152 lines
6.7 KiB
Python
152 lines
6.7 KiB
Python
"""从 math_learning_dashboard_v3.html 抽取的初始内容。"""
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KNOWLEDGE = [
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{"name": "函数概念", "mastery": 80, "position": 1},
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{"name": "函数值", "mastery": 85, "position": 2},
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{"name": "定义域", "mastery": 90, "position": 3},
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{"name": "零点", "mastery": 80, "position": 4},
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{"name": "正负性", "mastery": 70, "position": 5},
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{"name": "偶函数", "mastery": 40, "position": 6},
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{"name": "奇函数", "mastery": 40, "position": 7},
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{"name": "图像对称性", "mastery": 50, "position": 8},
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]
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QUESTIONS = [
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{
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"stem": "已知函数 f(x) 满足 f(-x)=f(x)，且 f(3)=5。下列哪个点一定在图像上？",
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"options": ["A. (-3,5)", "B. (-3,-5)", "C. (3,-5)", "D. (5,-3)"],
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"correct_index": 0,
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"knowledge_name": "偶函数",
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"explanation": "由偶函数定义 f(-x)=f(x) 可知，点 (3,5) 关于 y 轴的镜像点 (-3,5) 一定在图像上。",
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},
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{
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"stem": "已知函数 f(x) 满足 f(-x)=-f(x)，且 f(2)=4。下列哪个点一定在图像上？",
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"options": ["A. (-2,4)", "B. (-2,-4)", "C. (2,-4)", "D. (4,-2)"],
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"correct_index": 1,
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"knowledge_name": "奇函数",
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"explanation": "由奇函数定义 f(-x)=-f(x) 可知，(2,4) 关于原点的对称点 (-2,-4) 一定在图像上。",
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},
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{
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"stem": "下列函数的图像关于 $y$ 轴对称的是？",
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"options": ["A. $y=x+1$", "B. $y=x^3$", "C. $y=x^2+2$", "D. $y=x^2+x$"],
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"correct_index": 2,
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"knowledge_name": "偶函数",
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"explanation": "$y=x^2+2$ 满足 $f(-x)=f(x)$，因此图像关于 $y$ 轴对称，属于偶函数。",
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},
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{
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"stem": "函数 y=1/(x²-9) 的图像不可能经过哪个点？",
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"options": ["A. (0,-1/9)", "B. (4,1/7)", "C. (-4,1/7)", "D. (3,1)"],
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"correct_index": 3,
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"knowledge_name": "定义域",
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"explanation": "分母 x²-9 不能为 0，x=±3 不在定义域内，所以图像不可能经过 (3,1)。",
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},
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{
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"stem": "若函数图像关于原点对称，则下列关系一定成立的是？",
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"options": ["A. f(-x)=f(x)", "B. f(-x)=-f(x)", "C. f(x)=0", "D. f(x)>0"],
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"correct_index": 1,
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"knowledge_name": "奇函数",
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"explanation": "图像关于原点对称对应奇函数，即对任意定义域内的 x 有 f(-x)=-f(x)。",
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},
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{
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"stem": "已知 $f(-x)=f(x)$，则函数图像的几何性质是？",
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"options": ["A. 关于 x 轴对称", "B. 关于 y 轴对称", "C. 关于原点对称", "D. 无法判断"],
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"correct_index": 1,
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"knowledge_name": "偶函数",
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"explanation": "f(-x)=f(x) 是偶函数的定义式，几何上等价于图像关于 y 轴对称。",
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},
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{
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"stem": "函数 y=2/(x²-1) 的图像不可能经过哪个点？",
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"options": ["A. (0,-2)", "B. (2,2/3)", "C. (-2,2/3)", "D. (1,2)"],
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"correct_index": 3,
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"knowledge_name": "定义域",
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"explanation": "分母 x²-1 不能为 0，x=±1 不在定义域内，所以图像不可能经过 (1,2)。",
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},
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{
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"stem": "若 f(-x)=-f(x)，且图像经过 (5,-2)，则一定经过？",
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"options": ["A. (-5,-2)", "B. (-5,2)", "C. (5,2)", "D. (2,5)"],
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"correct_index": 1,
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"knowledge_name": "奇函数",
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"explanation": "奇函数图像关于原点对称，(5,-2) 关于原点的对称点是 (-5,2)。",
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},
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{
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"stem": "若 f(-x)=f(x)，且图像经过 (4,7)，则下列点一定在图像上的是？",
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"options": ["(-4,7)", "(-4,-7)", "(4,-7)", "(7,4)"],
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"correct_index": 0,
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"knowledge_name": "偶函数",
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"explanation": "由偶函数定义得到关于 y 轴的镜像点。",
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},
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{
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"stem": "若 f(-x)=-f(x)，且图像经过 (3,5)，则下列点一定在图像上的是？",
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"options": ["(-3,5)", "(-3,-5)", "(3,-5)", "(5,3)"],
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"correct_index": 1,
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"knowledge_name": "奇函数",
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"explanation": "由奇函数定义得到关于原点的对称点。",
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},
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{
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"stem": "下列函数中属于偶函数的是？",
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"options": ["$y=x^3$", "$y=x^2+1$", "$y=x+1$", "$y=x^3+x$"],
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"correct_index": 1,
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"knowledge_name": "偶函数",
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"explanation": "代入 $-x$ 检验 $f(-x)=f(x)$。",
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},
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{
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"stem": "下列函数中属于奇函数的是？",
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"options": ["y=x²", "y=x^3", "y=x²+1", "y=x^3+1"],
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"correct_index": 1,
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"knowledge_name": "奇函数",
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"explanation": "代入 -x 检验 f(-x)=-f(x)。",
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},
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{
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"stem": "函数 y=1/(x-2) 的图像不可能经过哪个点？",
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"options": ["(0,-1/2)", "(3,1)", "(2,0)", "(4,1/2)"],
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"correct_index": 2,
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"knowledge_name": "定义域",
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"explanation": "分母不能为 0，因此 x=2 不属于定义域。",
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},
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{
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"stem": "若一个函数图像关于 y 轴对称，则该函数通常满足？",
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"options": ["f(-x)=f(x)", "f(-x)=-f(x)", "f(x)=0", "f(x)>0"],
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"correct_index": 0,
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"knowledge_name": "图像对称性",
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"explanation": "关于 y 轴对称对应偶函数。",
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},
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{
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"stem": "若一个函数图像关于原点对称，则该函数通常满足？",
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"options": ["f(-x)=f(x)", "f(-x)=-f(x)", "f(x)=1", "f(x)>0"],
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"correct_index": 1,
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"knowledge_name": "图像对称性",
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"explanation": "关于原点对称对应奇函数。",
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},
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{
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"stem": "若 f(x) 是偶函数，且 f(-5)=9，则 f(5)=？",
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"options": ["-9", "0", "9", "无法确定"],
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"correct_index": 2,
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"knowledge_name": "偶函数",
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"explanation": "偶函数的两个对称自变量对应相同函数值。",
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},
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{
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"stem": "若 f(x) 是奇函数，且 f(-6)=4，则 f(6)=？",
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"options": ["4", "-4", "0", "无法确定"],
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"correct_index": 1,
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"knowledge_name": "奇函数",
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"explanation": "奇函数的两个相反自变量对应相反函数值。",
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},
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{
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"stem": "判断函数 y=x⁴-x² 的奇偶性。",
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"options": ["偶函数", "奇函数", "既奇又偶", "非奇非偶"],
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"correct_index": 0,
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"knowledge_name": "偶函数",
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"explanation": "偶次幂组成，代入 -x 后表达式不变。",
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},
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]
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SAMPLE_ERRORS = [
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("第1轮 · 1", "函数图像", "基础判断不稳定"),
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("第1轮 · 8", "定义域", "无定义点识别"),
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("第1轮 · 9", "偶函数", "核心概念混淆"),
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("第2轮 · 5", "偶函数", "点的对应关系错误"),
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("第2轮 · 7", "偶函数", "对称性判断错误"),
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("第2轮 · 8", "奇函数", "与偶函数混淆"),
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]
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SAMPLE_RECORDS = [
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("第一轮基础题", 10, 7),
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("第二轮专项题", 8, 5),
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]
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