{
  "section_id": "x2-ch5-numbering",
  "status": "reviewed-with-blocking-source-conflicts",
  "patch_file": "batch/patches/x2-ch5-numbering-proposed.json",
  "source": "sources/textbooks/教参与课标/沪教版教参选择性必修第二册/沪教版教参选择性必修第二册分章/01 选必第二册 第5章 导数及其应用.md",
  "source_sha256": "aa7eb6b92f5613a1edd135f4724e9d5923a85f9cc476df8a6e319460e37679bc",
  "coverage": {
    "renumber_proposals": 27,
    "anchors_exactly_match_source": 27,
    "numbering_review": "27/27 mappings approved. Each exact source anchor was checked; proposed answer index is the next number after the OCR-restarted prior answer. Mathematical rationale was recomputed across groups; numbering alone is not used to resolve the disputed source answers.",
    "focus_math_checks": [
      "5.1A2-A6, B3-B4: average rates, derivatives and graphic secant slope checked; A5 visual P(1,1), Q(5,4) gives 3/4, and secant steepens as Q approaches P.",
      "5.2A7-A12, B3-B5,B7: product/quotient/chain rules, rates, sphere radius derivative, tangent slope and error-detection results checked.",
      "5.3A5-A8,B2-B4: derivative signs, intervals, closed-interval extrema, dimensions and profit model checked.",
      "Chapter-end A4(3)(4), A7(1)-(3): original teacher answers compared against direct substitution/differentiation; blocking findings below."
    ],
    "mapping_detail": [
      {
        "line": 335,
        "number": 2,
        "group": "习题5.1A",
        "anchor": "1. (1) ${90}\\mathrm{\\;m}/\\mathrm{s}$ , ${70}\\mathrm{\\;m}/\\mathrm{s}$ . (2) ${80}\\mathrm{\\;m}/\\mathrm{s}$ . (3) ${10}\\mathrm{\\;s}$ .",
        "evidence": "H′=100-10t；区间均速90/70、t=2速度80、t=10停升",
        "anchor_matches_source": true,
        "independent_review": "approved"
      },
      {
        "line": 336,
        "number": 3,
        "group": "习题5.1A",
        "anchor": "2. (1) $3{\\mathrm{\\;m}}^{3}/\\mathrm{s}$ . (2)在 $t = a\\left( \\mathrm{\\;s}\\right)$ 附近,流水量大约以 $3{\\mathrm{\\;m}}^{3}/\\mathrm{s}$ 的速度增加.",
        "evidence": "y=3t，导数3且与a无关",
        "anchor_matches_source": true,
        "independent_review": "approved"
      },
      {
        "line": 340,
        "number": 4,
        "group": "习题5.1A",
        "anchor": "1. (1) ${2\\pi a} + {\\pi h}$ . (2) ${2\\pi a}$ .",
        "evidence": "π((a+h)²-a²)/h=2πa+πh，瞬时为2πa",
        "anchor_matches_source": true,
        "independent_review": "approved"
      },
      {
        "line": 341,
        "number": 5,
        "group": "习题5.1A",
        "anchor": "2. (1) $\\frac{3}{4}$ . (2)变大.",
        "evidence": "原图P(1,1),Q(5,4)，斜率3/4；Q趋P割线变陡",
        "anchor_matches_source": true,
        "independent_review": "approved"
      },
      {
        "line": 342,
        "number": 6,
        "group": "习题5.1A",
        "anchor": "3. (1) 4. (2) 2. (3) 0 . (4) -2. (5) -4.",
        "evidence": "(-x²)′=-2x，依次4,2,0,-2,-4",
        "anchor_matches_source": true,
        "independent_review": "approved"
      },
      {
        "line": 356,
        "number": 3,
        "group": "习题5.1B",
        "anchor": "1. $f\\left( 1\\right)  = 1,{f}^{\\prime }\\left( 1\\right)  = 4$ .",
        "evidence": "切线y=4x-3在x=1给f(1)=1及f′(1)=4",
        "anchor_matches_source": true,
        "independent_review": "approved"
      },
      {
        "line": 357,
        "number": 4,
        "group": "习题5.1B",
        "anchor": "2. $- \\frac{1}{3}$ .",
        "evidence": "原图切线通过(0,-1),(3,-2)，斜率-1/3",
        "anchor_matches_source": true,
        "independent_review": "approved"
      },
      {
        "line": 396,
        "number": 7,
        "group": "习题5.2A",
        "anchor": "1. ${y}^{\\prime } =  - {8x} + {11}$ . 提示:可以先用积的求导公式,也可以先把函数表达式展开.",
        "evidence": "(x-2)(3-4x)=-4x²+11x-6，导数-8x+11",
        "anchor_matches_source": true,
        "independent_review": "approved"
      },
      {
        "line": 397,
        "number": 8,
        "group": "习题5.2A",
        "anchor": "2. (1) $h\\left( 1\\right)  = \\frac{22}{3},{h}^{\\prime }\\left( 1\\right)  = 9$ . (2) $h\\left( 1\\right)  = \\frac{47}{3},{h}^{\\prime }\\left( 1\\right)  = {44}.$ (3) $h\\left( 1\\right)  = \\frac{7}{6}$ , ${h}^{\\prime }\\left( 1\\right)  =  - \\frac{1}{12}$",
        "evidence": "代入f=2,f′=3,g=4,g′=5，和积商法则分别得22/3,9；47/3,44；7/6,-1/12",
        "anchor_matches_source": true,
        "independent_review": "approved"
      },
      {
        "line": 398,
        "number": 9,
        "group": "习题5.2A",
        "anchor": "3. (1) ${y}^{\\prime } = \\frac{1}{\\sqrt{{2x} - 5}}$ . (2) ${y}^{\\prime } =  - \\frac{1}{2}\\sin \\frac{x}{2}.$ (3) ${y}^{\\prime } =  - \\frac{1}{{\\mathrm{e}}^{x + 1}}$ .",
        "evidence": "复合函数导数依次1/√(2x-5)、-sin(x/2)/2、-e^(-x-1)",
        "anchor_matches_source": true,
        "independent_review": "approved"
      },
      {
        "line": 399,
        "number": 10,
        "group": "习题5.2A",
        "anchor": "4. ${y}^{\\prime } =  - \\frac{2}{{\\left( x - 1\\right) }^{2}}$ . 提示: 既可以用商的求导公式,也可以看作复合函数求导.",
        "evidence": "(2/(x-1))′=-2/(x-1)²",
        "anchor_matches_source": true,
        "independent_review": "approved"
      },
      {
        "line": 400,
        "number": 11,
        "group": "习题5.2A",
        "anchor": "5. (1) $- {1.2}^{ \\circ  }\\mathrm{C}/\\mathrm{{min}}$ . (2) ${2.7}\\mathrm{\\;{min}}$ .",
        "evidence": "T′=-120/(t+5)²，T′(5)=-1.2；T′=-2得√60-5≈2.7",
        "anchor_matches_source": true,
        "independent_review": "approved"
      },
      {
        "line": 401,
        "number": 12,
        "group": "习题5.2A",
        "anchor": "6. $- \\frac{\\pi }{8}\\mathrm{\\;m}/\\mathrm{h},\\frac{\\pi }{8}\\mathrm{\\;m}/\\mathrm{h}$ .",
        "evidence": "y′=π/4*cos(πt/12+5π/6)，t=6/18分别-π/8,+π/8",
        "anchor_matches_source": true,
        "independent_review": "approved"
      },
      {
        "line": 409,
        "number": 3,
        "group": "习题5.2B",
        "anchor": "1. (1) $r = \\sqrt[3]{\\frac{3V}{4\\pi }}$ . (2) $\\frac{1}{3}\\sqrt[3]{\\frac{3}{4\\pi }}$ .",
        "evidence": "r=(3V/(4π))^(1/3)，r′(1)=(3/(4π))^(1/3)/3",
        "anchor_matches_source": true,
        "independent_review": "approved"
      },
      {
        "line": 410,
        "number": 4,
        "group": "习题5.2B",
        "anchor": "2. (1)不正确,第二个减号应改为加号. (2)不正确,应在结果前加负号.",
        "evidence": "商法则sin(x)/x第二项为+cos(x)/x；ln(2-x)导数为-1/(2-x)",
        "anchor_matches_source": true,
        "independent_review": "approved"
      },
      {
        "line": 411,
        "number": 5,
        "group": "习题5.2B",
        "anchor": "3. $y =  \\pm  2\\sqrt{2}x - 1$ . 提示: 可先设切点为 $\\left( {a,2{a}^{2}}\\right)$ .",
        "evidence": "切点(a,2a²)，截距-2a²=-1得a=±1/√2，斜率±2√2",
        "anchor_matches_source": true,
        "independent_review": "approved"
      },
      {
        "line": 415,
        "number": 7,
        "group": "习题5.2B",
        "anchor": "1. (1) ${y}^{\\prime } = {2x}\\sin {3x} + 3{x}^{2}\\cos {3x} + {x}^{-\\frac{3}{2}}$ . (2) ${y}^{\\prime } = \\frac{4}{{\\left( {\\mathrm{e}}^{x} + {\\mathrm{e}}^{-x}\\right) }^{2}}$ .",
        "evidence": "积与复合求导得2x sin3x+3x² cos3x+x^(-3/2)；商求导得4/(e^x+e^-x)²",
        "anchor_matches_source": true,
        "independent_review": "approved"
      },
      {
        "line": 457,
        "number": 5,
        "group": "习题5.3A",
        "anchor": "1. 提示: 在 $\\left( {-\\infty , + \\infty }\\right)$ 上导数恒大于零.",
        "evidence": "(x³+4x)′=3x²+4>0",
        "anchor_matches_source": true,
        "independent_review": "approved"
      },
      {
        "line": 458,
        "number": 6,
        "group": "习题5.3A",
        "anchor": "2. 单调减区间为 $\\left( {-\\infty , - 2}\\right)$ 及 $\\left( {2, + \\infty }\\right)$ .",
        "evidence": "(-x³+12x-1)′=12-3x²，负值区间为x<-2或x>2",
        "anchor_matches_source": true,
        "independent_review": "approved"
      },
      {
        "line": 459,
        "number": 7,
        "group": "习题5.3A",
        "anchor": "3. 提示: 函数在其定义域内导数恒大于零.",
        "evidence": "(x-1/x)′=1+1/x²>0，定义域内没有驻点也无极值",
        "anchor_matches_source": true,
        "independent_review": "approved"
      },
      {
        "line": 460,
        "number": 8,
        "group": "习题5.3A",
        "anchor": "4. $\\left\\lbrack  {-1,{15}}\\right\\rbrack$ .",
        "evidence": "区间[0,3]候选x=0,2,3给-1,15,8，值域[-1,15]",
        "anchor_matches_source": true,
        "independent_review": "approved"
      },
      {
        "line": 468,
        "number": 2,
        "group": "习题5.3B",
        "anchor": "1. -7 .",
        "evidence": "驻点条件3+4m-n=0及函数值1+3m-n=0给m=-2,n=-5，和-7",
        "anchor_matches_source": true,
        "independent_review": "approved"
      },
      {
        "line": 469,
        "number": 3,
        "group": "习题5.3B",
        "anchor": "2. 当长为 $2\\mathrm{\\;m}$ 、宽为 $1\\mathrm{\\;m}$ 、高为 $\\frac{3}{2}\\mathrm{\\;m}$ 时,体积最大,最大体积是 $3{\\mathrm{\\;m}}^{3}$ .",
        "evidence": "a=2b,a+b+c=4.5，V=2b²(4.5-3b)，b=1最大，尺寸2,1,1.5体积3",
        "anchor_matches_source": true,
        "independent_review": "approved"
      },
      {
        "line": 470,
        "number": 4,
        "group": "习题5.3B",
        "anchor": "3. 9 元.",
        "evidence": "L=(x-7)(12-x)²，L′=(12-x)(26-3x)，最优26/3≈9元",
        "anchor_matches_source": true,
        "independent_review": "approved"
      },
      {
        "line": 483,
        "number": 5,
        "group": "复习题A",
        "anchor": "1. (1) ${y}^{\\prime } =  - \\cos x$ . (2) ${y}^{\\prime } = \\frac{1}{3}{x}^{-\\frac{2}{3}} + 3{x}^{-4}$ . (3) ${y}^{\\prime } =  - {4x} + \\frac{43}{2}$ . (4) ${y}^{\\prime } =  - \\frac{\\sin x}{{x}^{2}} - \\frac{2\\cos x}{{x}^{3}}$ .",
        "evidence": "四个导数依次-cosx,(1/3)x^-2/3+3x^-4,-4x+43/2,-sinx/x²-2cosx/x³",
        "anchor_matches_source": true,
        "independent_review": "approved"
      },
      {
        "line": 484,
        "number": 6,
        "group": "复习题A",
        "anchor": "2. ( 1 )单调增区间为 $\\left( {-\\infty , + \\infty }\\right)$ ,无极值点.",
        "evidence": "线性增；二次驻点1/2极大；三次导数根-2与4/3且两外增中减",
        "anchor_matches_source": true,
        "independent_review": "approved"
      },
      {
        "line": 490,
        "number": 7,
        "group": "复习题A",
        "anchor": "1. (1)最大值是 1 ,最小值是 0 .",
        "evidence": "仅确认题型/位置，内容有数学冲突，必须另标blocking，不能称数学通过",
        "anchor_matches_source": true,
        "independent_review": "approved"
      }
    ]
  },
  "source_conflicts": [
    {
      "section": "x2-chapter-end-5",
      "source": "teacher",
      "line": 481,
      "question": "复习题A4(3)",
      "severity": "blocking",
      "finding": "h=e^x/ln x在x=1无定义，不能给出有限导数值1/e。",
      "disposition": "保留原文，显式校注，不伪造答案或静默改题。"
    },
    {
      "section": "x2-chapter-end-5",
      "source": "teacher",
      "line": 481,
      "question": "复习题A4(4)",
      "severity": "blocking",
      "finding": "h=e^(2x+1)+ln(3x-1)，h'(1)=2e^3+3/2；原列2/3+3e^2错误。",
      "disposition": "保留原文，显式校注。"
    },
    {
      "section": "x2-chapter-end-5",
      "source": "teacher",
      "line": 490,
      "question": "复习题A7(1)",
      "severity": "blocking",
      "finding": "[0,3]上线性函数2x/3-1最小值=-1、最大值=1；原答案只列最大1与最小0，其中0错误。",
      "disposition": "保留原文，显式校注。"
    },
    {
      "section": "x2-chapter-end-5",
      "source": "teacher",
      "line": 492,
      "question": "复习题A7(2)",
      "severity": "blocking",
      "finding": "f=2+x-x^2在[-1,1]顶点x=1/2取最大9/4；两端值0、2，最小值0。原文最大19错误，最小0正确。",
      "disposition": "保留原文，显式校注。"
    },
    {
      "section": "x2-5-3",
      "source": "teacher",
      "line": 437,
      "question": "练习5.3(4)第1题",
      "severity": "blocking",
      "finding": "教参原文仅有题号“1.”，答案内容缺失；不得用独立计算值冒充正式答案。",
      "disposition": "保留答案缺口并提示原源缺项。"
    }
  ],
  "nonissues_checked": [
    {
      "question": "复习题A4(1)",
      "finding": "3e-5正确。"
    },
    {
      "question": "复习题A4(2)",
      "finding": "e正确。"
    },
    {
      "question": "复习题A7(3)",
      "finding": "最大19、最小13/27均正确。"
    }
  ],
  "patches": [],
  "limits": [
    "题答编号映射通过不等于整册数学全通过。",
    "本轮仅审查5章编号补丁、所列重点数学和指定冲突；未全册逐题重算，未核最终网页视觉。"
  ]
}