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"04-applications.html#shape-practice3" ] }, { "id": "curvature", "title": "曲率：弯得有多急", "topics": [ "弧微分", "曲率", "曲率半径" ], "source": "chapters/04-applications.md", "scope": "大纲要求的教学展开", "kind": "技术讲解", "references": [], "prerequisites": [ "shape" ], "teaching": "04-applications.html#curvature", "examples": [ "04-applications.html#curvature-example1", "04-applications.html#curvature-example2" ], "exercises": [ "04-applications.html#curvature-practice1", "04-applications.html#curvature-practice2", "04-applications.html#curvature-practice3" ] }, { "id": "optimization", "title": "根、不等式与实际最优化", "topics": [ "根的个数", "不等式证明", "最优化建模" ], "source": "chapters/04-applications.md", "scope": "大纲要求的教学展开", "kind": "技术讲解", "references": [], "prerequisites": [ "curvature" ], "teaching": "04-applications.html#optimization", "examples": [ "04-applications.html#optimization-example1", "04-applications.html#optimization-example2" ], "exercises": [ "04-applications.html#optimization-practice1", "04-applications.html#optimization-practice2", "04-applications.html#optimization-practice3" ] }, { "id": "primitive", "title": "原函数与基本积分", "topics": [ "原函数", "不定积分", "基本积分公式", "线性性质" ], "source": "chapters/05-antiderivatives.md", "scope": "大纲要求的教学展开", "kind": "技术讲解", "references": [], "prerequisites": [ "derivative-rules" ], "teaching": "05-antiderivatives.html#primitive", "examples": [ "05-antiderivatives.html#primitive-example1", "05-antiderivatives.html#primitive-example2" ], "exercises": [ "05-antiderivatives.html#primitive-practice1", "05-antiderivatives.html#primitive-practice2", "05-antiderivatives.html#primitive-practice3" ] }, { "id": "substitution", "title": "换元积分：把复合结构拆开", "topics": [ "第一换元法", "第二换元法", "三角换元" ], "source": "chapters/05-antiderivatives.md", "scope": "大纲要求的教学展开", "kind": "技术讲解", "references": [], "prerequisites": [ "primitive" ], "teaching": "05-antiderivatives.html#substitution", "examples": [ "05-antiderivatives.html#substitution-example1", "05-antiderivatives.html#substitution-example2" ], "exercises": [ "05-antiderivatives.html#substitution-practice1", "05-antiderivatives.html#substitution-practice2", "05-antiderivatives.html#substitution-practice3" ] }, { "id": "parts", "title": "分部积分与递推", "topics": [ "分部积分法", "循环积分", "递推积分" ], "source": "chapters/05-antiderivatives.md", "scope": "大纲要求的教学展开", "kind": "技术讲解", "references": [], "prerequisites": [ "substitution" ], "teaching": "05-antiderivatives.html#parts", "examples": [ "05-antiderivatives.html#parts-example1", "05-antiderivatives.html#parts-example2" ], "exercises": [ "05-antiderivatives.html#parts-practice1", "05-antiderivatives.html#parts-practice2", "05-antiderivatives.html#parts-practice3" ] }, { "id": "rational-integrals", "title": "有理式、三角式与根式的处理", "topics": [ "有理函数积分", "部分分式", "三角有理式", "简单根式积分" ], "source": "chapters/05-antiderivatives.md", "scope": "大纲要求的教学展开", "kind": "技术讲解", "references": [], "prerequisites": [ "parts" ], "teaching": "05-antiderivatives.html#rational-integrals", "examples": [ "05-antiderivatives.html#rational-integrals-example1", "05-antiderivatives.html#rational-integrals-example2" ], "exercises": [ "05-antiderivatives.html#rational-integrals-practice1", "05-antiderivatives.html#rational-integrals-practice2", "05-antiderivatives.html#rational-integrals-practice3" ] }, { "id": "riemann", "title": "定积分：把很多小份加起来", "topics": [ "定积分定义", "可积性", "定积分性质", "积分中值定理" ], "source": "chapters/06-definite.md", "scope": "大纲要求的教学展开", "kind": "技术讲解", "references": [], "prerequisites": [ "rational-integrals" ], "teaching": "06-definite.html#riemann", "examples": [ "06-definite.html#riemann-example1", "06-definite.html#riemann-example2" ], "exercises": [ "06-definite.html#riemann-practice1", "06-definite.html#riemann-practice2", "06-definite.html#riemann-practice3" ] }, { "id": "fundamental", "title": "变限积分与牛顿—莱布尼茨公式", "topics": [ "变上限积分", "微积分基本定理", "定积分计算", "换元与分部" ], "source": "chapters/06-definite.md", "scope": "大纲要求的教学展开", "kind": "技术讲解", "references": [], "prerequisites": [ "riemann" ], "teaching": "06-definite.html#fundamental", "examples": [ "06-definite.html#fundamental-example1", "06-definite.html#fundamental-example2" ], "exercises": [ "06-definite.html#fundamental-practice1", "06-definite.html#fundamental-practice2", "06-definite.html#fundamental-practice3" ] }, { "id": "improper", "title": "反常积分与敛散性", "topics": [ "无穷区间积分", "无界函数积分", "比较判别", "p积分" ], "source": "chapters/06-definite.md", "scope": "大纲要求的教学展开", "kind": "技术讲解", "references": [], "prerequisites": [ "fundamental" ], "teaching": "06-definite.html#improper", "examples": [ "06-definite.html#improper-example1", "06-definite.html#improper-example2" ], "exercises": [ "06-definite.html#improper-practice1", "06-definite.html#improper-practice2", "06-definite.html#improper-practice3" ] }, { "id": "area-volume", "title": "面积与体积：先画一片", "topics": [ "平面面积", "极坐标面积", "截面体积", "旋转体体积" ], "source": "chapters/07-integral-applications.md", 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