end{center}", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第二册导数及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019104": { + "id": "019104", + "content": "如图, 将一根直径为$d$的圆木锯成截面为矩形的梁. 问: 矩形的高$h$和宽$b$应如何选择, 才能使梁的抗弯强度$W=\\dfrac{1}{6} b h^2$最大?\n\\begin{center}\n \\begin{tikzpicture}[>=latex]\n \\draw [dashed] (0,0) circle (1);\n \\filldraw (0,0) circle (0.03) node [left] {$d$};\n \\draw (230:1) rectangle (50:1);\n \\draw [dashed] (50:1) -- (230:1);\n \\draw ({cos(50)},0) node [right] {$h$};\n \\draw (0,{-sin(50)}) node [above] {$b$};\n \\end{tikzpicture}\n \\end{center}", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第二册导数及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019105": { + "id": "019105", + "content": "已知某商品的成本$C$与产量$q$满足函数关系$C=C(q)$, 其中$C(q)=100+\\dfrac{1}{4} q^2$, 并定义平均成本为$\\overline {C}=\\overline {C}(q)$, 其中$\\overline {C}(q)=\\dfrac{C(q)}{q}$.\\\\\n(1) 比较$C'(10)$和$C'(20)$, 解释两者的大小代表了怎样的实际意义;\\\\\n(2) 当产量为多少时, 平均成本最低?", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第二册导数及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019106": { + "id": "019106", + "content": "一艘船航行每小时所需的燃料费与船速的平方成正比. 如果船速是$10 \\text{km} / \\mathrm{h}$, 那么每小时的燃料费是$80$元. 已知该船航行的其他费用为每小时$480$元, 在$100 \\text{km}$的航程中, 保持怎样的船速可使航行总费用最少? (结果精确到$1 \\text{km} / \\mathrm{h}$)", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第二册导数及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019107": { + "id": "019107", + "content": "用长度为$l$的铁丝围成长方形, 求所围成的长方形的面积的最大值.", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第二册导数及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019108": { + "id": "019108", + "content": "已知某商品的生产成本$C$与产量$q$之间的关系为$C=100+4 q$, 单价$p$与产量$q$之间的关系为$p=25-\\dfrac{1}{8} q$. 假设生产的这些商品全被卖出去, 问: 产量$q$为何值时, 利润最大?", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第二册导数及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019109": { + "id": "019109", + "content": "定义在$\\mathbf{R}$上的函数$y=f(x)$为奇函数, 其导数为$y'=f'(x)$, 且当$x \\in(-\\infty, 0]$时, $f'(x)<1$, 则不等式$f(x)-f(2022) \\geq x-2022$的解集是\\bracket{20}.\n\\fourch{$(2022,+\\infty)$}{$[2022,+\\infty)$}{$(-\\infty, 2022]$}{$(-\\infty, 2022)$}", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第二册导数及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019110": { + "id": "019110", + "content": "在等腰梯形$ABCD$中, 已知上底$CD=40$, 腰$AD=40$, 问: $AB$为多少时, 等腰梯形的面积最大?", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第二册导数及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019111": { + "id": "019111", + "content": "某物体的运动路程$y$(单位: $\\mathrm{m}$) 与时间$x$(单位: $\\mathrm{s}$) 之间的关系为$y=2 x^2+\\dfrac{3}{2} x$.\\\\\n(1) 求$y$关于$x$的导数, 并解释它的实际意义;\\\\\n(2) 当$x=3$时, 求物体的运动速度.", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第二册导数及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019112": { + "id": "019112", + "content": "求曲线$y=2 x^2+\\dfrac{3}{2} x$在点$(2,11)$处的切线方程.", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第二册导数及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019113": { + "id": "019113", + "content": "求过点$A(0,-2)$且与曲线$y=2 x^2+\\dfrac{3}{2} x$相切的直线方程.", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第二册导数及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019114": { + "id": "019114", + "content": "已知$f(x)=\\ln x+x^2-3 x$. 求:\\\\\n(1) 函数$y=f(x)$的单调区间及极值;\\\\\n(2) 函数$y=f(x)$在区间$[\\dfrac{1}{4}, 4]$上的最大值与最小值.", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第二册导数及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019115": { + "id": "019115", + "content": "江轮逆水上行$300 \\text{km}$, 水速为$v \\text{km} / \\mathrm{h}$, 船在静水中的速度为$x \\text{km} / \\mathrm{h}$. 已知行船时每小时的耗油量为$c x^2$升, 即与船在静水中的速度的平方成正比, 问: $x$多大时, 全程的耗油量$H(x)$最小?", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第二册导数及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019116": { + "id": "019116", + "content": "已知$f(x)=x^3-x+1$, 则下列说法正确的是\\bracket{20}.\n\\twoch{函数$y=f(x)$有三个极值点}{函数$y=f(x)$有三个零点}{点$(0,1)$是曲线$y=f(x)$的对称中心}{直线$y=2 x$是曲线$y=f(x)$的切线}", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第二册导数及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019117": { + "id": "019117", + "content": "若曲线$y=(x+a) \\mathrm{e}^x$有两条过坐标原点的切线, 求实数$a$的取值范围.", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第二册导数及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, "020001": { "id": "020001", "content": "判断下列各组对象能否组成集合, 若能组成集合, 指出是有限集还是无限集.\\\\\n(1) 上海市控江中学$2022$年入学的全体高一年级新生;\\\\\n(2) 中国现有各省的名称;\\\\\n(3) 太阳、$2$、上海市;\\\\\n(4) 大于$10$且小于$15$的有理数;\\\\\n(5) 末位是$3$的自然数;\\\\\n(6) 影响力比较大的中国数学家;\\\\\n(7) 方程$x^2+x-3=0$的所有实数解;\\\\ \n(8) 函数$y=\\dfrac 1x$图像上所有的点;\\\\ \n(9) 在平面直角坐标系中, 到定点$(0, 0)$的距离等于$1$的所有点;\\\\\n(10) 不等式$3x-10<0$的所有正整数解;\\\\\n(11) 所有的平面四边形.",