From f044a03bfadc3f34f5653c7d2844e5a1d9cb604f Mon Sep 17 00:00:00 2001 From: "weiye.wang" Date: Sat, 29 Oct 2022 20:44:53 +0800 Subject: [PATCH] 20221029 evening --- 工具/寻找阶段末尾空闲题号.ipynb | 4 +- 工具/模板文件/题目编辑.aux | 3 + 工具/模板文件/题目编辑.log | 887 ++++++++ 工具/模板文件/题目编辑.synctex.gz | Bin 0 -> 2267 bytes 工具/添加题目到数据库.ipynb | 14 +- 工具/识别题库中尚未标注的题目类型.ipynb | 179 +- 工具/题号选题pdf生成.ipynb | 4 +- 题库0.3/Problems.json | 2744 ++++++++++++++++++++++- 8 files changed, 3778 insertions(+), 57 deletions(-) create mode 100644 工具/模板文件/题目编辑.aux create mode 100644 工具/模板文件/题目编辑.log create mode 100644 工具/模板文件/题目编辑.synctex.gz diff --git a/工具/寻找阶段末尾空闲题号.ipynb b/工具/寻找阶段末尾空闲题号.ipynb index e7d2e0a2..dce691d8 100644 --- a/工具/寻找阶段末尾空闲题号.ipynb +++ b/工具/寻找阶段末尾空闲题号.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "code", - "execution_count": 2, + "execution_count": 1, "metadata": {}, "outputs": [ { @@ -11,7 +11,7 @@ "text": [ "首个空闲id: 12030 , 直至 020000\n", "首个空闲id: 20227 , 直至 030000\n", - "首个空闲id: 30282 , 直至 999999\n" + "首个空闲id: 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"030286", + "content": "在一个平面内, $\\triangle ABC$为直角三角形, $A$为直角, $AB=3$, $BC=4$, 长为$10$的线段$PQ$以点$A$为中点, 当$\\overrightarrow{PQ}$与$\\overrightarrow{BC}$的夹角$\\theta$取何值时, $\\overrightarrow{BP}\\cdot \\overrightarrow{CQ}$的值最大? 并求出这个最大值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册复习课例题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030287": { + "id": "030287", + "content": "已知复平面上如图所示的平行四边形$ABCD$的顶点$A,B,D$三点所对应的复数分别是$2+3\\mathrm{i}$、$5-\\mathrm{i}$、$6+\\mathrm{i}$, 求向量$\\overrightarrow{AC}$所对应的复数.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册复习课例题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030288": { + "id": "030288", + "content": "已知复数$z_1=\\sqrt 3+\\mathrm{i}$, $|z_2|=1$, $z_1\\overline{z_2}$是虚部为负数的纯虚数, 求复数$z_2$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册复习课例题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030289": { + "id": "030289", + "content": "若虚数$z_1$、$z_2$满足$z_1^2=z_2$, 且$z_1$、$z_2$是一个实系数一元二次方程的两个根, 写出这样的一个实系数一元二次方程.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册复习课例题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030290": { + "id": "030290", + "content": "火箭发射$t\\text{s}$后, 其高度(单位: $\\text{m}$)为$h(t)=0.9t^2$. 求:\\\\\n(1) 在$1\\le t\\le 2$这段时间里, 火箭爬高的平均速度;\\\\ \n(2) 发射后第$10\\text{s}$时, 火箭爬高的瞬时速度.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + 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"", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030294": { + "id": "030294", + "content": "设$f(x)=\\dfrac 1x$, 求 $f'(1)$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030295": { + "id": "030295", + "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, 已知在第$x\\text{h}$时, 原油的温度(单位: $\\text{C}^\\circ$)为$y=f(x)=x^2-7x+15$($0\\le x\\le 8$). 计算第$2\\text{h}$与第$6\\text{h}$时, 原油温度的瞬时变化率, 并说明它们的意义.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030296": { + "id": "030296", + "content": "一辆汽车在公路上沿直线变速行驶, 假设$t\\text{s}$时汽车的速度(单位: $\\text{m}/\\text{s}$)为$y=v(t)=-t^2+6t+60$, 求汽车在第$2\\text{s}$与第$6\\text{s}$时的瞬时加速度, 并说明它们的意义.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030297": { + "id": "030297", + "content": "设$f(x)=x$, 求$f'(1)$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030298": { + "id": "030298", + "content": "一质点$A$沿直线运动, 位移$y$(单位: $\\text{m}$)与时间$t$(单位: $\\text{s}$)之间的关系为$y(t)=2t^2+1$, 求质点$A$在$t=2.7\\text{s}$时的瞬时速度.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030299": { + "id": "030299", + "content": "设函数$f(x)=x^2-1$, 求:\\\\ \n(1) 当自变量$x$由$1$变到$1.1$时, 函数的平均变化率;\\\\\n(2) 函数在$x=1$处的导数.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030300": { + "id": "030300", + "content": "函数$f(x)$的图像如图所示, 下列数值排序正确的是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw [->] (0,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,0) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (0,0.5) .. controls (1,2.5) and (2,2.8) .. (3.5,3);\n\\foreach \\i in {1,2,3}{\\draw (\\i,0.2) -- (\\i,0) node [below] {$\\i$};};\n\\end{tikzpicture}\n\\end{center}\n\\twoch{$f'(1)>f'(2)>f'(3)>0$}{$f'(1)f'(2)>0>f'(3)$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030301": { + "id": "030301", + "content": "求曲线$y=-2x^2+1$在点$(1, -1)$处的切线方程.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030302": { + "id": "030302", + "content": "吹气球时, 气球的半径$r$(单位: $\\text{dm}$)与体积$V$(单位:$\\text{L}$)之间的函数关系是$r(V)=\\sqrt[3]{\\dfrac{3V}{4\\pi}}$. 利用信息技术工具, 画出$0\\le V\\le 5$时函数的图象, 并根据其图象估计$V=0.6,1.2\\text{L}$时, 气球的瞬时膨胀率.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030303": { + "id": "030303", + "content": "一个物体从$10\\text{m}$高处做自由落体运动, $t\\text{s}$时该物体距离地面的高度(单位: $\\text{m}$)为$h(t)=-4.9t^2+10$. 求该物体在$t=1$时的瞬时速度, 并解释此时物体的运动状况.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030304": { + "id": "030304", + "content": "圆的面积$S$(单位: $\\text{cm}^2$)与半径$R$(单位: $\\text{m}$)的关系为$S=\\pi R^2$. 求$R=5\\text{cm}$时面积关于半径的瞬时变化率.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030305": { + "id": "030305", + "content": "某质点沿直线运动, 位移$y$(单位: $\\text{m}$)与时间$t$: (单位: $\\text{s}$)之间的关系为$y(t)=5t^2+6$. 求:\\\\\n(1) $2\\le t\\le 3$这段时间内的平均速度;\\\\\n(2) $t=2\\text{s}$时的瞬时速度.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030306": { + "id": "030306", + "content": "已知车轮旋转的角度$\\theta$(单位: $\\text{rad}$)与时间(单位: $\\text{s}$)之间的关系为$\\theta(t)=\\dfrac{25\\pi}{8}t^2$. 求车轮转动开始后第$3.2\\text{s}$时的瞬时角速度.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030307": { + "id": "030307", + "content": "小明骑车上学, 开始时匀速行驶, 途中因交通堵塞停留了一段时间, 后为了赶时间加快速度行驶, 与以上事件吻合得最好的图象是\\bracket{20}.\n\\fourch{\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (0,0) -- (3,0) node [below] {$t$};\n\\draw [->] (0,0) -- (0,2) node [right] {距学校的距离};\n\\draw (0,0) node [below left] {$O$};\n\\draw (0,0) -- (1,0.5) .. controls (1.8,0.6) .. (2,1.3);\n\\end{tikzpicture}\n}{\\begin{tikzpicture}[>=latex]\n\\draw [->] (0,0) -- (3,0) node [below] {$t$};\n\\draw [->] (0,0) -- (0,2) node [right] {距学校的距离};\n\\draw (0,0) node [below left] {$O$};\n\\draw (0,1.1) -- (0.6,0.8) -- (0.9,0.8) -- (2.5,0);\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex]\n\\draw [->] (0,0) -- (3,0) node [below] {$t$};\n\\draw [->] (0,0) -- (0,2) node [right] {距学校的距离};\n\\draw (0,0) node [below left] {$O$};\n\\draw (0,1.1) -- (0.6,0.8) -- (0.8,0.8) .. controls (1.7,0.8) .. (2.1,0);\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex]\n\\draw [->] (0,0) -- (3,0) node [below] {$t$};\n\\draw [->] (0,0) -- (0,2) node [right] {距学校的距离};\n\\draw (0,0) node [below left] {$O$};\n\\draw (0,1.1) -- (1,0.8) -- (2,0);\n\\end{tikzpicture}}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030308": { + "id": "030308", + "content": "求曲线$y=\\dfrac 12 x^2-2$在点$(1, -\\dfrac 32)$处的切线的倾斜角.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030309": { + "id": "030309", + "content": "一个质量为$3\\text{kg}$的物体做直线运动, 设位移$y$(单位: $\\text{m}$)与时间: (单位: $\\text{s}$)之间的关系为$y(t)=1+t^2$, 并且物体的动能$E_k=\\dfrac 12 mv^2$. 求物体开始运动后第$5\\text{s}$时的动能.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030310": { + "id": "030310", + "content": "根据下面的文字叙述, 画出相应的路程关于时间的函数图象的大致形状.\\\\ \n(1) 汽车在笔直的公路上匀速行驶;\\\\\n(2) 汽车在笔直的公路上不断加速行驶;\\\\\n(3) 汽车在笔直的公路上不断减速行驶.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030311": { + "id": "030311", + "content": "已知函数$f(x)$的图像, 试画出其导函数$f'(x)$图像的大致形状.\\\\\n(1) \n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (-0.5,1.5) -- (1.5,-0.5);\n\\draw (1,0.1) -- (1,0) node [below] {$1$};\n\\draw (0.1,1) -- (0,1) node [left] {$1$};\n\\end{tikzpicture}\n(2) \n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (0.1,0.5) .. controls (1.3,0.6) .. (1.8,1.3);\n\\draw (1,0.1) -- (1,0) node [below] {$1$};\n\\draw (0.1,1) -- (0,1) node [left] {$1$};\n\\end{tikzpicture}\n(3)\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -1.4:1.4] plot (\\x,{pow(\\x,2)});\n\\draw (1,0.1) -- (1,0) node [below] {$1$};\n\\draw (0.1,1) -- (0,1) node [left] {$1$};\n\\end{tikzpicture}", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030312": { + "id": "030312", + "content": "在高台跳水运动中, $t\\text{s}$时运动员的重心相对于水面的高度(单位: $\\text{m}$)是$h(t)=-4.9t^2+4.8t+11$. 高度$h$关于时间$t$的导数是速度$v$, 速度$v$关于时间$t$的导数$v'$的物理意义是什么? 试求$v$, $v'$关于时间$t$的函数解析式.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030313": { + "id": "030313", + "content": "根据下列条件, 分别画出函数$y=f(x)$的图像在这点附近的大致形状:\\\\\n(1) $f(1)=-5$, $f'(1)=-1$;\\\\\n(2) $f(5)=10$, $f'(5)=15$;\\\\\n(3) $f(10)=20$, $f'(10)=0$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030314": { + "id": "030314", + "content": "求下列函数的导数:\\\\\n(1) $y=x^{\\frac 23}$;\\\\\n(2) $y=\\log_2 x$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030315": { + "id": "030315", + "content": "假设某地在$20$年间的年均通货膨胀率为$5\\%$, 物价$p$(单位: 元)与时间$t$(单位: 年)之间的关系为$p(t)=p_0(1+5\\%)^t$, 其中$p_0$为$t=0$时的物价. 假定某种商品的$p_0=1$, 那么在第$10$个年头, 这种商品的价格上涨的速度大约是多少(精确到$0.01$元/年)?", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030316": { + "id": "030316", + "content": "求下列函数的导数:\\\\\n(1) $y=\\dfrac 1{x^4}$;\\\\\n(2) $y=\\sqrt[3]{x^4}$;\\\\\n(3) $y=3^x$;\\\\\n(4) $y=(\\dfrac 12)^x$;\\\\\n(5) $y=\\log_4 x$;\\\\\n(6) $y=\\log_{\\frac 12} x$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030317": { + "id": "030317", + "content": "求下列函数在给定点的导数:\\\\\n(1) $y=x^5$在$x=3$处的导数;\\\\\n(2) $y=\\ln x$在$x=\\dfrac 23$处的导数;\\\\\n(3) $y=\\sin x$在$x=2\\pi$处的导数;\\\\\n(4) $y=\\mathrm{e}^x$在$x=0$处的导数.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030318": { + "id": "030318", + "content": "求余弦曲线$y=\\cos x$在点$(\\dfrac \\pi 2,0)$处的切线方程.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030319": { + "id": "030319", + "content": "求曲线$y=x^{\\frac 12}$在点$(4,2)$处的切线方程.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030320": { + "id": "030320", + "content": "求下列函数的导数:\\\\\n(1) $y=x^3-x+3$;\\\\\n(2) $y=2^x+\\cos x$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030321": { + "id": "030321", + "content": "求下列函数的导数:\\\\\n(1) $y=x^3\\mathrm{e}^x$;\\\\\n(2) $y=\\dfrac{2\\sin x}{x^2}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030322": { + "id": "030322", + "content": "日常生活中的饮用水通常是经过净化的. 随着水的纯净度的提高, 所需净化费用不断增加. 已知将$1\\text{t}$水净化到纯净度为$x\\%$时所需费用(单位: 元)为\n$c(x)=\\dfrac{5284}{100-x}$($800$的解集.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030338": { + "id": "030338", + "content": "氡气是一种由地表自然散发的无味的放射性气体, 如果最初有$500\\text{g}$氡气, 那么$t$天后, 氡气的剩余量为$A(t)=500\\times 0.834^t\\text{g}$.\\\\\n(1) 氡气的散发速度是多少?\\\\\n(2) $A'(7)$的值是什么(精确到$0.1$)? 它表示什么意义?", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030339": { + "id": "030339", + "content": "设某高山滑雪运动员在一次滑雪训练中滑行的路程$l$(单位: $\\text{m}$)与时间: (单位: $\\text{s}$)之间的关系为$l(t)=2t^2+\\dfrac 32t$.\n(1) 求$l$关于$t$的导数, 并解释它的实际意义;\\\\\n(2) 当$t=3\\text{s}$时, 求运动员的滑雪速度;\\\\ \n(3) 当运动员的滑雪路程为$38\\text{m}$时, 求此时的滑雪速度.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030340": { + "id": "030340", + "content": "设曲线$y=\\mathrm{e}^{2ax}$在点$(0, 1)$处的切线与直线$2x-y+1=0$垂直, 求$a$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030341": { + "id": "030341", + "content": "请按步骤, 完成下面的任务.\\\\\n(1) 利用信息技术工具, 分别画出$h=1,0.5,0.1,0.05$时, 函数$y=\\dfrac{\\sin(x+h)-\\sin x}h$的图像;\\\\\n(2) 画出函数$y=\\cos x$的图像, 并与上面的四个图像比较, 当$h$越来越小时, 你观察到了什么?\\\\\n(3) 猜测$y=\\sin x$的导数, 它与基本初等函数的导数公式表中$\\sin x$的导数公式一样吗?", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030342": { + "id": "030342", + "content": "某海湾拥有世界上最大的海潮, 其高低水位之差可达到$15\\text{m}$. 假设在该海湾某一固定点, 大海水深$d$(单位: $\\text{m}$)与午夜后的时间: (单位: $\\text{h}$)之间的关系为$d(t)=10+4\\cos t$. 求下列时刻潮水的速度(精确到$0.01\\text{m}/\\text{h}$):\\\\\n(1) 上午$6:00$;\\\\ \n(2) 上午$9:00$;\\\\\n(3) 中午$12:00$;\\\\ \n(4) 下午$6:00$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030343": { + "id": "030343", + "content": "利用导数判断下列函数的单调性:\\\\\n(1) $f(x)=x^3+3x$;\\\\\n(2) $f(x)=\\sin x-x$, $x\\in (0,\\pi)$;\\\\\n(3) $f(x)=\\dfrac{x-1}x$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030344": { + "id": "030344", + "content": "已知导函数$f'(x)$的下列信息: 当$10$; 当$x<1$, 或$x>4$时, $f'(x)<0$; 当$x=1$, 或$x=4$时, $f'(x)=0$. 试画出函数$f(x)$图像的大致形状.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030345": { + "id": "030345", + "content": "判断下列函数的单调性:\\\\\n(1) $f(x)=x^2-2x+4$;\\\\\n(2) $f(x)=\\mathrm{e}^x-x$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030346": { + "id": "030346", + "content": "利用导数讨论二次函数$f(x)=ax^2+bx+c$的单调区间.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030347": { + "id": "030347", + "content": "求函数$f(x)=\\dfrac 13x^3-\\dfrac 12x^2-2x+1$的单调区间.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030348": { + "id": "030348", + "content": "判断下列函数的单调性, 并求出单调区间:\\\\\n(1) $f(x)=3x-x^3$;\\\\\n(2) $f(x)=x^3-x^2-x$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030349": { + "id": "030349", + "content": "证明函数$f(x)=2x^3-6x^2+7$在区间$(0,2)$上单调递减.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030350": { + "id": "030350", + "content": "求函数$f(x)=\\dfrac 13 x^3-4x+4$的极值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030351": { + "id": "030351", + "content": "求下列函数的极值:\\\\\n(1) $f(x)=6x^2-x-2$;\\\\\n(2) $f(x)=x^3-27x$;\\\\\n(3) $f(x)=6+12x-x^3$;\\\\\n(4) $f(x)=3x-x^3$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030352": { + "id": "030352", + "content": "求函数$f(x)=\\dfrac 13 x^3-4x+4$的最大值与最小值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030353": { + "id": "030353", + "content": "证明: 当$x>0$时, $1-\\dfrac 1x\\le \\ln x$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030354": { + "id": "030354", + "content": "求下列函数在给定区间上的最大值与最小值:\\\\\n(1) $f(x)=6x^2-x-2$, $x\\in [0,2]$;\\\\\n(2) $f(x)=x^3-27x$, $x\\in [-4,4]$;\\\\\n(3) $f(x)=6+12x-x^3$, $x\\in [-\\dfrac 13,3]$;\\\\\n(4) $f(x)=3x-x^3$, $x\\in [2,3]$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030355": { + "id": "030355", + "content": "证明不等式: $x-1\\ge \\ln x$, $x\\in (0,+\\infty)$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030356": { + "id": "030356", + "content": "给定函数$f(x)=(x+1)\\mathrm{e}^x$.\\\\\n(1) 判断函数$f(x)$的单调性, 并求出$f(x)$的极值;\\\\\n(2) 画出函数$f(x)$的大致图像;\\\\\n(3) 求出方程$f(x)=a$($a\\in \\mathbf{R}$)的解的个数.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030357": { + "id": "030357", + "content": "某制造商制造并出售球形瓶装的某种饮料, 瓶子的制造成本是$0.8\\pi r^2$\n分, 其中$r$(单位: $\\text{cm}$)是瓶子的半径, 已知每出售$1\\text{mL}$的饮料, 制造商可获利$0.2$分, 且制造商能制作的瓶子的最大半径为$6\\text{cm}$.\\\\\n(1) 瓶子半径多大时, 能使每瓶饮料的利润最大?\\\\\n(2) 瓶子半径多大时, 每瓶饮料的利润最小?", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030358": { + "id": "030358", + "content": "利用函数的单调性, 证明下列不等式, 并通过函数图像直观验证: $\\sin x=latex]\n\\draw (0,0) arc (0:180:1) --++ (0,-2) --++ (2,0) --++ (0,2);\n\\draw [dashed] (0,0) -- (-2,0);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030360": { + "id": "030360", + "content": "判断下列函数的单调性, 并求出单调区间:\\\\\n(1) $f(x)=-2x+1$;\\\\\n(2) $f(x)=x+\\cos x$, $x\\in (0,\\dfrac\\pi 2)$;\\\\\n(3) $f(x)=2x-4$;\\\\\n(4) $f(x)=2x^3+4x$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030361": { + "id": "030361", + "content": "判断下列函数的单调性, 并求出单调区间:\\\\\n(1) $f(x)=x^2+2x-4$;\\\\\n(2) $f(x)=2x^2-3x+3$;\\\\\n(3) $f(x)=3x+x^3$;\\\\\n(4) $f(x)=x^3+x^2-x$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030362": { + "id": "030362", + "content": "求下列函数的极值:\\\\\n(1) $f(x)=6x^2+x+2$;\\\\\n(2) $f(x)=x^3-12x$;\\\\\n(3) $f(x)=6-12x+x^3$;\\\\\n(4) $f(x)=48x-x^3$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030363": { + "id": "030363", + "content": "求下列函数在给定区间上的最大值与最小值:\\\\\n(1) $f(x)=6x^2+x+2$, $x\\in [-1,1]$;\\\\\n(2) $f(x)=x^3-12x$, $x\\in [-3,3]$;\\\\\n(3) $f(x)=6-12x+x^3$, $x\\in [-\\dfrac 13,1]$;\\\\\n(4) $f(x)=48x-x^3$, $x\\in [-3,5]$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030364": { + "id": "030364", + "content": "将一条长为$l$的铁丝截成两段, 分别弯成两个正方形, 要使两个正方形的面积和最小, 两段铁丝的长度分别是多少?", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030365": { + "id": "030365", + "content": "将一个边长为$a$的正方形铁片的四角截去四个边长均为$x$的小正方形, 做成一个无盖方盒.\\\\\n(1) 试把方盒的容积$V$表示为$x$的函数;\\\\ \n(2) $x$多大时, 方盒的容积$V$最大?", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030366": { + "id": "030366", + "content": "用测量工具测量某物体的长度, 由于工具的精度以及测量技术的原因, 测得$n$个数据$a_1,a_2,a_3,\\cdots,a_n$.\n证明: 用$n$个数据的平均值$x=\\dfrac 1n\\displaystyle\\sum_{i=1}^na_i$表示这个物体的长度, 能使$f(x)=\\dfrac 1n\\displaystyle\\sum_{i=1}^n(x-a_i)^2$最小.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030367": { + "id": "030367", + "content": "已知某商品的生产成本$C$与产量$q$之间的关系为$C=100+4q$, 单价$p$与产量$q$之间的关系为$p=25-\\dfrac 18q$. 产量$q$为何值时, 利润最大?", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030368": { + "id": "030368", + "content": "已知某商品进价为$a$元/件, 根据以往经验, 当售价是$b$($b\\ge \\dfrac 43 a$)元/件时, 可卖出$c$件. 市场调查表明, 当售价下降$10\\%$时, 销量可增加$40\\%$, 现决定一次性降价, 销售价为多少时, 可获得最大利润?", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030369": { + "id": "030369", + "content": "利用函数的单调性, 证明下列不等式, 并通过函数图象直观验证:\\\\\n(1) $\\mathrm{e}^x>1+x$, $x\\ne 0$;\\\\\n(2) $\\ln x0$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030370": { + "id": "030370", + "content": "已知点$P$和点$Q$是曲线$y=x^2-2x-3$上的两点, 且点$P$的横坐标是$1$, 点$Q$的横坐标是$4$. 求:\\\\ \n(1) 割线$PQ$的斜率;\\\\ \n(2) 点$P$处的切线方程.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030371": { + "id": "030371", + "content": "求下列函数的导数:\\\\\n(1) $y=2x\\tan x$;\\\\\n(2) $y=2^x\\ln x$;\\\\\n(3) $y=(x-2)^3(3x+1)^2$;\\\\\n(4) $y=\\dfrac{x^2}{(2x+1)^3}$;\\\\\n(5) $y=\\mathrm{e}^{-2x+1}\\cos(-x^2+x)$;\\\\\n(6) $y=\\dfrac{\\sin 2x}{\\sqrt{x}}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030372": { + "id": "030372", + "content": "已知函数$y=f(x)$的图像是下列四个图像之一, 且其导函数$y=f'(x)$的图像如图所示, 则该函数的图象是\\bracket{20}.\\\\\n\\begin{minipage}[t]{.25\\linewidth}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1.5,0) -- (1.5,0) node [below] {$x$};\n\\draw [->] (0,-0.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -1:1] plot (\\x,{1-\\x*\\x});\n\\draw (-1,0) node [below] {$-1$} (1,0) node [below] {$1$};\n\\draw (0,1.5) node [right] {$y=f'(x)$};\n\\end{tikzpicture}\n\\end{minipage}\n\\begin{minipage}[t]{.75\\linewidth}\n\\fourch{\\begin{tikzpicture}[>=latex,scale = 0.8]\n\\draw [->] (-1.5,0) -- (1.5,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,1) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw [domain = -1:1] plot (\\x,{\\x-pow(\\x,3)/3});\n\\end{tikzpicture}\n}{\\begin{tikzpicture}[>=latex,scale = 0.8]\n\\draw [->] (-1.5,0) -- (1.5,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,1) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw [domain = -1:1] plot ({(\\x-pow(\\x,3)/3)*3/2},{\\x*2/3});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex,scale = 0.8]\n\\draw [->] (-1.5,0) -- (1.5,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,1) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (-1,{-2/3}) .. controls (0.5,-0.4) .. (1,{2/3});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex,scale = 0.8]\n\\draw [->] (-1.5,0) -- (1.5,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,1) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw (-1,{-2/3}) .. controls (-0.5,0.4) .. (1,{2/3});\n\\end{tikzpicture}}\n\\end{minipage}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030373": { + "id": "030373", + "content": "一个距地心距离为$r$, 质量为$m$的人造卫星, 与地球之间的万有引力$F$由公式$F=\\dfrac{GMm}{r^2}$给出, 其中$M$为地球质量, $G$为引力常量, 求$F$对于$r$的瞬时变化率.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030374": { + "id": "030374", + "content": "一杯$80\\text{C}^\\circ$的热红茶置于$20\\text{C}^\\circ$的房间里, 它的温度会逐渐下降, 温度$T$(单位: $\\text{C}^\\circ$)与时间(单位: $\\text{min}$)之间的关系由函数$T=f(t)$给出.\\\\\n(1) 判断$f'(t)$的正负, 并说明理由;\\\\\n(2) $f'(3)=-4$的实际意义是什么? 如果$f(3)=65$, 你能画出函数$f(t)$在$t=3$时图像的大致形状吗?", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030375": { + "id": "030375", + "content": "求函数$f(x)=\\sqrt[3]{x^2}$的单调区间.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030376": { + "id": "030376", + "content": "已知函数$f(x)=x^2+px+q$, 试确定$p,q$的值, 使得当$x=1$时, $f(x)$有最小值$4$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030377": { + "id": "030377", + "content": "已知函数$f(x)=x(x-c)^2$在$x=2$处有极大值, 求$c$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030378": { + "id": "030378", + "content": "过点$P(1,1)$作直线$AB$, 分别与$x$轴的正半轴、$y$轴的正半轴交于点$A, B$. 当直线$AB$在什么位置时, $\\triangle AOB$的面积最小? 最小面积是多少?", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030379": { + "id": "030379", + "content": "如图, 直线$l$和圆$P$, 当$l$从$l_0$开始在平面上按逆时针方向绕点$O$匀速转动(转动角度不超过$90^\\circ$时, 它扫过的圆内阴影部分的面积$S$是时间$t$的函数, 这个函数的图像大数是\\bracket{20}.\\\\\n\\begin{minipage}[t]{.25\\linewidth}\n\\begin{tikzpicture}[>=latex]\n\\begin{scope}\n\\draw [clip] (0,0) circle (1); \n\\draw (0.71,0.71) node [above right] {$P$};\n\\filldraw [pattern = north east lines] (1.2,0) -- (-1,-1) -- (1,-1);\n\\end{scope}\n\\draw (0,0) circle (1);\n\\draw (-1,-1) node [left] {$O$}-- (1,-1) node [right] {$l_0$};\n\\draw (-1,-1) -- (1.2,0) node [right] {$l$};\n\\end{tikzpicture}\n\\end{minipage}\n\\begin{minipage}[t]{.75\\linewidth}\n\\fourch{\\begin{tikzpicture}[>=latex, scale = {2/pi}]\n\\draw [->] (0,0) -- ({2*pi/3},0) node [below] {$x$};\n\\draw [->] (0,0) -- (0,pi) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (0,0) -- (pi/2,pi);\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = {2/pi}]\n\\draw [->] (0,0) -- ({2*pi/3},0) node [below] {$x$};\n\\draw [->] (0,0) -- (0,pi) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (0,0) ..controls (pi/4,0.2) .. (pi/2,pi);\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = {2/pi}]\n\\draw [->] (0,0) -- ({2*pi/3},0) node [below] {$x$};\n\\draw [->] (0,0) -- (0,pi) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (0,0) ..controls (0.2,0.8) .. (pi/4,pi/2) ..controls ({pi/2-0.2},{pi-0.8}) .. (pi/2,pi);\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = {2/pi}]\n\\draw [->] (0,0) -- ({2*pi/3},0) node [below] {$x$};\n\\draw [->] (0,0) -- (0,pi) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0:pi/2,samples = 100] plot (\\x,{acos(cos(\\x/pi*180)-sin(\\x/pi*180))/180*pi-0.5*sin(2*acos(cos(\\x/pi*180)-sin(\\x/pi*180))});\n\\end{tikzpicture}}\n\\end{minipage}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030380": { + "id": "030380", + "content": "当室内的有毒细菌开始增加时, 就要使用杀菌剂, 刚开始使用的时候, 细菌数量还会继续增加, 随着时间的增加, 它增加的幅度逐渐变小, 到一定时间, 细菌数量开始减少, 已知使用杀菌剂$t\\text{h}$后的细菌数量为$b(t)=10^5+10^4t-10^3t^2$.\\\\\n(1) 求细菌数量在$t=5$与$t=10$时的瞬时变化率;\\\\\n(2) 细菌数量在哪段时间增加, 在哪段时间减少? 为什么?", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030381": { + "id": "030381", + "content": "已知曲线$y=x+\\ln x$在点$(1,1)$处的切线与曲线$y=ax^2+(2a+3)x+1$只有一个公共点, 求$a$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030382": { + "id": "030382", + "content": "用总长$14.8\\text{m}$的钢条制作一个长方体容器的框架, 如果所制容器底面一边的长比另一边的长多$0.5\\text{m}$, 那么高为多少时容器的容积最大? 最大容积是多少?", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030383": { + "id": "030383", + "content": "用半径为$R$的圆形铁皮剪出一个圆心角为$\\alpha$的扇形, 制成一个圆锥形容器, 扇形的圆心角$\\alpha$为多大时, 容器的容积最大?", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030384": { + "id": "030384", + "content": "已知$A$, $B$两地的距离是$130\\text{km}$, 根据交通法规, 两地之间的公路车速应限制在$50$至$100\\text{km}$, 假设油价是$7元$/$\\text{L}$, 以$x\\text{km}/\\text{h}$的速度行驶时, 汽车的耗油率为$(3+\\dfrac{x^2}{360})\\text{L}/\\text{h}$, 司机每小时的工资是$35$元, 那么最经济的车速是多少? 如果不考虑其他费用, 这次行车的总费用是多少?", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030385": { + "id": "030385", + "content": "作函数$y=\\dfrac{\\mathrm{e}^x(2x-1)}{x-1}$的大致图像.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030386": { + "id": "030386", + "content": "已知函数$f(x)=\\mathrm{e}^x-\\ln (x+m)$. 当$x\\le 2$时, 求证$f(x)>0$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030387": { + "id": "030387", + "content": "已知函数$f(x)=a\\cdot \\mathrm{e}^{2x}+(a-2)\\mathrm{e}^x-x$.\\\\\n(1) 讨论$f(x)$的单调性;\\\\\n(2) 若$f(x)$有两个零点, 求$a$的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题", + "edit": [ + "20221028\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030388": { + "id": "030388", + "content": "(1) 已知$f'(x_0)=2$, 则$\\displaystyle \\lim_{h\\to 0}\\dfrac{f(x_0-4h)-f(x_0)}{h}=$\\blank{50}.\\\\\n(2) 已知某物体的运动方程为$s=24t-3t^2$($s$的单位: $\\text{m}$, $t$的单位: $\\text{s}$), 则物体在$t=$\\blank{50}$\\text{s}$时的瞬时速度为$12\\text{m}/\\text{s}$.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学教与学例题与习题", + "edit": [ + "20221029\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030389": { + "id": "030389", + "content": "已知曲线$y=2x^2-7$在点$P$处的切线方程为$8x-y-15=0$, 则切点$P$的坐标为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学教与学例题与习题", + "edit": [ + "20221029\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030390": { + "id": "030390", + "content": "已知二次函数$f(x)=ax^2+bx+c$, 求:\\\\\n(1) 函数从$x_1=1$到$x_2=3$的平均变化率;\\\\\n(2) 函数在$x=2$处的瞬时变化率;\\\\ \n(3) 当$x$为何值时, 函数在$x$处的瞬时变化率等于从$x_1=1$到$x_2=3$的平均变化率?", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学教与学例题与习题", + "edit": [ + "20221029\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030391": { + "id": "030391", + "content": "已知函数$f(x)=x^6-x$, 直线$l$是曲线$y=f(x)$的一条切线.\\\\\n(1) 若$l$与直线$16x-13y=0$垂直, 求$l$与曲线$y=f(x)$的切点坐标;\\\\\n(2) 若$l$经过点$(0,-5)$, 求$l$的方程.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学教与学例题与习题", + "edit": [ + "20221029\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030392": { + "id": "030392", + "content": "已知函数$f(x)=\\dfrac 13x^3-\\dfrac 12 ax^2+(a-1)x+2$($a\\in \\mathbf{R}$), $y=f(x)$满足$f(x)+f(-x)=4$, 已知点$M$是曲线$y=f(x)$上任意一点, 曲线在$M$处的切线为$l$.\\\\\n(1) 求切线$l$的倾斜角$\\alpha$的取值范围;\\\\ \n(2) 若过点$P(1,m)$($m\\ne \\dfrac 43$)可作曲线$y=f(x)$的三条切线, 求实数$m$的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学教与学例题与习题", + "edit": [ + "20221029\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030393": { + "id": "030393", + "content": "等比数列$\\{a_n\\}$中, $a_2a_7=8$, 函数$f(x)=x(x+a_1)(x+a_2)\\cdots (x+a_8)$, 则$f'(0)=$\\bracket{20}.\n\\fourch{$2^6$}{$2^9$}{$2^{12}$}{$2^{15}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学教与学例题与习题", + "edit": [ + "20221029\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030394": { + "id": "030394", + "content": "已知函数$y=f(x)$的导数为$y=f'(x)$, 而且$f(x)=x^2+3xf'(2)+\\ln x$, 求$f'(2)$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学教与学例题与习题", + "edit": [ + "20221029\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030395": { + "id": "030395", + "content": "求下列函数的导数:\\\\\n(1) $f(x)=\\dfrac{(x-2)^2}{x+1}$;\\\\\n(2) $f(x)=(x^2+9)(x-\\dfrac 3x)$;\\\\\n(3) $f(x)=\\dfrac{\\ln x}{x}$;\\\\\n(4) $f(x)=x^2\\cos x$;\\\\\n(5) $f(x)=\\ln (2x+1)$;\\\\\n(6) $f(x)=\\cos(\\dfrac \\pi 3-2x)$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学教与学例题与习题", + "edit": [ + "20221029\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030396": { + "id": "030396", + "content": "设函数$f(x)=\\sin x-\\cos x$, $x\\in \\mathbf{R}$.\\\\\n(1) 求$y=f^2(x)$的最大值和图像的对称中心;\\\\\n(2) $y=f'(x)$为$y=f(x)$的导数, 若$f'(x_0)=2f(x_0)$, 求$\\sin 2x_0+\\cos^2 x_0+\\tan(\\dfrac\\pi 4+x_0)$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学教与学例题与习题", + "edit": [ + "20221029\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030397": { + "id": "030397", + "content": "曲线$y=x\\ln x-2x$在$P(1,-2)$处的切线方程为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学教与学例题与习题", + "edit": [ + "20221029\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030398": { + "id": "030398", + "content": "已知函数$f_0(x)=\\sin x+\\cos x$, $f_1(x)=f_0'(x)$, $f_2(x)=f_1'(x)$, $\\cdots$, $f_{n+1}(x)=f_n'(x)$, $n$为正整数, 则$f_{2022}(x)=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学教与学例题与习题", + "edit": [ + "20221029\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030399": { + "id": "030399", + "content": "拉格朗日中值定理是微分学中的基本定理之一, 定理内容: 如果函数$y=f(x)$\n在闭区间$[a,b]$上的图像连续不间断, 在开区间$(a,b)$内的导数为$y=f'(x)$, 那么在区间$(a,b)$内至少存在一点$c$, 使得$f(b)-f(a)=f'(c)(b-a)$成立, 其中$c$称为$y=f(x)$在$[[a, b]$的``拉格朗日中值点''. 根据这个定理, 可得函数$f(x)=x^3-3x$在$[-2,2]$上的``拉格朗日中值点''的个数为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学教与学例题与习题", + "edit": [ + "20221029\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030400": { + "id": "030400", + "content": "给出定义: 若函数$y=f(x)$在$D$上可导, 即$y=f'(x)$存在, 且导数$y=f'(x)$在$D$上也可导, 则称$y=f(x)$在$D$上存在二阶导数, 记$f''(x)=(f'(x))'$, 若$f''(x)<0$在$D$上恒成立, 则称$y=f(x)$在$D$上为严格凸函数, 以下四个函数在$(0,\\dfrac \\pi 2)$上是严格凸函数的是\\bracket{20}.\n\\fourch{$f(x)=\\sin x+\\cos x$}{$f(x)=-\\ln x-2x$}{$f(x)=x^3+2x-1$}{$f(x)=x\\mathrm{e}^x$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学教与学例题与习题", + "edit": [ + "20221029\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030401": { + "id": "030401", + "content": "已知曲线$f(x)=(ax+b)\\mathrm{e}^x$在点$(0,2)$处的切线方程为$x+y-2=0$, 则$a=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学教与学例题与习题", + "edit": [ + "20221029\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030402": { + "id": "030402", + "content": "写出一个同时满足性质\\textcircled{1}, \\textcircled{2}的函数$f(x)=$\\blank{50}. ($y=f(x)$不是常值函数)\\\\\n\\textcircled{1} $y=f'(x)$为偶函数; \\textcircled{2} $f(x+\\pi)=f'(x)$.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学教与学例题与习题", + "edit": [ + "20221029\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030403": { + "id": "030403", + "content": "求下列函数的导数.\\\\\n(1) $f(x)=x^2\\sin x+2\\cos x$;\\\\\n(2) $f(x)=\\dfrac{\\mathrm{e}^x+1}{\\mathrm{e}^x-1}$;\\\\\n(3) $f(x)=\\cos(3x+4)$;\\\\\n(4) $f(x)=(2x-1)^5$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学教与学例题与习题", + "edit": [ + "20221029\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030404": { + "id": "030404", + "content": "已知曲线$y=x^3+x-2$在点$P_0$处的切线$l_1$平行于直线$4x-y-1=0$, \n且点$P_0$在第三象限.\\\\\n(1) 求$P_0$的坐标;\\\\ \n(2) 若直线$l\\perp l_1$, 且$l$也过切点$P_0$, 求直线$l$的方程.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学教与学例题与习题", + "edit": [ + "20221029\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030405": { + "id": "030405", + "content": "直线$y=kx+2$交抛物线$C:x^2=4y$于$A,B$两点, 分别过点$A,B$作抛物线$C$的切线$l_1,l_2$, 若$l_1,l_2$分别交$x$轴于点$M,N$, 求四边形$ABNM$面积的最小值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学教与学例题与习题", + "edit": [ + "20221029\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030406": { + "id": "030406", + "content": "当$x\\ne 1$时, 在恒等式$1+x+x^2+\\cdots+x^n=\\dfrac{1-x^{n+1}}{1-x}$的两边对$x$求导, 得$1+2x+3x^2+\\cdots+nx^{n-1}=\\dfrac{1+(nx-n-1)x^n}{(1-x)^2}$. 利用上述方法, 试由等式$(1+x)^n=\\mathrm{C}_n^0+\\mathrm{C}_n^1x+\\cdots+\\mathrm{C}_n^nx^n$($x\\in \\mathbf{R}$, 正整数$n\\ge 2$),\\\\\n(1) 证明: $n((1+x)^{n-1}-1)=\\sum_{k=2}^{n}k\\mathrm{C}_n^kx^{k-1}$;(提示: $\\sum_{i=1}^n a_i=a_1+a_2+\\cdots+a_n$)\\\\\n(2) 求$\\mathrm{C}_{10}^1+2\\mathrm{C}_{10}^2+3\\mathrm{C}_{10}^3+\\cdots+10\\mathrm{C}_{10}^{10}$;\\\\\n(3) 求$1^2\\mathrm{C}_{100}^1+3^2\\mathrm{C}_{100}^3+5^2\\mathrm{C}_{100}^5+\\cdots+99^2\\mathrm{C}_{100}^{99}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学教与学例题与习题", + "edit": [ + "20221029\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030407": { + "id": "030407", + "content": "设$a$为实数, 函数$f(x)=x^3+ax^2+(a-3)x$, 且$y=f'(x)$是偶函数, 则$y=f(x)$得单调递增区间为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学教与学例题与习题", + "edit": [ + "20221029\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030408": { + "id": "030408", + "content": "若函数$f(x)=2x^2-\\ln x$在定义域内的一个子区间$(k-1,k+1)$上不是单调函数, 则实数$k$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学教与学例题与习题", + "edit": [ + "20221029\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030409": { + "id": "030409", + "content": "已知函数$f(x)=x^3-3ax^2+9x+5$($a<0$), 若曲线$y=f(x)$的的切线斜率最小时与直线$3x+y-2=0$平行.\\\\\n(1) 求$a$的值;\\\\\n(2) 求函数$y=f(x)$的单调区间.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学教与学例题与习题", + "edit": [ + "20221029\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030410": { + "id": "030410", + "content": "已知函数$f(x)=ax^3+bx^2+cx+d$($a+b<0$)在$\\mathbf{R}$上是增函数, 求\n$\\dfrac c{a+b}$的最大值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学教与学例题与习题", + "edit": [ + "20221029\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030411": { + "id": "030411", + "content": "已知函数$f(x)=ax^3-9x+1$, $a>0$.\\\\\n(1) 若$a=3$, 求函数$y=f(x)$的极值;\\\\\n(2) 若函数$y=f(x)$恰有三个零点, 求实数$a$的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学教与学例题与习题", + "edit": [ + "20221029\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030412": { + "id": "030412", + "content": "已知函数$f(x)=x^3+ax^2-2$在$x=2$时取得极值.\\\\\n(1) 求$y=f(x)$在$(-1,f(-1))$处的切线方程;\\\\\n(2) 求$y=f(x)$在区间$[-1,2]$上的最大值与最小值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学教与学例题与习题", + "edit": [ + "20221029\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030413": { + "id": "030413", + "content": "已知函数$f(x)=ax-\\ln x$($a$为常数).\\\\\n(1) 当$a=1$时, 求函数$y=f(x)$的最值;\\\\\n(2) 求函数$y=f(x)$在$[1,+\\infty)$上的最值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学教与学例题与习题", + "edit": [ + "20221029\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030414": { + "id": "030414", + "content": "(1) 设函数$f(x)=\\ln x-x+1$, 求$y=f(x)$的极大值;\\\\\n(2) 求证: 当$x>1$时, $x-10$且$a\\ne 1$时, $y=f(x)$存在一个极小值点$x_0$, 若$x_0>3$, 求实数$a$的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学教与学例题与习题", + "edit": [ + "20221029\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030426": { + "id": "030426", + "content": "已知函数$f(x)=\\ln x-kx+1$.\n(1) 若$f(x)\\le 0$恒成立, 求实数$k$的取值范围;\\\\\n(2) 证明: $(1+\\dfrac 13)(1+\\dfrac 1{3^2})(1+\\dfrac 1{3^3})\\cdots(1+\\dfrac 1{3^n})<\\sqrt{\\mathrm{e}}$($n\\in \\mathbf{N}$, $n>0$).", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学教与学例题与习题", + "edit": [ + "20221029\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" } } \ No newline at end of file