From 14f538767706b8e0844ab0d2f72384520110061b Mon Sep 17 00:00:00 2001 From: wangweiye7840 Date: Wed, 2 Aug 2023 10:04:46 +0800 Subject: [PATCH] =?UTF-8?q?=E6=B7=BB=E5=8A=A026=E5=B1=8A=E9=AB=98=E4=B8=80?= =?UTF-8?q?=E4=B8=8A=E5=AD=A6=E6=9C=9F=E7=AC=AC=E4=B8=89=E3=80=81=E7=AC=AC?= =?UTF-8?q?=E5=9B=9B=E7=AB=A0=E5=89=A9=E4=BD=99=E7=9A=84=E7=9F=A5=E8=AF=86?= =?UTF-8?q?=E6=A2=B3=E7=90=86?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- 题库0.3/BasicKnowledge.json | 148 ++++++++++++++++++++++++++++++++++++ 1 file changed, 148 insertions(+) diff --git a/题库0.3/BasicKnowledge.json b/题库0.3/BasicKnowledge.json index ae20d1e3..0f86b43e 100644 --- a/题库0.3/BasicKnowledge.json +++ b/题库0.3/BasicKnowledge.json @@ -901,5 +901,153 @@ "K0410001X" ], "content": "计算$\\sqrt{2}$的巴比伦算法所构造的递推公式是\\blank{90}." + }, + "B00127": { + "lesson": "K0202", + "objs": [ + "K0202001B", + "K0201004B", + "K0202003B" + ], + "content": "有理数指数幂:\\\\\n(1) 指数为零: $a^0=$\\blank{50}($a \\neq 0$);\\\\\n(2) 指数为负整数: $a^{-n}=$\\blank{50}($a \\neq 0$, $n$ 是正整数);\\\\\n(3) 指数为正分数: $a^{\\frac{m}{n}}=$\\blank{50}($a \\geq 0$, $m, n$是正整数, $n \\geq 2$, $(m, n)=1$);\\\\\n(4) 指数为负分数: $a^{-\\frac{m}{n}}=$\\blank{50}($a \\neq 0$, $m, n$是正整数, $n \\geq 2$, $(m, n)=1$)." + }, + "B00128": { + "lesson": "K0203", + "objs": [ + "K0203001B", + "K0203002B" + ], + "content": "实数指数幂的性质(已知$a$\\blank{20}, $s,t\\in \\mathbf{R}$):\\\\\n(1) $a^s a^t=$\\blank{50};\\\\\n(2) $(a^s)^t=$\\blank{50};\\\\\n(3) $(ab)^t=$\\blank{50}." + }, + "B00129": { + "lesson": "K0203", + "objs": [ + "K0203003B" + ], + "content": "幂的基本不等式: 当$a>$\\blank{20}, $s>$\\blank{20}时, \\blank{20}$>$\\blank{20}." + }, + "B00130": { + "lesson": "K0204", + "objs": [ + "K0204001B", + "K0204002B" + ], + "content": "对数的定义:\\\\\n(1) 在 $a>0$, $a \\neq 1$, 且 $N>0$ 的条件下, 唯一满足\\blank{50}的数 $x$, 称为 $N$ 以 $a$ 为底的对数, 并用符号\\blank{50}表示, 而 $N$ 称为\\blank{30}.\\\\\n(2) 从定义出发可以自然地得到一些常用的对数等式($a>0$, $a\\ne 1$, $N>0$, $b\\in \\mathbf{R}$):\\\\\n\\textcircled{1} $a^{\\log _a N}=$\\blank{30}; \\textcircled{2} $\\log_a a^b=$\\blank{30}; \\textcircled{3} $\\log _a 1=$\\blank{50}; \\textcircled{4} $\\log _a a=$\\blank{50}." + }, + "B00131": { + "lesson": "K0204", + "objs": [ + "K0204003B" + ], + "content": "常用对数与自然对数:\\\\\n(1) \\blank{100}称为常用对数, 记作\\blank{50};\\\\\n(2) 常数 $\\mathrm{e}$ 是\\blank{30}数, $e \\approx$\\blank{50}, \\blank{100}称为自然对数, 记作\\blank{50}." + }, + "B00132": { + "lesson": "K0205", + "objs": [ + "K0205001B" + ], + "content": "对数运算的基本性质($a>0$, $a\\ne 1$, $M,N\\in (0,+\\infty)$, $c\\in \\mathbf{R}$):\\\\\n(1) 对数性质 1: $\\log_a(MN)=$\\blank{100};\\\\\n(2) 对数性质 2: $\\log_a\\dfrac{M}{N}=$\\blank{100};\\\\\n(3) 对数性质 3: $\\log_aN^c=$\\blank{100}. 特别地, $\\log _a \\sqrt[n]{M}=$\\blank{50}($n$ 为大于 $1$ 的正整数)." + }, + "B00133": { + "lesson": "K0206", + "objs": [ + "K0206001B", + "K0206003B" + ], + "content": "对数换底公式(已知$a>0$, $a\\ne 1$, $b>0$, $b\\ne 1$, $N>0$): $\\log_a N=\\dfrac{\\ \\blank{50}\\ }{\\ \\blank{50}\\ }$.\\\\\n推论1: $\\log _a b\\cdot$\\blank{50}$=1$;\\\\\n推论2: $\\log_{a^m}N^n=$\\blank{50}($m,n\\in \\mathbf{R}$)." + }, + "B00134": { + "lesson": "K0208", + "objs": [ + "K0208002B" + ], + "content": "当幂函数的指数为正数时, 它在 $[0,+\\infty)$上是\\blank{50}(单调性); 当幂函数的指数为负数时, 它在 ($0,+\\infty$)上是\\blank{50}.(单调性)" + }, + "B00135": { + "lesson": "K0208", + "objs": [ + "K0208003B" + ], + "content": "幂函数的图像必过定点\\blank{50}." + }, + "B00136": { + "lesson": "K0208", + "objs": [ + "K0208005B" + ], + "content": "通过图像的平移可以直观地分析与幂函数密切相关的函数的一些性质. 例如: 函数 $y=\\dfrac{2 x+7}{x+3}$ 的图像可视为函数 $y=\\dfrac{1}{x}$ 的图像按\\blank{200}平移所得, 因此它的单调减区间为\\blank{100}, 值域为\\blank{100}." + }, + "B00137": { + "lesson": "K0209", + "objs": [ + "K0209001B" + ], + "content": "当底数 $a$ 固定, 且\\blank{100}时, 等式 $y=a^x$ 确定了变量 $y$ 随变量变化的规律, 称为底为 $a$ 的指数函数." + }, + "B00138": { + "lesson": "K0209", + "objs": [ + "K0209002B" + ], + "content": "指数函数的定义域为\\blank{50}." + }, + "B00139": { + "lesson": "K0210", + "objs": [ + "K0210001B", + "K0210002B", + "K0210005B" + ], + "content": "函数的图像与性质密切相关, 对于指数函数而言:\n\\begin{center}\n\\begin{tabular}{|c|>{\\centering\\arraybackslash}p{5cm}|>{\\centering\\arraybackslash}p{5cm}|}\n\\hline\n$y=a^x$ & $a>1$ & $0=latex]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [dashed] (-2,0.5) node [above] {$y=$\\blank{10}}-- (2,0.5);\n\\end{tikzpicture} & \\begin{tikzpicture}[>=latex]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [dashed] (-2,0.5) node [above] {$y=$\\blank{10}}-- (2,0.5);\n\\end{tikzpicture} \\\\\n\\hline\n\\multirow{3}{*}{图像特征} & \\multicolumn{2}{l|}{\\textcircled{1} 图像都在$x$轴\\blank{30}, \\blank{80}于$x$轴, 但永不\\blank{30}.}\\\\ \n\\cline{2-3} & \\multicolumn{2}{l|}{\\textcircled{2} 过点\\blank{50}.} \\\\ \n\\cline{2-3} & \\multicolumn{1}{l|}{\\textcircled{3} 由左至右图像\\blank{30}.} & \\multicolumn{1}{l|}{\\textcircled{3} 由左至右图像\\blank{30}.} \\\\ \\hline\n\\multirow{3}{*}{函数性质} & \\multicolumn{2}{l|}{\\textcircled{1} 定义域为\\blank{50}, 函数值恒\\blank{30}.}\\\\ \n\\cline{2-3} & \\multicolumn{2}{l|}{\\textcircled{2} 当$x=$\\blank{30}时, $y=$\\blank{30}.} \\\\ \n\\cline{2-3} & \\multicolumn{1}{l|}{\\textcircled{3} 在$\\mathbf{R}$上是\\blank{50}函数.} & \\multicolumn{1}{l|}{\\textcircled{3} 在$\\mathbf{R}$上是\\blank{50}函数.} \\\\ \\hline\n\\end{tabular}\n\\end{center}" + }, + "B00140": { + "lesson": "K0211", + "objs": [ + "K0211002B" + ], + "content": "如图表示几个地区的某物种关于时间的图形, 找出与下面每个描述相符的图形并描述剩余图形.\\\\\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (0,0) -- (4,0) node [right] {时间(年)};\n\\draw [->] (0,0) -- (0,4) node [above] {物种数量};\n\\draw [domain = 0:{ln(4)/ln(1.08)/8}] plot (\\x,{exp(8*\\x*ln(1.08))}) node [above] {\\textcircled{1}};\n\\draw [domain = 0:{ln(4)/ln(1.05)/8}] plot (\\x,{exp(8*\\x*ln(1.05))}) node [above] {\\textcircled{2}};\n\\draw (0,1.2) -- (4,2.5) node [right] {\\textcircled{3}};\n\\draw [domain = 0:4] plot (\\x,{3.5*exp(8*\\x*ln(0.95))}) node [right] {\\textcircled{4}};\n\\draw (0,1.8) -- (4,1.8) node [right] {\\textcircled{5}};\n\\draw (0,2.3) -- (4,1.3) node [right] {\\textcircled{6}};\n\\end{tikzpicture}\n\\end{center}\n(1) 物种数量每年增长 $5 \\%$: \\blank{30};\\\\\n(2) 物种数量每年增长 $8 \\%$: \\blank{30};\\\\\n(3) 物种数量每年增加 $5000$ 人: \\blank{30};\\\\\n(4) 物种数量保持不变的: \\blank{30};\\\\\n剩余两个图形分别可以描述为:\\blank{150}, \\blank{150}." + }, + "B00141": { + "lesson": "K0212", + "objs": [ + "K0212001B" + ], + "content": "当底数 $a$ 固定, 且\\blank{80}时, $x$ 以 $a$ 为底的对数确定了变量 $y$ 随变量\\blank{30}变化的规律, 称为底为 $a$ 的对数函数." + }, + "B00142": { + "lesson": "K0212", + "objs": [ + "K0212002B" + ], + "content": "对数函数的定义域为\\blank{50}." + }, + "B00143": { + "lesson": "K0213", + "objs": [ + "K0213005B", + "K0213006B" + ], + "content": "对数函数与同底的指数函数互为\\blank{50}, 它们的图像间的关系是\\blank{100}." + }, + "B00144": { + "lesson": "K0213", + "objs": [ + "K0213007B" + ], + "content": "函数的图像与性质密切相关, 对于对数函数而言:\n\\begin{center}\n\\begin{tabular}{|c|>{\\centering\\arraybackslash}p{5cm}|>{\\centering\\arraybackslash}p{5cm}|}\n\\hline\n$y=\\log_a x$ & $a>1$ & $0=latex]\n\\draw [->] (-1,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\filldraw (0.5,0) node [below] {$1$};\n\\end{tikzpicture} & \\begin{tikzpicture}[>=latex]\n\\draw [->] (-1,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\filldraw (0.5,0) node [below] {$1$};\n\\end{tikzpicture} \\\\\n\\hline\n\\multirow{3}{*}{图像特征} & \\multicolumn{2}{l|}{\\textcircled{1} 图像都在$y$轴\\blank{30}, \\blank{80}于$y$轴, 但永不\\blank{30}.}\\\\ \n\\cline{2-3} & \\multicolumn{2}{l|}{\\textcircled{2} 过点\\blank{50}.} \\\\ \n\\cline{2-3} & \\multicolumn{1}{l|}{\\textcircled{3} 由左至右图像\\blank{30}.} & \\multicolumn{1}{l|}{\\textcircled{3} 由左至右图像\\blank{30}.} \\\\ \\hline\n\\multirow{3}{*}{函数性质} & \\multicolumn{2}{l|}{\\textcircled{1} 定义域为\\blank{50}.}\\\\ \n\\cline{2-3} & \\multicolumn{2}{l|}{\\textcircled{2} 当$x=$\\blank{30}时, $y=$\\blank{30}.} \\\\ \n\\cline{2-3} & \\multicolumn{1}{l|}{\\textcircled{3} 在区间\\blank{30}上是\\blank{30}函数.} & \\multicolumn{1}{l|}{\\textcircled{3} 在区间\\blank{30}上是\\blank{30}函数.} \\\\ \\hline\n\\end{tabular}\n\\end{center}" + }, + "B00145": { + "lesson": "K0214", + "objs": [ + "K0214002B" + ], + "content": "当$a>1$, $b\\in \\mathbf{R}$时, 关于$x$的不等式$\\log_a x>b$的解集为\\blank{50}.\\\\\n这是因为当$\\log_a$\\blank{50}$=b$. 当$x\\in $\\blank{50}时, 因为函数$y=\\log_a x$是\\blank{50}函数, 所以$\\log_a x>b$, 这表明区间\\blank{50}中的实数$x$都是解; 而当$x\\in $\\blank{50}时, 同样因为函数$y=\\log_a x$是\\blank{50}函数, 所以$\\log_a x\\le b$, 这表明区间\\blank{50}中的实数$x$都不是解." + }, + "B00146": { + "lesson": "K0214", + "objs": [ + "K0214002B" + ], + "content": "当$0b$的解集为\\blank{50}.\\\\\n这是因为当$\\log_a$\\blank{50}$=b$. 当$x\\in $\\blank{50}时, 因为函数$y=\\log_a x$是\\blank{50}函数, 所以$\\log_a x>b$, 这表明区间\\blank{50}中的实数$x$都是解; 而当$x\\in $\\blank{50}时, 同样因为函数$y=\\log_a x$是\\blank{50}函数, 所以$\\log_a x\\le b$, 这表明区间\\blank{50}中的实数$x$都不是解." } } \ No newline at end of file