{"id":26410419,"url":"https://github.com/tangnpc/math","last_synced_at":"2026-01-30T00:44:39.613Z","repository":{"id":181333081,"uuid":"370316355","full_name":"TangNPC/Math","owner":"TangNPC","description":null,"archived":false,"fork":false,"pushed_at":"2021-05-24T10:45:58.000Z","size":3322,"stargazers_count":0,"open_issues_count":0,"forks_count":0,"subscribers_count":1,"default_branch":"main","last_synced_at":"2025-03-17T20:14:31.794Z","etag":null,"topics":[],"latest_commit_sha":null,"homepage":null,"language":null,"has_issues":true,"has_wiki":null,"has_pages":null,"mirror_url":null,"source_name":null,"license":null,"status":null,"scm":"git","pull_requests_enabled":true,"icon_url":"https://github.com/TangNPC.png","metadata":{"files":{"readme":"README.md","changelog":null,"contributing":null,"funding":null,"license":null,"code_of_conduct":null,"threat_model":null,"audit":null,"citation":null,"codeowners":null,"security":null,"support":null,"governance":null}},"created_at":"2021-05-24T10:40:26.000Z","updated_at":"2021-05-24T10:46:00.000Z","dependencies_parsed_at":"2023-07-15T02:36:29.477Z","dependency_job_id":null,"html_url":"https://github.com/TangNPC/Math","commit_stats":null,"previous_names":["heytang233/math","tangnpc/math"],"tags_count":0,"template":false,"template_full_name":null,"repository_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/TangNPC%2FMath","tags_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/TangNPC%2FMath/tags","releases_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/TangNPC%2FMath/releases","manifests_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/TangNPC%2FMath/manifests","owner_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/owners/TangNPC","download_url":"https://codeload.github.com/TangNPC/Math/tar.gz/refs/heads/main","host":{"name":"GitHub","url":"https://github.com","kind":"github","repositories_count":244102855,"owners_count":20398386,"icon_url":"https://github.com/github.png","version":null,"created_at":"2022-05-30T11:31:42.601Z","updated_at":"2022-07-04T15:15:14.044Z","host_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub","repositories_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories","repository_names_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repository_names","owners_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/owners"}},"keywords":[],"created_at":"2025-03-17T20:14:35.329Z","updated_at":"2026-01-30T00:44:39.586Z","avatar_url":"https://github.com/TangNPC.png","language":null,"funding_links":[],"categories":[],"sub_categories":[],"readme":"# 函数\n\n## 1. 函数的定义\n\n![函数的定义](.\\src\\img\\1.1.png)\n\n- 万有引力公式：\n\n$$\nS = \\frac{1}{2}gt^2\n$$\n\n- 可以写成\n\n$$\nS= 5t^2 \\longrightarrow y=f(x)\n$$\n\n$$\nx\\longrightarrow 自变量\\\\\ny\\longrightarrow 因变量\\\\\nf\\longrightarrow 对应法则(表达式)\n$$\n\n### 两函数相同的条件\n\n$$\nf(x)=\n\\begin{cases}\n定义域\u0026 (不化简)\\\\\n对应法则\u0026 (化简)\n\\end{cases}\n$$\n\n![Screenshot_20210522-170016_哔哩哔哩](.\\src\\img\\1.1.1.png)\n\n## 2. 函数的表达形式\n\n![函数的表现形式](.\\src\\img\\1.2.1.png)\n\n1. **显函数**\n   $$\n   y = x,y=x^2\n   $$\n\n2. **隐函数**\n   $$\n   f(x,y)=0\n   $$\n\n3. **参数函数(方程)**\n   $$\n   f(x)=\n   \\begin{cases}\n   x = \\sin t\\\\\n   y = \\cos t\n   \\end{cases}\n   $$\n\n4. **分段函数(求导)**\n\n$$\ny=|x|=\\begin{cases}\nx, \u0026x = x\u003e0\\\\\n0, \u0026x = 0\u0026分段点\\\\\n-x, \u0026x\u003c0\n\\end{cases}\\\\\nx\\in R\n$$\n\n![分段函数](.\\src\\img\\1.2.2.png)\n$$\nf(x)=\\begin{cases}\n\\sin x, \u00260\u003cx\u003c1\\\\\nx^2+1, \u0026x \\geq 1\u0026 x=1为分段点\\\\\n\\end{cases}\\\\\n考点：求分段函数在分段点的极限、连续、可导性\n$$\n\n## 3. 基本初等函数\n\n### 常函数\n\n$$\ny=f(x)=C\\\\\ny=2，定义域:x \\in R，y\\in R\n图像\n$$\n\n![save (1)](.\\src\\svg\\3.1.svg)\n\n### 幂函数\n\n$$\ny=x^{-1},\\frac{1}{x},(x\\neq 0)\n$$\n\n![1](.\\src\\svg\\3.2.1.svg)\n\n$$\nx^{\\frac{1}{2}},(x\\geq 0)\n$$\n![x^{\\frac{1}{2}}](.\\src\\svg\\3.2.2.svg)\n\n$$\ny=x\n$$\n![x](.\\src\\svg\\3.2.3.svg)\n\n$$\ny=x^2\n$$\n![x](.\\src\\svg\\3.2.4.svg)\n$$\ny=x^3\n$$\n![x^3](.\\src\\svg\\3.2.5.svg)\n\n### 指数函数\n\n$$\ny=a^x,(a\u003e0且a\\neq 1.x\\in R)\n$$\n\n![image-20210524133132961](.\\src\\img\\image-20210524133132961.png)\n\n### 指数函数的关系试\n\n1. $$\n   e^m \\cdot e^n = e^{m+e}\n   $$\n\n2. $$\n   \\frac{e^{m}}{e^{n}}=e^{m-n}\n   $$\n\n3. $$\n   (e^m) = e^{m}\\cdot n\n   $$\n\n4. $$\n   (ab)^m = a^m \\cdot a^m\n   $$\n\n5. $$\n   (\\frac{a}{b})^m = \\frac{a^m}{b^m}\n   $$\n\n### 对数函数的关系试\n\n1. $$\n   \\ln a\\cdot b = \\ln a + \\ln b\n   $$\n\n2. $$\n   \\ln \\frac{a}{b} = \\ln a - \\ln b\n   $$\n\n3. $$\n   lnx^k = k\\cdot\\ln x(只有对数函数可以做到)\n   $$\n\n4. $$\n   e^{\\ln x} =x\\\\\n   \\ln e^x = x\n   $$\n\n5. $$\n   \\ln e = 1\n   $$\n\n$$\n\\log _{a}x(a\u003e0且a\\neq1,真数x恒大于0,a是底数)\\\\\n\\log _{e}x=\\ln x\\\\\n\\begin{cases}\n\\ln 1 =0\\\\\n\\ln e =1\n\\end{cases}\n$$\n\n![desmos-graph (9)](.\\src\\svg\\desmos-graph9.svg)\n$$\n\\log _{a}x(a\u003e1)\n$$\n![desmos-graph (10)](.\\src\\svg\\desmos-graph10.svg)\n\n$$\n\\log _{e}x=\\ln x\\\\\n\\begin{cases}\n\\ln +\\infty \u0026 \\to +\\infty\\\\\n\\ln 0 \u0026 \\to -\\infty\n\\end{cases}\n$$\n![desmos-graph (11)](.\\src\\svg\\desmos-graph11.svg)\n$$\n\\lg x = \\log_{10}x\n$$\n\n### 三角函数\n\n#### sin正弦\n\n![sin](.\\src\\img\\sin.png)\n\n![save (2)](src\\svg\\sin.svg)\n\n#### cos余弦\n\n![cos](.\\src\\img\\cos.png)\n\n![cos](.\\src\\svg\\cos.svg)\n\n#### tan正切\n\n![image-20210524144310104](.\\src\\img\\tan.png)\n\n![save (4)](.\\src\\svg\\tan.svg)\n\n#### cot余切\n\n$$\ny=\\cot x = \\frac{1}{\\tan x}\\\\\n\\cot 30^{\\circ} = \\cot \\frac{\\pi}{6} = \\frac{1}{\\tan\\frac{\\pi}{6}}= \\frac{3}{\\sqrt{3}} = \\frac{3\\sqrt{3}}{3} = \\sqrt{3}\n$$\n\n#### sec正割\n\n$$\ny=\\sec x\\\\\nsecx = \\frac{1}{\\cos x}\\\\\n$$\n\n#### csc余割\n\n$$\ny=\\csc x\\\\\ncscx = \\frac{1}{\\sin x}\\\\\n$$\n\n### 倍角公式\n\n$$\n\\sin2x=2\\sin x \\cdot\\cos x\\\\\n\\cos 2x = \\cos^2x - \\sin^2x\\\\\n降幂公式\\\\\n=1-2\\sin^2x \\to \\sin^2 = \\frac{1-\\cos2x}{2}\\\\\n=2cos^2x-1 \\to cos^2x = \\frac{1+\\cos2x}{2}\n$$\n\n### 反三角函数\n\n$$\n\\left.\\begin{matrix}\ny=\\arcsin x\\\\\ny=\\arccos x\n\\end{matrix}\\right\\}\n定义域-1\\leqslant x \\leqslant 1\n$$\n\n![image-20210524153153855](.\\src\\img\\arc.png)\n$$\ny=\\arctan x\\\\\n\\left\\{\\begin{matrix}\n\\arctan+\\infty \\to \\frac{\\pi}{2}\\\\\n\\arctan-\\infty \\to -\\frac{\\pi}{2}\n\\end{matrix}\\right.\n$$\n![arctan_x](.\\src\\img\\arctan_x.svg)\n$$\n\u003cEmpty \\space Math \\space Block\u003e\n$$\n","project_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Ftangnpc%2Fmath","html_url":"https://awesome.ecosyste.ms/projects/github.com%2Ftangnpc%2Fmath","lists_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Ftangnpc%2Fmath/lists"}