• 核心认知：这门课其实是两门几乎不相干的课共用一个分数：
  • 微积分（Calculus，周一主线）
  • 线性代数（Linear Algebra，周三主线）
    它们并行推进、互相几乎不说话，所以不能只押一条线；Quiz A 和 Final 都会同时抽样两条线。[1]Source: asksia-bible-math1061-bilingual.pdf1 ·学 You haven't seen the lecture yet. Read a chapter top to bottom. Every topic is an AHA-unit - a formula or picture - plain-English idea - the method in steps - a worked example with real numbers - the common trap. The diagrams are original drawings of the standard maths - learn the idea cold. 你还没上过这节课。把一章从头读到 尾。每个主题都是一个 AHA 单元 -- 公式或图→通俗白话的概念 → 分步方法→ 用真实数字的例题 → 常见陷阱。图都是标准数学的原 创手绘 -- 把概念彻底学会。 B 2 . DRILL 2 · 练 You've done the lecture and tutorial. Cover the worked steps and re-derive each answer by hand. Quiz A and most of the final are no- calculator, so exact values - fractions in lowest terms, surds left as surds - on fresh numbers is the whole game. 你已上完课和习题课。遮住解题步 骤,亲手重新推出每个答案。Quiz A 和大部分期末都不许用计算器,所 以在全新数字上给出精确值 -- 化 到最简的分数、保留为根号的根式 -- 才是全部要义。 C 3 . EXAM 3 ·考 It's STUVAC. The AHA-units, trap boxes and the blueprint overleaf are your map. Quiz A (Week 8, 15%) samples the first half of both streams; the final (60%) samples the lot. Walk in knowing the recurring chains. 到了 STUVAC 复习周。AHA 单 元、陷阱框和下一页的蓝图就是你的 地图。Quiz A(第8周,15%)抽 样两条主线的前半;期末(60%) 抽样全部。带着对反复出现题链的熟 悉走进考场。 ! The single most important thing to understand about MATH1061 关于 MATH1061 最需要理解的一件事 This is two disjoint courses sharing one mark - Calculus (Monday lectures) and Linear Algebra (Wednesday lectures) run in parallel from Week 1 with their own lecturers, tutorials and notes. They barely talk to each other, so you cannot coast on one: the Week-8 in-person Quiz A samples BOTH streams (sets, functions, limits, differentiation, complex numbers, vectors, planes) and the 60% final samples the whole year. Budget your revision across both halves - do not leave a stream blank. 这是共享一个成绩的两门互不相干的课 -- 微积分(周一讲课)与线性代数(周三讲课)从第1周起并行推进,各有自己的讲 师、习题课和讲义。它们几乎不交流,所以你不能靠一门混过去:第 8周的线下 Quiz A 对两条主线都抽样(集合、函数、极限、 求导、复数、向量、平面),而 60% 的期末抽样全年。把复习时间分配到两半 -- 不要让任何一条主线留空。 MATH1061 . Mathematics 1A i How this book was built - and the two-layer rule 本书是如何搭建的 -- 以及两层规则 Standard mathematical definitions, theorems and formulas are stated plainly (they are universal - a limit law is a fact). The unit's own framing and any lecturer example numbers are paraphrased and re-numbered; every worked example here uses our own fresh numbers, never copied from slides or past papers. The Calculus stream follows the in-house Calculus of One Variable notes; Linear Algebra follows the MATH1061 Linear Algebra notes (Poole, Linear Algebra: A Modern Introduction, 4th ed. , as reference). Verify dates and weights against your own Canvas - details shift between cohorts. 标准的数学定义、定理和公式都如实陈述(它们是普适的 -- 极限定律是事实)。本单元自身的表述框架以及任何讲师的例题数字 都被改述并重新编号;这里每道例题都使用我们自己的全新数字,绝不照抄幻灯片或往年试卷。微积分主线遵循校内的 Calculus of One Variable 讲义;线性代数遵循 MATH1061 线性代数讲义(以 Poole, Linear Algebra: A Modern Introduction, 第4版 作为参考)。请对照你自己的 Canvas 核实日期与权重 -- 细节在不同届之间会变动。 MATH1061 . Mathematics 1A THE BLUEPRINT FINAL 60% - THE EXAM BLUEPRINT The final is 60% - everything points at it 期末占 60% ––一切都指向它 Quizzes 8% . A1 5% . AZ 10% · Quiz A 15% · tutorials 2% · final exam 60% 小测 8% · A1 5% · A2 10% · Quiz A 15% · 习题课 2% · 期末考 60% Your mark is six pieces, but one dominates. The final exam is 60% - more than the other five combined - and it samples both streams across the whole year. The in-person Quiz A (15%, Week 8) is the next biggest single hit and the one with the strictest conditions.
• 评分结构（材料里明确给的）：期末 Final 60%，影响最大；Quiz A 15%（第8周线下MCQ、闭卷、无计算器）是第二大项。其余还有 weekly quizzes、A1、A2、tutorial 等。[2]Source: asksia-bible-math1061-bilingual.pdf你的成绩由六个部分构成,但有一项占据主导。期末考占 60% -- 超过其余五项之和 -- 它在全年范围内对两条主线都抽样考查。 线下的 Quiz A (15%,第8周)是下一个最大的单项,也是条件最严格的一项。 60% FINAL EXAM 期末考试 15% QUIZ A (WK 8, MCQ) Quiz A (第8周,选择题) 89 WEEKLY QUIZZES 每周小测 100% WEIGHTS CHECK / 权重核对 √ The six assessment pieces 六项评估构成 Component Weight When / detail Final examination - both streams, full year 60% Exam period In-person Quiz A - 12 MCQ, 40 15% Week 8 (LA tutorial) min, no calculators Weekly online quizzes (best 8 of 10) 8% Weeks 2-12, Sun 11:59 pm Assignment 2 (written working) 10% later semester Assignment 1 (written working) 5% ~ Week 4 Tutorial contribution (8 + 8 tutes) 2% weekly, from Wk 2 The final exam - what we do and don't know 期末考 -- 我们知道与不知道的 i Book status: not officially stated 本书状态:官方未明确说明 The unit materials we have document the weight (60%) but not the length or open/closed-book status of the final. Treat it as closed-book and revise accordingly - that is the safe default and matches Quiz A's conditions - but check the unit outline / Canvas "Information on Final Exam" page for the official format before exam period. 我们手头的单元材料记录了权重(60%),但没有说明期末 的时长或开/闭卷状态。当作闭卷来相应复习 -- 这是安全 的默认值,也与 Quiz A 的条件一致 -- 但在考试期之 前,请查阅单元大纲/Canvas 的‘Information on Final Exam’页面了解官方形式。

0）你最该先搞清的“考试游戏规则”（不然复习会白忙）

0.1 考试风格一句话总结

• 高度“程序化 / 模板化”：拿到一个函数/极限/矩阵/复数 → 选对方法 → 按步骤做 → 给出精确值。[3]Source: asksia-bible-math1061-bilingual.pdfThe strategy this dictates 由此决定的策略 MATH1061 . Mathematics 1A ★ Quiz A - the closed-book MCQ in Week 8 Quiz A ––第8周的闭卷选择题 Held in the Week-8 Linear-Algebra tutorial: 40 minutes, 12 MCQ, 1 mark each, NO calculators, no notes, no extra paper, one correct answer per question. It samples the first half of BOTH streams - sets & functions, limits, differentiation, complex numbers, vectors, lines & planes, cross product. Exact-value arithmetic by hand under time is the whole test. There is no Special Consideration for the online quizzes (a better-mark rule covers them) - but Quiz A and the final are the marks you sit once. 在第 8周的线性代数习题课进行:40 分钟,12 道选择题, 每题1分,不许用计算器、不许带笔记、不许用额外草稿 纸,每题只有一个正确答案。它对两条主线的前半抽样 -- 集合与函数、极限、求导、复数、向量、直线与平 面、叉积。限时下手算精确值是全部考点。在线小测没有特 殊考量(由取较好成绩的规则覆盖) -- 但 Quiz A 和期末 是你只考一次的分数。 ✓ Drill the recurring chains, both streams 通刷反复出现的题链,两条主线 Every question is procedural: take a function, limit, matrix or complex number, apply the right technique, give the exact value. The chains that recur - Calculus: factor/cancel - limit; squeeze-bound - limit; differentiate + f'=0 - classify; Linear Algebra: realise denominator - complex division; dot product - angle/orthogonality; row- reduce - back-substitute; det(A-MI)=O - eigen. Show every line - method marks are real. Drill the chains and fresh numbers can't surprise you. 每道题都是程序化的:拿一个函数、极限、矩阵或复数,运 用正确的技巧,给出精确值。反复出现的题链 -- 微积 分:因式分解/约分→极限;夹逼定界→极限;求导→ f'=0→分类;线性代数:分母实数化→复数除法;点积 →夹角/正交;行化简→回代;det(A-入)=0→ 特征。写 出每一行 -- 步骤分是实打实的。把题链练熟,全新数字 便无法让你措手不及。 MATH1061 . Mathematics 1A CONTENTS - CONTENTS Both streams, one exam-ready book 两条主线,一本应考之书 Calculus first, then Linear Algebra - the order you actually meet them 先微积分,再线性代数 -- 按你实际遇到它们的顺序 Ch Topic Core methods Stream 1 . Calculus (Monday lectures) 1 Functions & limits function families . limit laws . squeeze . continuity . IVT → 2 Differentiation difference quotient . rules . chain . implicit . L'Hôpital → 3 Applications & curve sketching extrema . concavity · optimisation . MVT → 4 Taylor polynomials T_n about a . remainder . standard series → 5 Integration Riemann sum . FTC . substitution . parts . applications
• 方法分是真实存在的：一定写出每一行步骤；算术小错还有机会保住分。[3]Source: asksia-bible-math1061-bilingual.pdfThe strategy this dictates 由此决定的策略 MATH1061 . Mathematics 1A ★ Quiz A - the closed-book MCQ in Week 8 Quiz A ––第8周的闭卷选择题 Held in the Week-8 Linear-Algebra tutorial: 40 minutes, 12 MCQ, 1 mark each, NO calculators, no notes, no extra paper, one correct answer per question. It samples the first half of BOTH streams - sets & functions, limits, differentiation, complex numbers, vectors, lines & planes, cross product. Exact-value arithmetic by hand under time is the whole test. There is no Special Consideration for the online quizzes (a better-mark rule covers them) - but Quiz A and the final are the marks you sit once. 在第 8周的线性代数习题课进行:40 分钟,12 道选择题, 每题1分,不许用计算器、不许带笔记、不许用额外草稿 纸,每题只有一个正确答案。它对两条主线的前半抽样 -- 集合与函数、极限、求导、复数、向量、直线与平 面、叉积。限时下手算精确值是全部考点。在线小测没有特 殊考量(由取较好成绩的规则覆盖) -- 但 Quiz A 和期末 是你只考一次的分数。 ✓ Drill the recurring chains, both streams 通刷反复出现的题链,两条主线 Every question is procedural: take a function, limit, matrix or complex number, apply the right technique, give the exact value. The chains that recur - Calculus: factor/cancel - limit; squeeze-bound - limit; differentiate + f'=0 - classify; Linear Algebra: realise denominator - complex division; dot product - angle/orthogonality; row- reduce - back-substitute; det(A-MI)=O - eigen. Show every line - method marks are real. Drill the chains and fresh numbers can't surprise you. 每道题都是程序化的:拿一个函数、极限、矩阵或复数,运 用正确的技巧,给出精确值。反复出现的题链 -- 微积 分:因式分解/约分→极限;夹逼定界→极限;求导→ f'=0→分类;线性代数:分母实数化→复数除法;点积 →夹角/正交;行化简→回代;det(A-入)=0→ 特征。写 出每一行 -- 步骤分是实打实的。把题链练熟,全新数字 便无法让你措手不及。 MATH1061 . Mathematics 1A CONTENTS - CONTENTS Both streams, one exam-ready book 两条主线,一本应考之书 Calculus first, then Linear Algebra - the order you actually meet them 先微积分,再线性代数 -- 按你实际遇到它们的顺序 Ch Topic Core methods Stream 1 . Calculus (Monday lectures) 1 Functions & limits function families . limit laws . squeeze . continuity . IVT → 2 Differentiation difference quotient . rules . chain . implicit . L'Hôpital → 3 Applications & curve sketching extrema . concavity · optimisation . MVT → 4 Taylor polynomials T_n about a . remainder . standard series → 5 Integration Riemann sum . FTC . substitution . parts . applications[5]Source: asksia-bible-math1061-bilingual.pdf4 1 Let A = 2 . Find A-1 and use it to solve Ax = 1 2 ] Do CT ] ★ Exam-morning reminders - the recurring chains 考前提醒 -- 反复出现的题链 Calculus: factor/cancel for & limits; LIATE for parts; change limits in substitution; check Rn -> 0 before claiming a series. Linear algebra: projection divides by | u|2; det (cA) = c" det A; eigenvectors are nonzero; match P and D order. Always show working - method marks survive an arithmetic slip. 微积分:极限要因式分解约分;分部用 LIATE;换元要换积分限;断言级数收敛前先检验 lim an=0。线性代数:投影要除以 lull2; det(AB)=det A·det B; 特征向量非零;P与 D 的次序要对应。务必写出过程 -- 即便算术出错,步骤分仍能保住。 MATH1061 . Mathematics 1A AskSia Library VISUAL STUDY BIBLE . ASKSIA SCHOOL OF MATHEMATICS & STATISTICS SEMESTER 1 . 2026 0 3 r THE COMPLETE FIRST -YEAR BIBLE Mathematics 1A 数学 1A TWO STREAMS, ONE MARK - SINGLE-VARIABLE CALCULUS AND LINEAR ALGEBRA, DRILLED BY HAND ON FRESH NUMBERS. 悉尼大学 MATH1061 · 双语视觉精读 · LaTeX 公式排版 · 微积分 + 线性代数 · 期末 60% MATH1061 . THE UNIVERSITY OF SYDNEY 中英双语版 · BILINGUAL EDITION 英文主讲,中文随行 一 考试要点与术语保留英文原词 The final exam is 60% of your mark - the single biggest lever. This book covers both parallel streams: the Calculus half (functions - limits - differentiation - integration & Taylor) and the Linear Algebra half (complex numbers - vectors - systems - matrices - eigenvalues). Every definition is stated plainly, every method shown on a worked example with real arithmetic. Independent study companion. Not affiliated with or endorsed by The University of Sydney. Corrections: takedowns@asksia. ai PREFACE - HOW TO USE THIS BOOK Two streams, by hand, on fresh numbers 两条主线,纯手算,全新数字 Calculus and Linear Algebra run in parallel all semester - this book follows both 微积分与线性代数整学期并行推进 -- 本书两条线都跟 This is not a transcript of the lecture slides. It is a self-contained course in every technique MATH1061 examines, organised by the two streams you sit each week - Calculus (functions, limits, differentiation, integration, Taylor) and Linear Algebra (complex numbers, vectors, lines/planes, systems, matrices, eigenvalues). Each topic is built the same way so you always know where you are. 这并不是讲义幻灯片的誊录。它是一门自成体系的课程,涵盖 MATH1061 所考查的每一项技巧,按你每周分别上的两条主线组织 -- 微积分(函数、极限、求导、积分、泰勒)与线性代数(复数、向量、直线/平面、方程组、矩阵、特征值)。每个主题都以相同方式搭 建,让你始终清楚自己身在何处。 A 1 . LEARN
• 精确值为王：Quiz A 和大部分期末按材料口径是“无计算器”风格训练，答案要写最简分数、根式保留根式、$\pi$ 保留 $\pi$。[1]Source: asksia-bible-math1061-bilingual.pdf1 ·学 You haven't seen the lecture yet. Read a chapter top to bottom. Every topic is an AHA-unit - a formula or picture - plain-English idea - the method in steps - a worked example with real numbers - the common trap. The diagrams are original drawings of the standard maths - learn the idea cold. 你还没上过这节课。把一章从头读到 尾。每个主题都是一个 AHA 单元 -- 公式或图→通俗白话的概念 → 分步方法→ 用真实数字的例题 → 常见陷阱。图都是标准数学的原 创手绘 -- 把概念彻底学会。 B 2 . DRILL 2 · 练 You've done the lecture and tutorial. Cover the worked steps and re-derive each answer by hand. Quiz A and most of the final are no- calculator, so exact values - fractions in lowest terms, surds left as surds - on fresh numbers is the whole game. 你已上完课和习题课。遮住解题步 骤,亲手重新推出每个答案。Quiz A 和大部分期末都不许用计算器,所 以在全新数字上给出精确值 -- 化 到最简的分数、保留为根号的根式 -- 才是全部要义。 C 3 . EXAM 3 ·考 It's STUVAC. The AHA-units, trap boxes and the blueprint overleaf are your map. Quiz A (Week 8, 15%) samples the first half of both streams; the final (60%) samples the lot. Walk in knowing the recurring chains. 到了 STUVAC 复习周。AHA 单 元、陷阱框和下一页的蓝图就是你的 地图。Quiz A(第8周,15%)抽 样两条主线的前半;期末(60%) 抽样全部。带着对反复出现题链的熟 悉走进考场。 ! The single most important thing to understand about MATH1061 关于 MATH1061 最需要理解的一件事 This is two disjoint courses sharing one mark - Calculus (Monday lectures) and Linear Algebra (Wednesday lectures) run in parallel from Week 1 with their own lecturers, tutorials and notes. They barely talk to each other, so you cannot coast on one: the Week-8 in-person Quiz A samples BOTH streams (sets, functions, limits, differentiation, complex numbers, vectors, planes) and the 60% final samples the whole year. Budget your revision across both halves - do not leave a stream blank. 这是共享一个成绩的两门互不相干的课 -- 微积分(周一讲课)与线性代数(周三讲课)从第1周起并行推进,各有自己的讲 师、习题课和讲义。它们几乎不交流,所以你不能靠一门混过去:第 8周的线下 Quiz A 对两条主线都抽样(集合、函数、极限、 求导、复数、向量、平面),而 60% 的期末抽样全年。把复习时间分配到两半 -- 不要让任何一条主线留空。 MATH1061 . Mathematics 1A i How this book was built - and the two-layer rule 本书是如何搭建的 -- 以及两层规则 Standard mathematical definitions, theorems and formulas are stated plainly (they are universal - a limit law is a fact). The unit's own framing and any lecturer example numbers are paraphrased and re-numbered; every worked example here uses our own fresh numbers, never copied from slides or past papers. The Calculus stream follows the in-house Calculus of One Variable notes; Linear Algebra follows the MATH1061 Linear Algebra notes (Poole, Linear Algebra: A Modern Introduction, 4th ed. , as reference). Verify dates and weights against your own Canvas - details shift between cohorts. 标准的数学定义、定理和公式都如实陈述(它们是普适的 -- 极限定律是事实)。本单元自身的表述框架以及任何讲师的例题数字 都被改述并重新编号;这里每道例题都使用我们自己的全新数字,绝不照抄幻灯片或往年试卷。微积分主线遵循校内的 Calculus of One Variable 讲义;线性代数遵循 MATH1061 线性代数讲义(以 Poole, Linear Algebra: A Modern Introduction, 第4版 作为参考)。请对照你自己的 Canvas 核实日期与权重 -- 细节在不同届之间会变动。 MATH1061 . Mathematics 1A THE BLUEPRINT FINAL 60% - THE EXAM BLUEPRINT The final is 60% - everything points at it 期末占 60% ––一切都指向它 Quizzes 8% . A1 5% . AZ 10% · Quiz A 15% · tutorials 2% · final exam 60% 小测 8% · A1 5% · A2 10% · Quiz A 15% · 习题课 2% · 期末考 60% Your mark is six pieces, but one dominates. The final exam is 60% - more than the other five combined - and it samples both streams across the whole year. The in-person Quiz A (15%, Week 8) is the next biggest single hit and the one with the strictest conditions.[3]Source: asksia-bible-math1061-bilingual.pdfThe strategy this dictates 由此决定的策略 MATH1061 . Mathematics 1A ★ Quiz A - the closed-book MCQ in Week 8 Quiz A ––第8周的闭卷选择题 Held in the Week-8 Linear-Algebra tutorial: 40 minutes, 12 MCQ, 1 mark each, NO calculators, no notes, no extra paper, one correct answer per question. It samples the first half of BOTH streams - sets & functions, limits, differentiation, complex numbers, vectors, lines & planes, cross product. Exact-value arithmetic by hand under time is the whole test. There is no Special Consideration for the online quizzes (a better-mark rule covers them) - but Quiz A and the final are the marks you sit once. 在第 8周的线性代数习题课进行:40 分钟,12 道选择题, 每题1分,不许用计算器、不许带笔记、不许用额外草稿 纸,每题只有一个正确答案。它对两条主线的前半抽样 -- 集合与函数、极限、求导、复数、向量、直线与平 面、叉积。限时下手算精确值是全部考点。在线小测没有特 殊考量(由取较好成绩的规则覆盖) -- 但 Quiz A 和期末 是你只考一次的分数。 ✓ Drill the recurring chains, both streams 通刷反复出现的题链,两条主线 Every question is procedural: take a function, limit, matrix or complex number, apply the right technique, give the exact value. The chains that recur - Calculus: factor/cancel - limit; squeeze-bound - limit; differentiate + f'=0 - classify; Linear Algebra: realise denominator - complex division; dot product - angle/orthogonality; row- reduce - back-substitute; det(A-MI)=O - eigen. Show every line - method marks are real. Drill the chains and fresh numbers can't surprise you. 每道题都是程序化的:拿一个函数、极限、矩阵或复数,运 用正确的技巧,给出精确值。反复出现的题链 -- 微积 分:因式分解/约分→极限;夹逼定界→极限;求导→ f'=0→分类;线性代数:分母实数化→复数除法;点积 →夹角/正交;行化简→回代;det(A-入)=0→ 特征。写 出每一行 -- 步骤分是实打实的。把题链练熟,全新数字 便无法让你措手不及。 MATH1061 . Mathematics 1A CONTENTS - CONTENTS Both streams, one exam-ready book 两条主线,一本应考之书 Calculus first, then Linear Algebra - the order you actually meet them 先微积分,再线性代数 -- 按你实际遇到它们的顺序 Ch Topic Core methods Stream 1 . Calculus (Monday lectures) 1 Functions & limits function families . limit laws . squeeze . continuity . IVT → 2 Differentiation difference quotient . rules . chain . implicit . L'Hôpital → 3 Applications & curve sketching extrema . concavity · optimisation . MVT → 4 Taylor polynomials T_n about a . remainder . standard series → 5 Integration Riemann sum . FTC . substitution . parts . applications

0.2 最常考的“题位（slots）”长什么样

你可以把 Final 想成固定的题位在抽样：

• 微积分：1道极限、1道导数/最优化、1道积分、1道泰勒（级数）、外加1道“复数题”也常出现于整体抽样叙述中（材料这样概括）。[7]Source: asksia-bible-math1061-bilingual.pdf一句话要点。MATH1061 期末以计算为主,对两条主线都抽样:微积分出一道极限、一道导数/最优化、一道积分、一道泰勒级数和一 道复数题;线性代数出一道向量/平面、一道高斯消元方程组、一道行列式和一道特征值/特征向量题。本题库为每个题位提供一道全新 题目,完整解答。遮住解答,手算一遍,再核对。 ★ Fresh numbers, exam style - not the real stems 全新数字,考试风格 -- 并非真实题干 These are AskSia-authored questions written in the MATH1061 exam and Quiz-A style - they are not copied from any real paper. Give answers as exact values (fractions in lowest terms, exact surds/Tt) unless a decimal is asked for, and show every line - method marks are real. 这些是按 MATH1061 考试与 Quiz A 风格编写的 AskSia 原创题目 -- 并非抄自任何真实试卷。除非要求小数,答案应给出精确 值(最简分数、精确根式/π),并写出每一步 -- 步骤分是实打实的。 Q1-Q4 Calculus - limit, optimisation, parts, substitution Q1-Q4 微积分––极限、最优化、分部积分、换元 Q1 LIMIT 3 marks . Calculus Evaluate lim x2 - x-2 22 x2 - 4 . 02 OPTIMISATION 4 marks . Calculus A rectangle has perimeter 40. Maximise its area, classifying the stationary point. INTEGRATION BY PARTS Evaluate xe2ª dx. 3 marks . Calculus Q4 SUBSTITUTION Evaluate ∫ x 0 Vx2+1 =dx. 3 marks . Calculus MATH1061 . Mathematics 1A Q1-Q4 Worked solutions - calculus I 1 Q1. Factor: x2 - x -2 (x-2)(x+1) Cancel (x - 2) (the source of the g): lim2-+2 x+1 3 = = . x2 - 4 (x-2)(x+2) x+2 4 Q1. 因式分解: 约去(导致0/0的因子) :…。 2 Q2. Sides x and 20 - x (since 2x + 2y = 40). Area A(x) = x(20 - x) = 20x - x2. A'(x) = 20 -2x = 0=> x = 10. A" (x) = - 2 < 0, so it is a maximum: A = 10 . 10 = 100 (a square).
• 线代：1道向量/平面、1道高斯消元方程组、1道行列式、1道特征值/特征向量（甚至对角化）。[7]Source: asksia-bible-math1061-bilingual.pdf一句话要点。MATH1061 期末以计算为主,对两条主线都抽样:微积分出一道极限、一道导数/最优化、一道积分、一道泰勒级数和一 道复数题;线性代数出一道向量/平面、一道高斯消元方程组、一道行列式和一道特征值/特征向量题。本题库为每个题位提供一道全新 题目,完整解答。遮住解答,手算一遍,再核对。 ★ Fresh numbers, exam style - not the real stems 全新数字,考试风格 -- 并非真实题干 These are AskSia-authored questions written in the MATH1061 exam and Quiz-A style - they are not copied from any real paper. Give answers as exact values (fractions in lowest terms, exact surds/Tt) unless a decimal is asked for, and show every line - method marks are real. 这些是按 MATH1061 考试与 Quiz A 风格编写的 AskSia 原创题目 -- 并非抄自任何真实试卷。除非要求小数,答案应给出精确 值(最简分数、精确根式/π),并写出每一步 -- 步骤分是实打实的。 Q1-Q4 Calculus - limit, optimisation, parts, substitution Q1-Q4 微积分––极限、最优化、分部积分、换元 Q1 LIMIT 3 marks . Calculus Evaluate lim x2 - x-2 22 x2 - 4 . 02 OPTIMISATION 4 marks . Calculus A rectangle has perimeter 40. Maximise its area, classifying the stationary point. INTEGRATION BY PARTS Evaluate xe2ª dx. 3 marks . Calculus Q4 SUBSTITUTION Evaluate ∫ x 0 Vx2+1 =dx. 3 marks . Calculus MATH1061 . Mathematics 1A Q1-Q4 Worked solutions - calculus I 1 Q1. Factor: x2 - x -2 (x-2)(x+1) Cancel (x - 2) (the source of the g): lim2-+2 x+1 3 = = . x2 - 4 (x-2)(x+2) x+2 4 Q1. 因式分解: 约去(导致0/0的因子) :…。 2 Q2. Sides x and 20 - x (since 2x + 2y = 40). Area A(x) = x(20 - x) = 20x - x2. A'(x) = 20 -2x = 0=> x = 10. A" (x) = - 2 < 0, so it is a maximum: A = 10 . 10 = 100 (a square).

1）总复习策略：按“三层学习法”走（省时间且稳）

• A：LEARN（学）：每个主题按 “公式/图 → 白话概念 → 分步方法 → 例题（新数字）→ 常见陷阱” 学会。
• B：DRILL（练）：遮住例题步骤，亲手重新推一遍；因为考试要你在全新数字上做出精确值。[1]Source: asksia-bible-math1061-bilingual.pdf1 ·学 You haven't seen the lecture yet. Read a chapter top to bottom. Every topic is an AHA-unit - a formula or picture - plain-English idea - the method in steps - a worked example with real numbers - the common trap. The diagrams are original drawings of the standard maths - learn the idea cold. 你还没上过这节课。把一章从头读到 尾。每个主题都是一个 AHA 单元 -- 公式或图→通俗白话的概念 → 分步方法→ 用真实数字的例题 → 常见陷阱。图都是标准数学的原 创手绘 -- 把概念彻底学会。 B 2 . DRILL 2 · 练 You've done the lecture and tutorial. Cover the worked steps and re-derive each answer by hand. Quiz A and most of the final are no- calculator, so exact values - fractions in lowest terms, surds left as surds - on fresh numbers is the whole game. 你已上完课和习题课。遮住解题步 骤,亲手重新推出每个答案。Quiz A 和大部分期末都不许用计算器,所 以在全新数字上给出精确值 -- 化 到最简的分数、保留为根号的根式 -- 才是全部要义。 C 3 . EXAM 3 ·考 It's STUVAC. The AHA-units, trap boxes and the blueprint overleaf are your map. Quiz A (Week 8, 15%) samples the first half of both streams; the final (60%) samples the lot. Walk in knowing the recurring chains. 到了 STUVAC 复习周。AHA 单 元、陷阱框和下一页的蓝图就是你的 地图。Quiz A(第8周,15%)抽 样两条主线的前半;期末(60%) 抽样全部。带着对反复出现题链的熟 悉走进考场。 ! The single most important thing to understand about MATH1061 关于 MATH1061 最需要理解的一件事 This is two disjoint courses sharing one mark - Calculus (Monday lectures) and Linear Algebra (Wednesday lectures) run in parallel from Week 1 with their own lecturers, tutorials and notes. They barely talk to each other, so you cannot coast on one: the Week-8 in-person Quiz A samples BOTH streams (sets, functions, limits, differentiation, complex numbers, vectors, planes) and the 60% final samples the whole year. Budget your revision across both halves - do not leave a stream blank. 这是共享一个成绩的两门互不相干的课 -- 微积分(周一讲课)与线性代数(周三讲课)从第1周起并行推进,各有自己的讲 师、习题课和讲义。它们几乎不交流,所以你不能靠一门混过去:第 8周的线下 Quiz A 对两条主线都抽样(集合、函数、极限、 求导、复数、向量、平面),而 60% 的期末抽样全年。把复习时间分配到两半 -- 不要让任何一条主线留空。 MATH1061 . Mathematics 1A i How this book was built - and the two-layer rule 本书是如何搭建的 -- 以及两层规则 Standard mathematical definitions, theorems and formulas are stated plainly (they are universal - a limit law is a fact). The unit's own framing and any lecturer example numbers are paraphrased and re-numbered; every worked example here uses our own fresh numbers, never copied from slides or past papers. The Calculus stream follows the in-house Calculus of One Variable notes; Linear Algebra follows the MATH1061 Linear Algebra notes (Poole, Linear Algebra: A Modern Introduction, 4th ed. , as reference). Verify dates and weights against your own Canvas - details shift between cohorts. 标准的数学定义、定理和公式都如实陈述(它们是普适的 -- 极限定律是事实)。本单元自身的表述框架以及任何讲师的例题数字 都被改述并重新编号;这里每道例题都使用我们自己的全新数字,绝不照抄幻灯片或往年试卷。微积分主线遵循校内的 Calculus of One Variable 讲义;线性代数遵循 MATH1061 线性代数讲义(以 Poole, Linear Algebra: A Modern Introduction, 第4版 作为参考)。请对照你自己的 Canvas 核实日期与权重 -- 细节在不同届之间会变动。 MATH1061 . Mathematics 1A THE BLUEPRINT FINAL 60% - THE EXAM BLUEPRINT The final is 60% - everything points at it 期末占 60% ––一切都指向它 Quizzes 8% . A1 5% . AZ 10% · Quiz A 15% · tutorials 2% · final exam 60% 小测 8% · A1 5% · A2 10% · Quiz A 15% · 习题课 2% · 期末考 60% Your mark is six pieces, but one dominates. The final exam is 60% - more than the other five combined - and it samples both streams across the whole year. The in-person Quiz A (15%, Week 8) is the next biggest single hit and the one with the strictest conditions.
• C：EXAM（考）：用“题链 + 陷阱框 + 蓝图”当地图，去刷一整套跨两条主线的题。[1]Source: asksia-bible-math1061-bilingual.pdf1 ·学 You haven't seen the lecture yet. Read a chapter top to bottom. Every topic is an AHA-unit - a formula or picture - plain-English idea - the method in steps - a worked example with real numbers - the common trap. The diagrams are original drawings of the standard maths - learn the idea cold. 你还没上过这节课。把一章从头读到 尾。每个主题都是一个 AHA 单元 -- 公式或图→通俗白话的概念 → 分步方法→ 用真实数字的例题 → 常见陷阱。图都是标准数学的原 创手绘 -- 把概念彻底学会。 B 2 . DRILL 2 · 练 You've done the lecture and tutorial. Cover the worked steps and re-derive each answer by hand. Quiz A and most of the final are no- calculator, so exact values - fractions in lowest terms, surds left as surds - on fresh numbers is the whole game. 你已上完课和习题课。遮住解题步 骤,亲手重新推出每个答案。Quiz A 和大部分期末都不许用计算器,所 以在全新数字上给出精确值 -- 化 到最简的分数、保留为根号的根式 -- 才是全部要义。 C 3 . EXAM 3 ·考 It's STUVAC. The AHA-units, trap boxes and the blueprint overleaf are your map. Quiz A (Week 8, 15%) samples the first half of both streams; the final (60%) samples the lot. Walk in knowing the recurring chains. 到了 STUVAC 复习周。AHA 单 元、陷阱框和下一页的蓝图就是你的 地图。Quiz A(第8周,15%)抽 样两条主线的前半;期末(60%) 抽样全部。带着对反复出现题链的熟 悉走进考场。 ! The single most important thing to understand about MATH1061 关于 MATH1061 最需要理解的一件事 This is two disjoint courses sharing one mark - Calculus (Monday lectures) and Linear Algebra (Wednesday lectures) run in parallel from Week 1 with their own lecturers, tutorials and notes. They barely talk to each other, so you cannot coast on one: the Week-8 in-person Quiz A samples BOTH streams (sets, functions, limits, differentiation, complex numbers, vectors, planes) and the 60% final samples the whole year. Budget your revision across both halves - do not leave a stream blank. 这是共享一个成绩的两门互不相干的课 -- 微积分(周一讲课)与线性代数(周三讲课)从第1周起并行推进,各有自己的讲 师、习题课和讲义。它们几乎不交流,所以你不能靠一门混过去:第 8周的线下 Quiz A 对两条主线都抽样(集合、函数、极限、 求导、复数、向量、平面),而 60% 的期末抽样全年。把复习时间分配到两半 -- 不要让任何一条主线留空。 MATH1061 . Mathematics 1A i How this book was built - and the two-layer rule 本书是如何搭建的 -- 以及两层规则 Standard mathematical definitions, theorems and formulas are stated plainly (they are universal - a limit law is a fact). The unit's own framing and any lecturer example numbers are paraphrased and re-numbered; every worked example here uses our own fresh numbers, never copied from slides or past papers. The Calculus stream follows the in-house Calculus of One Variable notes; Linear Algebra follows the MATH1061 Linear Algebra notes (Poole, Linear Algebra: A Modern Introduction, 4th ed. , as reference). Verify dates and weights against your own Canvas - details shift between cohorts. 标准的数学定义、定理和公式都如实陈述(它们是普适的 -- 极限定律是事实)。本单元自身的表述框架以及任何讲师的例题数字 都被改述并重新编号;这里每道例题都使用我们自己的全新数字,绝不照抄幻灯片或往年试卷。微积分主线遵循校内的 Calculus of One Variable 讲义;线性代数遵循 MATH1061 线性代数讲义(以 Poole, Linear Algebra: A Modern Introduction, 第4版 作为参考)。请对照你自己的 Canvas 核实日期与权重 -- 细节在不同届之间会变动。 MATH1061 . Mathematics 1A THE BLUEPRINT FINAL 60% - THE EXAM BLUEPRINT The final is 60% - everything points at it 期末占 60% ––一切都指向它 Quizzes 8% . A1 5% . AZ 10% · Quiz A 15% · tutorials 2% · final exam 60% 小测 8% · A1 5% · A2 10% · Quiz A 15% · 习题课 2% · 期末考 60% Your mark is six pieces, but one dominates. The final exam is 60% - more than the other five combined - and it samples both streams across the whole year. The in-person Quiz A (15%, Week 8) is the next biggest single hit and the one with the strictest conditions.

2）你必须练到“闭眼能写”的反复出现题链（最高性价比）

A）微积分（Calculus）复习重点（按高频→低频）

A1. 函数 & 极限（Functions & Limits）

你必须会的定义（考试可直接写）

• 函数：每个输入 $x$ 对应唯一输出 $f(x)$；有 domain/codomain/range。[10]Source: asksia-bible-math1061-bilingual.pdfP-1 = 1 [3k + 1 3k - 17 5 4 2 3k -1 3k + 1 4 5 幂运算的回报:A^n=PD^nP-1 -- 例如 . . . 。 ! Eigenvectors are nonzero; match P and D order 特征向量非零;让 P 与 D 顺序匹配 An eigenvector must be nonzero (0 is never one). When building P and D, the k-th column of P must be an eigenvector for the k-th diagonal entry of D. A mismatch breaks A = PDP-1. Also: diagonalisable <> geometric mult = algebraic mult for every eigenvalue. 特征向量必须非零(0永远不是特征向量)。构造P与 D时,P的第 i列必须是 D第 i个对角元所对应的特征向量。错配会破坏 A=PDP-1。另外:可对角化 ⇒)每个特征值的几何重数=代数重数。 MATH1061 . Mathematics 1A - e. g. A2 = V. GLOSSARY . CALCULUS I - CHAPTER . GLOSSARY & KEY TERMS EN + 中文 Every examinable term, one line each 每个可考术语,每条一行 English term . X . crisp meaning - grouped by stream and topic 英文术语 · 中文 · 精炼释义 -- 按主线与主题分组 A fast reference for the vocabulary MATH1061 actually tests, across both streams - calculus and linear algebra. About 55 terms, each with a one-line meaning and small formula where it helps. Cover the right column and recite from the term; the +X column is filled during the bilingual pass. 一份针对 MATH1061 实际考查词汇的快速参考,横跨两条主线 -- 微积分与线性代数。约55个术语,每个配一行释义,有帮助处附 小公式。遮住右栏,对着术语背诵;中文一栏在双语整理阶段填入。 Term (EN) 中文 One-line meaning Part A - Calculus: functions, limits, derivatives Function f: A-+B — Rule giving each input x exactly one output f(x); has a domain, codomain and range. Natural domain All x for which the formula makes sense (no /O, no V of negatives, etc. ). Injective / surjective / bijective — One-to-one / onto / both; a bijection is invertible (Horizontal Line Test 'exactly once'). Inverse function f-1
• 极限的意思：$\lim_{x\to a} f(x)=L$ 是“$x$ 足够靠近 $a$（但不等于）时，$f(x)$ 可被迫任意接近 $L$”；$f(a)$ 本身甚至可以不存在。[13]Source: asksia-bible-math1061-bilingual.pdfEvery limit, derivative and integral this semester is a question about a function near a point or over an interval. The natural domain (where the formula makes sense) tells you where you are even allowed to ask the limit question - so pin it down before you compute anything. 本学期的每个极限、导数和积分都是关于某点附近或某区间上的函数的问题。自然定义域(公式有意义之处)告诉你在哪里才被允 许提出极限问题 -- 所以在计算任何东西之前先把它确定下来。 MATH1061 . Mathematics 1A C1 . LIMITS - THE LIMIT What f (x) approaches - not what it equals 趋近的是什么 -- 而非它等于什么 The single most important idea in the calculus stream 微积分主线中最重要的一个概念 lim f(x) = L means: we can force f(x) as close to L as we like, just by taking a close enough to a - but not equal to a. The value f (a) itself is irrelevant; it can even be undefined. 意思是:只要取得足够靠近,我们就能迫使任意接近 -- 但不等于。函数值本身无关紧要;它甚至可以无定义。 i The & idea, informally (full E-d is the Advanced course) 非正式的概念阐述(完整版在 Advanced 课程) Pick any tolerance & > 0 on the output. The limit is L if there is a window 8 > 0 on the input so that staying within o of a (but off a) keeps f(x) within & of L: 在输出端任取一个容差。极限是,如果在输入端存在一个窗口,使得保持在距内(但不等于)就能让保持在距内: 0 < |x -a| < 8 -> |f(x) - LE. 1. 3 The limit laws - build big limits from small ones 1. 3 极限定律–––由小极限搭建大极限 If limx-+a f and limx +a g both exist, limits pass straight through the arithmetic: 若与都存在,极限可直接穿过算术运算: lim (kf) = k lim f, lim (f + g) = lim f + lim g, 2-a lim(fg) =(lim f) (lim g), lim & = 17 limetaf f lim g + 0). 1 Try direct substitution first. If f is built from continuous families and a is in the domain, limx-+a f(x) = f(a) - just plug in. 先尝试直接代入。若函数由连续族构成且该点在定义域内,则极 限就等于函数值 -- 直接代入即可。 2 If you hit &, that is a signal, not an answer. Factor and cancel the common factor causing the zero, then substitute. 若得到 0/0,那是一个信号,而非答案。把导致零的公因式因式 分解并约去,然后再代入。 3 Never use the quotient law when the bottom limit is 0. Fix the form first. 当分母极限为 0时,绝不可使用商的极限法则。先把形式整理 好。 EX 1 A & rational limit no calculator
• 自然定义域（natural domain）：公式有意义的所有 $x$（不能除以0、不能开负数平方根等）；先确定它，因为这决定你“能不能问这个极限”。[13]Source: asksia-bible-math1061-bilingual.pdfEvery limit, derivative and integral this semester is a question about a function near a point or over an interval. The natural domain (where the formula makes sense) tells you where you are even allowed to ask the limit question - so pin it down before you compute anything. 本学期的每个极限、导数和积分都是关于某点附近或某区间上的函数的问题。自然定义域(公式有意义之处)告诉你在哪里才被允 许提出极限问题 -- 所以在计算任何东西之前先把它确定下来。 MATH1061 . Mathematics 1A C1 . LIMITS - THE LIMIT What f (x) approaches - not what it equals 趋近的是什么 -- 而非它等于什么 The single most important idea in the calculus stream 微积分主线中最重要的一个概念 lim f(x) = L means: we can force f(x) as close to L as we like, just by taking a close enough to a - but not equal to a. The value f (a) itself is irrelevant; it can even be undefined. 意思是:只要取得足够靠近,我们就能迫使任意接近 -- 但不等于。函数值本身无关紧要;它甚至可以无定义。 i The & idea, informally (full E-d is the Advanced course) 非正式的概念阐述(完整版在 Advanced 课程) Pick any tolerance & > 0 on the output. The limit is L if there is a window 8 > 0 on the input so that staying within o of a (but off a) keeps f(x) within & of L: 在输出端任取一个容差。极限是,如果在输入端存在一个窗口,使得保持在距内(但不等于)就能让保持在距内: 0 < |x -a| < 8 -> |f(x) - LE. 1. 3 The limit laws - build big limits from small ones 1. 3 极限定律–––由小极限搭建大极限 If limx-+a f and limx +a g both exist, limits pass straight through the arithmetic: 若与都存在,极限可直接穿过算术运算: lim (kf) = k lim f, lim (f + g) = lim f + lim g, 2-a lim(fg) =(lim f) (lim g), lim & = 17 limetaf f lim g + 0). 1 Try direct substitution first. If f is built from continuous families and a is in the domain, limx-+a f(x) = f(a) - just plug in. 先尝试直接代入。若函数由连续族构成且该点在定义域内,则极 限就等于函数值 -- 直接代入即可。 2 If you hit &, that is a signal, not an answer. Factor and cancel the common factor causing the zero, then substitute. 若得到 0/0,那是一个信号,而非答案。把导致零的公因式因式 分解并约去,然后再代入。 3 Never use the quotient law when the bottom limit is 0. Fix the form first. 当分母极限为 0时,绝不可使用商的极限法则。先把形式整理 好。 EX 1 A & rational limit no calculator

必背极限定律（会用就行）

• 若 $\lim f$、$\lim g$ 都存在：
  • $\lim (kf)=k\lim f$
  • $\lim (f+g)=\lim f+\lim g$
  • $\lim (fg)=(\lim f)(\lim g)$
  • $\lim (f/g)=\dfrac{\lim f}{\lim g}$ 前提：$\lim g\neq 0$ [13]Source: asksia-bible-math1061-bilingual.pdfEvery limit, derivative and integral this semester is a question about a function near a point or over an interval. The natural domain (where the formula makes sense) tells you where you are even allowed to ask the limit question - so pin it down before you compute anything. 本学期的每个极限、导数和积分都是关于某点附近或某区间上的函数的问题。自然定义域(公式有意义之处)告诉你在哪里才被允 许提出极限问题 -- 所以在计算任何东西之前先把它确定下来。 MATH1061 . Mathematics 1A C1 . LIMITS - THE LIMIT What f (x) approaches - not what it equals 趋近的是什么 -- 而非它等于什么 The single most important idea in the calculus stream 微积分主线中最重要的一个概念 lim f(x) = L means: we can force f(x) as close to L as we like, just by taking a close enough to a - but not equal to a. The value f (a) itself is irrelevant; it can even be undefined. 意思是:只要取得足够靠近,我们就能迫使任意接近 -- 但不等于。函数值本身无关紧要;它甚至可以无定义。 i The & idea, informally (full E-d is the Advanced course) 非正式的概念阐述(完整版在 Advanced 课程) Pick any tolerance & > 0 on the output. The limit is L if there is a window 8 > 0 on the input so that staying within o of a (but off a) keeps f(x) within & of L: 在输出端任取一个容差。极限是,如果在输入端存在一个窗口,使得保持在距内(但不等于)就能让保持在距内: 0 < |x -a| < 8 -> |f(x) - LE. 1. 3 The limit laws - build big limits from small ones 1. 3 极限定律–––由小极限搭建大极限 If limx-+a f and limx +a g both exist, limits pass straight through the arithmetic: 若与都存在,极限可直接穿过算术运算: lim (kf) = k lim f, lim (f + g) = lim f + lim g, 2-a lim(fg) =(lim f) (lim g), lim & = 17 limetaf f lim g + 0). 1 Try direct substitution first. If f is built from continuous families and a is in the domain, limx-+a f(x) = f(a) - just plug in. 先尝试直接代入。若函数由连续族构成且该点在定义域内,则极 限就等于函数值 -- 直接代入即可。 2 If you hit &, that is a signal, not an answer. Factor and cancel the common factor causing the zero, then substitute. 若得到 0/0,那是一个信号,而非答案。把导致零的公因式因式 分解并约去,然后再代入。 3 Never use the quotient law when the bottom limit is 0. Fix the form first. 当分母极限为 0时,绝不可使用商的极限法则。先把形式整理 好。 EX 1 A & rational limit no calculator

极限最常考的“操作模板”

• 先代入：如果 $f$ 在 $a$ 连续且 $a$ 在定义域内，常常直接 $\lim_{x\to a} f(x)=f(a)$。[13]Source: asksia-bible-math1061-bilingual.pdfEvery limit, derivative and integral this semester is a question about a function near a point or over an interval. The natural domain (where the formula makes sense) tells you where you are even allowed to ask the limit question - so pin it down before you compute anything. 本学期的每个极限、导数和积分都是关于某点附近或某区间上的函数的问题。自然定义域(公式有意义之处)告诉你在哪里才被允 许提出极限问题 -- 所以在计算任何东西之前先把它确定下来。 MATH1061 . Mathematics 1A C1 . LIMITS - THE LIMIT What f (x) approaches - not what it equals 趋近的是什么 -- 而非它等于什么 The single most important idea in the calculus stream 微积分主线中最重要的一个概念 lim f(x) = L means: we can force f(x) as close to L as we like, just by taking a close enough to a - but not equal to a. The value f (a) itself is irrelevant; it can even be undefined. 意思是:只要取得足够靠近,我们就能迫使任意接近 -- 但不等于。函数值本身无关紧要;它甚至可以无定义。 i The & idea, informally (full E-d is the Advanced course) 非正式的概念阐述(完整版在 Advanced 课程) Pick any tolerance & > 0 on the output. The limit is L if there is a window 8 > 0 on the input so that staying within o of a (but off a) keeps f(x) within & of L: 在输出端任取一个容差。极限是,如果在输入端存在一个窗口,使得保持在距内(但不等于)就能让保持在距内: 0 < |x -a| < 8 -> |f(x) - LE. 1. 3 The limit laws - build big limits from small ones 1. 3 极限定律–––由小极限搭建大极限 If limx-+a f and limx +a g both exist, limits pass straight through the arithmetic: 若与都存在,极限可直接穿过算术运算: lim (kf) = k lim f, lim (f + g) = lim f + lim g, 2-a lim(fg) =(lim f) (lim g), lim & = 17 limetaf f lim g + 0). 1 Try direct substitution first. If f is built from continuous families and a is in the domain, limx-+a f(x) = f(a) - just plug in. 先尝试直接代入。若函数由连续族构成且该点在定义域内,则极 限就等于函数值 -- 直接代入即可。 2 If you hit &, that is a signal, not an answer. Factor and cancel the common factor causing the zero, then substitute. 若得到 0/0,那是一个信号,而非答案。把导致零的公因式因式 分解并约去,然后再代入。 3 Never use the quotient law when the bottom limit is 0. Fix the form first. 当分母极限为 0时,绝不可使用商的极限法则。先把形式整理 好。 EX 1 A & rational limit no calculator[20]Source: asksia-cheatsheet-math1061.pdfMATH1061 Mathematics 1A UNIVERSITY OF SYDNEY . SCHOOL OF MATHEMATICS & STATISTICS EXAM REVISION Sem 1 2026 . SIDE 1 OF 2 Calculus · limits -> series SIDE 1/2 Taylor / Maclaurin series 0 · Exam Blueprint READ FIRST * MATH1061 runs two parallel streams: this side is Calculus (limits > derivatives -> integrals -> series); flip for Linear Algebra . Assessment: weekly quizzes 8% · A1 5% . A2 10% · in-person Quiz A 15% . tutorials 2% . final exam 60%. Most-tested moves: 0/0 limits (factor or L'Hôpital); differentiate with chain + product/quotient; classify critical points; evaluate a definite integral via FTC + a technique (sub / parts / partial fractions); write a Maclaurin series and use it. Method marks: show the working - state the rule, then the substitution, then the answer. A dropped chain-rule factor or a missed +C is the standard mark- loss. -- SIA > Two reflexes: name the form before you compute (is it 0/0? is it a product?), and always check the hypotheses - L'Hôpital needs 0/0 or w/c, FTC needs continuity. 1 . Functions & Limits WK 1-2 Function f:A-> B, one output per input; range = image £ codomain. Injective (1-1), surjective (onto, range = codomain), bijective = both => f-1 exists. Composition (g · f)(x)=g(f(x)). Limit. limx> f(x)=L: f(x) is forced arbitrarily close to L by taking x close to (#) a. Two-sided limit exists iff both one-sided limits exist and agree. LIMIT LAWS (IF BOTH LIMITS EXIST) ALGEBRA OF LIMITS lim(kf)=k·lim f . lim(f+g)=lim f + lim g Lim(fg)=(lim f) (lim g) lim(f/g)=lim f / lim g only if lim g=0 If f is continuous at a, limx-> a f(x)=f(a) - so most limits are "plug in"; only the joints of piecewise functions need care. Never use the quotient law when the denominator limit is 0 - factor, rationalise or use L'Hôpital instead. 1b . Squeeze & Standard Limits ★ MEMORISE gsfsh near a, lim g = lim h = L = lim f = L STANDARD LIMITS Limx-@ sin x / x = 1 Limx-0 (1-cos x)/x = 0 Limx-co (1+1/x)X = e Classic squeeze: - |x| ≤ x. sin(1/x) ≤ |x| => limx>0x sin(1/x)=0. 1c . 0/0 Limits . Worked FACTOR FIRST limx=>1 (x2-1)/(x2+x-2):
• 遇到 $0/0$：不是答案，是信号
  • 优先：因式分解→约去导致 $0/0$ 的因子→再代入。[13]Source: asksia-bible-math1061-bilingual.pdfEvery limit, derivative and integral this semester is a question about a function near a point or over an interval. The natural domain (where the formula makes sense) tells you where you are even allowed to ask the limit question - so pin it down before you compute anything. 本学期的每个极限、导数和积分都是关于某点附近或某区间上的函数的问题。自然定义域(公式有意义之处)告诉你在哪里才被允 许提出极限问题 -- 所以在计算任何东西之前先把它确定下来。 MATH1061 . Mathematics 1A C1 . LIMITS - THE LIMIT What f (x) approaches - not what it equals 趋近的是什么 -- 而非它等于什么 The single most important idea in the calculus stream 微积分主线中最重要的一个概念 lim f(x) = L means: we can force f(x) as close to L as we like, just by taking a close enough to a - but not equal to a. The value f (a) itself is irrelevant; it can even be undefined. 意思是:只要取得足够靠近,我们就能迫使任意接近 -- 但不等于。函数值本身无关紧要;它甚至可以无定义。 i The & idea, informally (full E-d is the Advanced course) 非正式的概念阐述(完整版在 Advanced 课程) Pick any tolerance & > 0 on the output. The limit is L if there is a window 8 > 0 on the input so that staying within o of a (but off a) keeps f(x) within & of L: 在输出端任取一个容差。极限是,如果在输入端存在一个窗口,使得保持在距内(但不等于)就能让保持在距内: 0 < |x -a| < 8 -> |f(x) - LE. 1. 3 The limit laws - build big limits from small ones 1. 3 极限定律–––由小极限搭建大极限 If limx-+a f and limx +a g both exist, limits pass straight through the arithmetic: 若与都存在,极限可直接穿过算术运算: lim (kf) = k lim f, lim (f + g) = lim f + lim g, 2-a lim(fg) =(lim f) (lim g), lim & = 17 limetaf f lim g + 0). 1 Try direct substitution first. If f is built from continuous families and a is in the domain, limx-+a f(x) = f(a) - just plug in. 先尝试直接代入。若函数由连续族构成且该点在定义域内,则极 限就等于函数值 -- 直接代入即可。 2 If you hit &, that is a signal, not an answer. Factor and cancel the common factor causing the zero, then substitute. 若得到 0/0,那是一个信号,而非答案。把导致零的公因式因式 分解并约去,然后再代入。 3 Never use the quotient law when the bottom limit is 0. Fix the form first. 当分母极限为 0时,绝不可使用商的极限法则。先把形式整理 好。 EX 1 A & rational limit no calculator[20]Source: asksia-cheatsheet-math1061.pdfMATH1061 Mathematics 1A UNIVERSITY OF SYDNEY . SCHOOL OF MATHEMATICS & STATISTICS EXAM REVISION Sem 1 2026 . SIDE 1 OF 2 Calculus · limits -> series SIDE 1/2 Taylor / Maclaurin series 0 · Exam Blueprint READ FIRST * MATH1061 runs two parallel streams: this side is Calculus (limits > derivatives -> integrals -> series); flip for Linear Algebra . Assessment: weekly quizzes 8% · A1 5% . A2 10% · in-person Quiz A 15% . tutorials 2% . final exam 60%. Most-tested moves: 0/0 limits (factor or L'Hôpital); differentiate with chain + product/quotient; classify critical points; evaluate a definite integral via FTC + a technique (sub / parts / partial fractions); write a Maclaurin series and use it. Method marks: show the working - state the rule, then the substitution, then the answer. A dropped chain-rule factor or a missed +C is the standard mark- loss. -- SIA > Two reflexes: name the form before you compute (is it 0/0? is it a product?), and always check the hypotheses - L'Hôpital needs 0/0 or w/c, FTC needs continuity. 1 . Functions & Limits WK 1-2 Function f:A-> B, one output per input; range = image £ codomain. Injective (1-1), surjective (onto, range = codomain), bijective = both => f-1 exists. Composition (g · f)(x)=g(f(x)). Limit. limx> f(x)=L: f(x) is forced arbitrarily close to L by taking x close to (#) a. Two-sided limit exists iff both one-sided limits exist and agree. LIMIT LAWS (IF BOTH LIMITS EXIST) ALGEBRA OF LIMITS lim(kf)=k·lim f . lim(f+g)=lim f + lim g Lim(fg)=(lim f) (lim g) lim(f/g)=lim f / lim g only if lim g=0 If f is continuous at a, limx-> a f(x)=f(a) - so most limits are "plug in"; only the joints of piecewise functions need care. Never use the quotient law when the denominator limit is 0 - factor, rationalise or use L'Hôpital instead. 1b . Squeeze & Standard Limits ★ MEMORISE gsfsh near a, lim g = lim h = L = lim f = L STANDARD LIMITS Limx-@ sin x / x = 1 Limx-0 (1-cos x)/x = 0 Limx-co (1+1/x)X = e Classic squeeze: - |x| ≤ x. sin(1/x) ≤ |x| => limx>0x sin(1/x)=0. 1c . 0/0 Limits . Worked FACTOR FIRST limx=>1 (x2-1)/(x2+x-2):
  • 必须记住一句禁令：分母极限为0时，不要用商的极限法则，先把形式修好。[13]Source: asksia-bible-math1061-bilingual.pdfEvery limit, derivative and integral this semester is a question about a function near a point or over an interval. The natural domain (where the formula makes sense) tells you where you are even allowed to ask the limit question - so pin it down before you compute anything. 本学期的每个极限、导数和积分都是关于某点附近或某区间上的函数的问题。自然定义域(公式有意义之处)告诉你在哪里才被允 许提出极限问题 -- 所以在计算任何东西之前先把它确定下来。 MATH1061 . Mathematics 1A C1 . LIMITS - THE LIMIT What f (x) approaches - not what it equals 趋近的是什么 -- 而非它等于什么 The single most important idea in the calculus stream 微积分主线中最重要的一个概念 lim f(x) = L means: we can force f(x) as close to L as we like, just by taking a close enough to a - but not equal to a. The value f (a) itself is irrelevant; it can even be undefined. 意思是:只要取得足够靠近,我们就能迫使任意接近 -- 但不等于。函数值本身无关紧要;它甚至可以无定义。 i The & idea, informally (full E-d is the Advanced course) 非正式的概念阐述(完整版在 Advanced 课程) Pick any tolerance & > 0 on the output. The limit is L if there is a window 8 > 0 on the input so that staying within o of a (but off a) keeps f(x) within & of L: 在输出端任取一个容差。极限是,如果在输入端存在一个窗口,使得保持在距内(但不等于)就能让保持在距内: 0 < |x -a| < 8 -> |f(x) - LE. 1. 3 The limit laws - build big limits from small ones 1. 3 极限定律–––由小极限搭建大极限 If limx-+a f and limx +a g both exist, limits pass straight through the arithmetic: 若与都存在,极限可直接穿过算术运算: lim (kf) = k lim f, lim (f + g) = lim f + lim g, 2-a lim(fg) =(lim f) (lim g), lim & = 17 limetaf f lim g + 0). 1 Try direct substitution first. If f is built from continuous families and a is in the domain, limx-+a f(x) = f(a) - just plug in. 先尝试直接代入。若函数由连续族构成且该点在定义域内,则极 限就等于函数值 -- 直接代入即可。 2 If you hit &, that is a signal, not an answer. Factor and cancel the common factor causing the zero, then substitute. 若得到 0/0,那是一个信号,而非答案。把导致零的公因式因式 分解并约去,然后再代入。 3 Never use the quotient law when the bottom limit is 0. Fix the form first. 当分母极限为 0时,绝不可使用商的极限法则。先把形式整理 好。 EX 1 A & rational limit no calculator
• 夹逼（squeeze）：背住标准夹逼思路；材料强调“先认形式再算”。[20]Source: asksia-cheatsheet-math1061.pdfMATH1061 Mathematics 1A UNIVERSITY OF SYDNEY . SCHOOL OF MATHEMATICS & STATISTICS EXAM REVISION Sem 1 2026 . SIDE 1 OF 2 Calculus · limits -> series SIDE 1/2 Taylor / Maclaurin series 0 · Exam Blueprint READ FIRST * MATH1061 runs two parallel streams: this side is Calculus (limits > derivatives -> integrals -> series); flip for Linear Algebra . Assessment: weekly quizzes 8% · A1 5% . A2 10% · in-person Quiz A 15% . tutorials 2% . final exam 60%. Most-tested moves: 0/0 limits (factor or L'Hôpital); differentiate with chain + product/quotient; classify critical points; evaluate a definite integral via FTC + a technique (sub / parts / partial fractions); write a Maclaurin series and use it. Method marks: show the working - state the rule, then the substitution, then the answer. A dropped chain-rule factor or a missed +C is the standard mark- loss. -- SIA > Two reflexes: name the form before you compute (is it 0/0? is it a product?), and always check the hypotheses - L'Hôpital needs 0/0 or w/c, FTC needs continuity. 1 . Functions & Limits WK 1-2 Function f:A-> B, one output per input; range = image £ codomain. Injective (1-1), surjective (onto, range = codomain), bijective = both => f-1 exists. Composition (g · f)(x)=g(f(x)). Limit. limx> f(x)=L: f(x) is forced arbitrarily close to L by taking x close to (#) a. Two-sided limit exists iff both one-sided limits exist and agree. LIMIT LAWS (IF BOTH LIMITS EXIST) ALGEBRA OF LIMITS lim(kf)=k·lim f . lim(f+g)=lim f + lim g Lim(fg)=(lim f) (lim g) lim(f/g)=lim f / lim g only if lim g=0 If f is continuous at a, limx-> a f(x)=f(a) - so most limits are "plug in"; only the joints of piecewise functions need care. Never use the quotient law when the denominator limit is 0 - factor, rationalise or use L'Hôpital instead. 1b . Squeeze & Standard Limits ★ MEMORISE gsfsh near a, lim g = lim h = L = lim f = L STANDARD LIMITS Limx-@ sin x / x = 1 Limx-0 (1-cos x)/x = 0 Limx-co (1+1/x)X = e Classic squeeze: - |x| ≤ x. sin(1/x) ≤ |x| => limx>0x sin(1/x)=0. 1c . 0/0 Limits . Worked FACTOR FIRST limx=>1 (x2-1)/(x2+x-2):
• 标准极限要背（cheatsheet 明确标“MEMORISE”）：
  • $\displaystyle \lim_{x\to 0}\frac{\sin x}{x}=1$
  • （材料还给了更多标准极限/夹逼例子，你复习时按cheatsheet那页背）[20]Source: asksia-cheatsheet-math1061.pdfMATH1061 Mathematics 1A UNIVERSITY OF SYDNEY . SCHOOL OF MATHEMATICS & STATISTICS EXAM REVISION Sem 1 2026 . SIDE 1 OF 2 Calculus · limits -> series SIDE 1/2 Taylor / Maclaurin series 0 · Exam Blueprint READ FIRST * MATH1061 runs two parallel streams: this side is Calculus (limits > derivatives -> integrals -> series); flip for Linear Algebra . Assessment: weekly quizzes 8% · A1 5% . A2 10% · in-person Quiz A 15% . tutorials 2% . final exam 60%. Most-tested moves: 0/0 limits (factor or L'Hôpital); differentiate with chain + product/quotient; classify critical points; evaluate a definite integral via FTC + a technique (sub / parts / partial fractions); write a Maclaurin series and use it. Method marks: show the working - state the rule, then the substitution, then the answer. A dropped chain-rule factor or a missed +C is the standard mark- loss. -- SIA > Two reflexes: name the form before you compute (is it 0/0? is it a product?), and always check the hypotheses - L'Hôpital needs 0/0 or w/c, FTC needs continuity. 1 . Functions & Limits WK 1-2 Function f:A-> B, one output per input; range = image £ codomain. Injective (1-1), surjective (onto, range = codomain), bijective = both => f-1 exists. Composition (g · f)(x)=g(f(x)). Limit. limx> f(x)=L: f(x) is forced arbitrarily close to L by taking x close to (#) a. Two-sided limit exists iff both one-sided limits exist and agree. LIMIT LAWS (IF BOTH LIMITS EXIST) ALGEBRA OF LIMITS lim(kf)=k·lim f . lim(f+g)=lim f + lim g Lim(fg)=(lim f) (lim g) lim(f/g)=lim f / lim g only if lim g=0 If f is continuous at a, limx-> a f(x)=f(a) - so most limits are "plug in"; only the joints of piecewise functions need care. Never use the quotient law when the denominator limit is 0 - factor, rationalise or use L'Hôpital instead. 1b . Squeeze & Standard Limits ★ MEMORISE gsfsh near a, lim g = lim h = L = lim f = L STANDARD LIMITS Limx-@ sin x / x = 1 Limx-0 (1-cos x)/x = 0 Limx-co (1+1/x)X = e Classic squeeze: - |x| ≤ x. sin(1/x) ≤ |x| => limx>0x sin(1/x)=0. 1c . 0/0 Limits . Worked FACTOR FIRST limx=>1 (x2-1)/(x2+x-2):

极限高频陷阱（你会丢的1分）

• 把 $0/0$ 当成0或无穷：错。你要把它当作“需要变形”的提示。[13]Source: asksia-bible-math1061-bilingual.pdfEvery limit, derivative and integral this semester is a question about a function near a point or over an interval. The natural domain (where the formula makes sense) tells you where you are even allowed to ask the limit question - so pin it down before you compute anything. 本学期的每个极限、导数和积分都是关于某点附近或某区间上的函数的问题。自然定义域(公式有意义之处)告诉你在哪里才被允 许提出极限问题 -- 所以在计算任何东西之前先把它确定下来。 MATH1061 . Mathematics 1A C1 . LIMITS - THE LIMIT What f (x) approaches - not what it equals 趋近的是什么 -- 而非它等于什么 The single most important idea in the calculus stream 微积分主线中最重要的一个概念 lim f(x) = L means: we can force f(x) as close to L as we like, just by taking a close enough to a - but not equal to a. The value f (a) itself is irrelevant; it can even be undefined. 意思是:只要取得足够靠近,我们就能迫使任意接近 -- 但不等于。函数值本身无关紧要;它甚至可以无定义。 i The & idea, informally (full E-d is the Advanced course) 非正式的概念阐述(完整版在 Advanced 课程) Pick any tolerance & > 0 on the output. The limit is L if there is a window 8 > 0 on the input so that staying within o of a (but off a) keeps f(x) within & of L: 在输出端任取一个容差。极限是,如果在输入端存在一个窗口,使得保持在距内(但不等于)就能让保持在距内: 0 < |x -a| < 8 -> |f(x) - LE. 1. 3 The limit laws - build big limits from small ones 1. 3 极限定律–––由小极限搭建大极限 If limx-+a f and limx +a g both exist, limits pass straight through the arithmetic: 若与都存在,极限可直接穿过算术运算: lim (kf) = k lim f, lim (f + g) = lim f + lim g, 2-a lim(fg) =(lim f) (lim g), lim & = 17 limetaf f lim g + 0). 1 Try direct substitution first. If f is built from continuous families and a is in the domain, limx-+a f(x) = f(a) - just plug in. 先尝试直接代入。若函数由连续族构成且该点在定义域内,则极 限就等于函数值 -- 直接代入即可。 2 If you hit &, that is a signal, not an answer. Factor and cancel the common factor causing the zero, then substitute. 若得到 0/0,那是一个信号,而非答案。把导致零的公因式因式 分解并约去,然后再代入。 3 Never use the quotient law when the bottom limit is 0. Fix the form first. 当分母极限为 0时,绝不可使用商的极限法则。先把形式整理 好。 EX 1 A & rational limit no calculator
• 分母趋于0还硬套商法则：错。[13]Source: asksia-bible-math1061-bilingual.pdfEvery limit, derivative and integral this semester is a question about a function near a point or over an interval. The natural domain (where the formula makes sense) tells you where you are even allowed to ask the limit question - so pin it down before you compute anything. 本学期的每个极限、导数和积分都是关于某点附近或某区间上的函数的问题。自然定义域(公式有意义之处)告诉你在哪里才被允 许提出极限问题 -- 所以在计算任何东西之前先把它确定下来。 MATH1061 . Mathematics 1A C1 . LIMITS - THE LIMIT What f (x) approaches - not what it equals 趋近的是什么 -- 而非它等于什么 The single most important idea in the calculus stream 微积分主线中最重要的一个概念 lim f(x) = L means: we can force f(x) as close to L as we like, just by taking a close enough to a - but not equal to a. The value f (a) itself is irrelevant; it can even be undefined. 意思是:只要取得足够靠近,我们就能迫使任意接近 -- 但不等于。函数值本身无关紧要;它甚至可以无定义。 i The & idea, informally (full E-d is the Advanced course) 非正式的概念阐述(完整版在 Advanced 课程) Pick any tolerance & > 0 on the output. The limit is L if there is a window 8 > 0 on the input so that staying within o of a (but off a) keeps f(x) within & of L: 在输出端任取一个容差。极限是,如果在输入端存在一个窗口,使得保持在距内(但不等于)就能让保持在距内: 0 < |x -a| < 8 -> |f(x) - LE. 1. 3 The limit laws - build big limits from small ones 1. 3 极限定律–––由小极限搭建大极限 If limx-+a f and limx +a g both exist, limits pass straight through the arithmetic: 若与都存在,极限可直接穿过算术运算: lim (kf) = k lim f, lim (f + g) = lim f + lim g, 2-a lim(fg) =(lim f) (lim g), lim & = 17 limetaf f lim g + 0). 1 Try direct substitution first. If f is built from continuous families and a is in the domain, limx-+a f(x) = f(a) - just plug in. 先尝试直接代入。若函数由连续族构成且该点在定义域内,则极 限就等于函数值 -- 直接代入即可。 2 If you hit &, that is a signal, not an answer. Factor and cancel the common factor causing the zero, then substitute. 若得到 0/0,那是一个信号,而非答案。把导致零的公因式因式 分解并约去,然后再代入。 3 Never use the quotient law when the bottom limit is 0. Fix the form first. 当分母极限为 0时,绝不可使用商的极限法则。先把形式整理 好。 EX 1 A & rational limit no calculator

A2. 导数（Differentiation）与“最优化/曲线分析”

定义与几何意义（必会写）

• 导数定义：
  $$f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$$
  是切线斜率。[29]Source: asksia-cheatsheet-math1061.pdf" 3 cE[a, b] with f(c)=N Use IVT to show a root exists: f(a)<0<f(b) => some c with f(c)=0. Needs a closed interval; gives existence, not the value. Inverse & hyperbolic. Restrict the domain so f is injective before inverting; f-1 is the reflection of f in y=x. Key examples: cosh x=(e}+e"})/2, sinh x=(ex-e -* )/2, with cosh2x - sinh2x = 1. 3 . The Derivative DEFINITION WK 3-4 f'(a)=Limn-@ [f(a+h)-f(a)] / h = slope of tangent at (a, f(a)) Tangent line: y = f(a) + f'(a)(x-a) . Differentiable = continuous (not conversely - |x| has a corner at 0). RULES LINEARITY / PRODUCT / QUOTIENT / CHAIN (kf)'=kf' . (f+g)'=f'±g' (fg)' = f'g + fg' (product) (f/g)' = (f'g - fg')/g2 (quotient) (gof)' (x) = g'(f(x)) . f' (x) (chain) Implicit: differentiate F(x,y)=0 in x, chain-rule the y- terms, solve y'. e. g. x2+y2=1 => 2x+2yy'=0 => y' =- x/y. Logarithmic: y=x* = > In y = x In x => y'/y = In x +1 => y' = x x (ln x + 1). Quotient worked. d/dx (sin x / x) = (x cos x - sin x)/x2. Chain worked. d/dx sin(x2) = cos(x2)-2x. Higher derivatives f", f" feed the second-derivative test and Taylor coefficients; for a product use the product rule repeatedly (or Leibniz's formula). Differentiable => continuous, but not the reverse (corners and cusps). 4 . Standard Derivatives D/DX . * TABLE F(X) F'(X) x kxk-1 e e aª a* In a ln x
• 切线方程：
  $$y=f(a)+f'(a)(x-a)$$ [29]Source: asksia-cheatsheet-math1061.pdf" 3 cE[a, b] with f(c)=N Use IVT to show a root exists: f(a)<0<f(b) => some c with f(c)=0. Needs a closed interval; gives existence, not the value. Inverse & hyperbolic. Restrict the domain so f is injective before inverting; f-1 is the reflection of f in y=x. Key examples: cosh x=(e}+e"})/2, sinh x=(ex-e -* )/2, with cosh2x - sinh2x = 1. 3 . The Derivative DEFINITION WK 3-4 f'(a)=Limn-@ [f(a+h)-f(a)] / h = slope of tangent at (a, f(a)) Tangent line: y = f(a) + f'(a)(x-a) . Differentiable = continuous (not conversely - |x| has a corner at 0). RULES LINEARITY / PRODUCT / QUOTIENT / CHAIN (kf)'=kf' . (f+g)'=f'±g' (fg)' = f'g + fg' (product) (f/g)' = (f'g - fg')/g2 (quotient) (gof)' (x) = g'(f(x)) . f' (x) (chain) Implicit: differentiate F(x,y)=0 in x, chain-rule the y- terms, solve y'. e. g. x2+y2=1 => 2x+2yy'=0 => y' =- x/y. Logarithmic: y=x* = > In y = x In x => y'/y = In x +1 => y' = x x (ln x + 1). Quotient worked. d/dx (sin x / x) = (x cos x - sin x)/x2. Chain worked. d/dx sin(x2) = cos(x2)-2x. Higher derivatives f", f" feed the second-derivative test and Taylor coefficients; for a product use the product rule repeatedly (or Leibniz's formula). Differentiable => continuous, but not the reverse (corners and cusps). 4 . Standard Derivatives D/DX . * TABLE F(X) F'(X) x kxk-1 e e aª a* In a ln x

四大求导法则（考试核心）

• 线性：$(kf)'=kf'$，$(f\pm g)'=f'\pm g'$
• 乘积：$(fg)'=f'g+fg'$
• 商：$\displaystyle (f/g)'=\frac{f'g-fg'}{g^2}$
• 链式：$(g\circ f)'(x)=g'(f(x))f'(x)$ [29]Source: asksia-cheatsheet-math1061.pdf" 3 cE[a, b] with f(c)=N Use IVT to show a root exists: f(a)<0<f(b) => some c with f(c)=0. Needs a closed interval; gives existence, not the value. Inverse & hyperbolic. Restrict the domain so f is injective before inverting; f-1 is the reflection of f in y=x. Key examples: cosh x=(e}+e"})/2, sinh x=(ex-e -* )/2, with cosh2x - sinh2x = 1. 3 . The Derivative DEFINITION WK 3-4 f'(a)=Limn-@ [f(a+h)-f(a)] / h = slope of tangent at (a, f(a)) Tangent line: y = f(a) + f'(a)(x-a) . Differentiable = continuous (not conversely - |x| has a corner at 0). RULES LINEARITY / PRODUCT / QUOTIENT / CHAIN (kf)'=kf' . (f+g)'=f'±g' (fg)' = f'g + fg' (product) (f/g)' = (f'g - fg')/g2 (quotient) (gof)' (x) = g'(f(x)) . f' (x) (chain) Implicit: differentiate F(x,y)=0 in x, chain-rule the y- terms, solve y'. e. g. x2+y2=1 => 2x+2yy'=0 => y' =- x/y. Logarithmic: y=x* = > In y = x In x => y'/y = In x +1 => y' = x x (ln x + 1). Quotient worked. d/dx (sin x / x) = (x cos x - sin x)/x2. Chain worked. d/dx sin(x2) = cos(x2)-2x. Higher derivatives f", f" feed the second-derivative test and Taylor coefficients; for a product use the product rule repeatedly (or Leibniz's formula). Differentiable => continuous, but not the reverse (corners and cusps). 4 . Standard Derivatives D/DX . * TABLE F(X) F'(X) x kxk-1 e e aª a* In a ln x

两个“考试最爱抓”的法则陷阱（必须背到条件反射）

• 链式法则漏乘内层导数：例如 $\dfrac{d}{dx}\sin(x^2)=\cos(x^2)\cdot 2x$，不是只有 $\cos(x^2)$。[12]Source: asksia-bible-math1061-bilingual.pdf考试钟爱的两个法则陷阱 (1) Chain rule: after differentiating the outer function, you must multiply by the derivative of the inside. ₫ sin(x2) = cos(x2) . 2x, not just cos(x2). (2) Quotient rule sign & order: the numerator is f'g - fg' (top-derivative first), not fg' - f'g. Swapping them flips the whole sign. (1)链式法则:对外层函数求导后,你必须乘以内部的导数。,而不只是。(2)商法则的符号与顺序:分子是(分子导数在前),而 非。把它们互换会翻转整个符号。 EX 3. 2 Chain x product x quotient together composite Differentiate y = x2 e 32 COS x . 求导。 1 Top is a product. Let u = x2e3x. Then u' = 2x e3x + x2 . 3e3x = xe3x (2 + 3x) (product rule; the 3 is the chain factor from e3x). 分子是一个乘积。令 . . ,则由乘积法则(其中含因子来自对内层求导的链式因子)。 2 Bottom. v = cos x, v' = - sin x. 分母。 3 Quotient rule. y' = u'v - uv' 02 xe3x (2 + 3x) cos x + x2e3ª sin x cos2 x = 商的法则。 4 Tidy. Factor xe32: y' xe3x (2+ 3x) cos x + x sin x] . cos2 x 整理。提取公因子: MATH1061 . Mathematics 1A tangent EX 3. 3 Tangent line Find the tangent line to f(x) = x3 - 4x at x = 1. 求在处的切线。 1 Point. f(1) = 1-4 =- 3, so the point is (1, -3). 点。故该点为 …. . 。 Slope. f'(x) =3x2-4, so f'(1) =3-4 =- 1. 斜率。故 . . . 。 Assemble. y = f(1) + f'(1)(x-1) = - 3- (x-1), i. e. y =- 2-2. 组装。即得切线方程。 MATH1061 . Mathematics 1A 3 . DERIVATIVE 3. 3 The standard-derivative table - memorise cold
• 商法则分子顺序/符号写反：一定是 $f'g-fg'$（“上导在前”），顺序一换整体符号就反了。[12]Source: asksia-bible-math1061-bilingual.pdf考试钟爱的两个法则陷阱 (1) Chain rule: after differentiating the outer function, you must multiply by the derivative of the inside. ₫ sin(x2) = cos(x2) . 2x, not just cos(x2). (2) Quotient rule sign & order: the numerator is f'g - fg' (top-derivative first), not fg' - f'g. Swapping them flips the whole sign. (1)链式法则:对外层函数求导后,你必须乘以内部的导数。,而不只是。(2)商法则的符号与顺序:分子是(分子导数在前),而 非。把它们互换会翻转整个符号。 EX 3. 2 Chain x product x quotient together composite Differentiate y = x2 e 32 COS x . 求导。 1 Top is a product. Let u = x2e3x. Then u' = 2x e3x + x2 . 3e3x = xe3x (2 + 3x) (product rule; the 3 is the chain factor from e3x). 分子是一个乘积。令 . . ,则由乘积法则(其中含因子来自对内层求导的链式因子)。 2 Bottom. v = cos x, v' = - sin x. 分母。 3 Quotient rule. y' = u'v - uv' 02 xe3x (2 + 3x) cos x + x2e3ª sin x cos2 x = 商的法则。 4 Tidy. Factor xe32: y' xe3x (2+ 3x) cos x + x sin x] . cos2 x 整理。提取公因子: MATH1061 . Mathematics 1A tangent EX 3. 3 Tangent line Find the tangent line to f(x) = x3 - 4x at x = 1. 求在处的切线。 1 Point. f(1) = 1-4 =- 3, so the point is (1, -3). 点。故该点为 …. . 。 Slope. f'(x) =3x2-4, so f'(1) =3-4 =- 1. 斜率。故 . . . 。 Assemble. y = f(1) + f'(1)(x-1) = - 3- (x-1), i. e. y =- 2-2. 组装。即得切线方程。 MATH1061 . Mathematics 1A 3 . DERIVATIVE 3. 3 The standard-derivative table - memorise cold

最优化与曲线分析：你要背的“得分步骤模板”

• 这部分材料把“方法分”写得很明确：[21]Source: asksia-cheatsheet-math1061.pdfcosh x cosh x sinh x cosh2x - sinh2x = 1. Combine with the chain rule for any composite, e. g. d/dx In(f(x)) = f'(x)/f(x). 5 . L'Hôpital's Rule INDETERMINATE 0/0 OR 00/00 Limx-a f/g = limx-a f' /g' (when the RHS exists) Check the form first. Other forms: 0. 00 -> rewrite as 0/0 or co/co; for 100, 00, 00° take logs, then apply. Worked. limx->0 (e-1-x)/x2 is 0/0 -> (ex-1)/2x still 0/0 > ex/2 -> 1/2. 0. 00 worked. limx->0+ x In x = lim (In x)/(1/x) = (00/00) -> (1/x)/(-1/x2) =- x -> 0. Warning: never apply L'Hôpital to a determinate form (e. g. 3/0 or 5/2) - re-check the form after every step. 6 . Extrema & Curve Sketching WK 5- 6 FIRST DERIVATIVE TEST · f'>0 = increasing; f'<0 = decreasing · Critical point f'(c)=0: necessary, not sufficient (x3 at 0) · f' :- >+ at c = local min; +>- = > local max SECOND DERIVATIVE TEST · f">0 => concave up; f"<0 => concave down · f'(c)=0, f"(c)>0=min; f"(c)<0=max Sketch checklist: domain . intercepts . sign of f' (incr/decr, crit pts) . sign of f" (concavity, inflections) . end behaviour x-> +00, 6b · Optimisation · Worked METHOD MARKS Method: write the quantity & its domain -> f'(x)=0 for critical points -> classify (1st/2nd test) -> compare with endpoints for the global optimum. Eg. max area of a rectangle of perimeter 20: A=x(10-x), A'=10-2x=0=>x=5; A" =- 2<0=> max. A=25 (a square). Trap: f''(c)=0 does not force an inflection; never forget endpoint values on a closed interval. 7 . Riemann Integral DEFINITE INTEGRAL = SIGNED AREA
  • 写出要最大/最小的量（目标函数）与它的定义域/约束
  • 求导 $f'(x)=0$ 找临界点
  • 用一阶/二阶判别做分类
  • 如果是闭区间问题，必须比较端点（很多人忘了端点丢分）[21]Source: asksia-cheatsheet-math1061.pdfcosh x cosh x sinh x cosh2x - sinh2x = 1. Combine with the chain rule for any composite, e. g. d/dx In(f(x)) = f'(x)/f(x). 5 . L'Hôpital's Rule INDETERMINATE 0/0 OR 00/00 Limx-a f/g = limx-a f' /g' (when the RHS exists) Check the form first. Other forms: 0. 00 -> rewrite as 0/0 or co/co; for 100, 00, 00° take logs, then apply. Worked. limx->0 (e-1-x)/x2 is 0/0 -> (ex-1)/2x still 0/0 > ex/2 -> 1/2. 0. 00 worked. limx->0+ x In x = lim (In x)/(1/x) = (00/00) -> (1/x)/(-1/x2) =- x -> 0. Warning: never apply L'Hôpital to a determinate form (e. g. 3/0 or 5/2) - re-check the form after every step. 6 . Extrema & Curve Sketching WK 5- 6 FIRST DERIVATIVE TEST · f'>0 = increasing; f'<0 = decreasing · Critical point f'(c)=0: necessary, not sufficient (x3 at 0) · f' :- >+ at c = local min; +>- = > local max SECOND DERIVATIVE TEST · f">0 => concave up; f"<0 => concave down · f'(c)=0, f"(c)>0=min; f"(c)<0=max Sketch checklist: domain . intercepts . sign of f' (incr/decr, crit pts) . sign of f" (concavity, inflections) . end behaviour x-> +00, 6b · Optimisation · Worked METHOD MARKS Method: write the quantity & its domain -> f'(x)=0 for critical points -> classify (1st/2nd test) -> compare with endpoints for the global optimum. Eg. max area of a rectangle of perimeter 20: A=x(10-x), A'=10-2x=0=>x=5; A" =- 2<0=> max. A=25 (a square). Trap: f''(c)=0 does not force an inflection; never forget endpoint values on a closed interval. 7 . Riemann Integral DEFINITE INTEGRAL = SIGNED AREA
• 判别要点（cheatsheet 写得很清楚）：[21]Source: asksia-cheatsheet-math1061.pdfcosh x cosh x sinh x cosh2x - sinh2x = 1. Combine with the chain rule for any composite, e. g. d/dx In(f(x)) = f'(x)/f(x). 5 . L'Hôpital's Rule INDETERMINATE 0/0 OR 00/00 Limx-a f/g = limx-a f' /g' (when the RHS exists) Check the form first. Other forms: 0. 00 -> rewrite as 0/0 or co/co; for 100, 00, 00° take logs, then apply. Worked. limx->0 (e-1-x)/x2 is 0/0 -> (ex-1)/2x still 0/0 > ex/2 -> 1/2. 0. 00 worked. limx->0+ x In x = lim (In x)/(1/x) = (00/00) -> (1/x)/(-1/x2) =- x -> 0. Warning: never apply L'Hôpital to a determinate form (e. g. 3/0 or 5/2) - re-check the form after every step. 6 . Extrema & Curve Sketching WK 5- 6 FIRST DERIVATIVE TEST · f'>0 = increasing; f'<0 = decreasing · Critical point f'(c)=0: necessary, not sufficient (x3 at 0) · f' :- >+ at c = local min; +>- = > local max SECOND DERIVATIVE TEST · f">0 => concave up; f"<0 => concave down · f'(c)=0, f"(c)>0=min; f"(c)<0=max Sketch checklist: domain . intercepts . sign of f' (incr/decr, crit pts) . sign of f" (concavity, inflections) . end behaviour x-> +00, 6b · Optimisation · Worked METHOD MARKS Method: write the quantity & its domain -> f'(x)=0 for critical points -> classify (1st/2nd test) -> compare with endpoints for the global optimum. Eg. max area of a rectangle of perimeter 20: A=x(10-x), A'=10-2x=0=>x=5; A" =- 2<0=> max. A=25 (a square). Trap: f''(c)=0 does not force an inflection; never forget endpoint values on a closed interval. 7 . Riemann Integral DEFINITE INTEGRAL = SIGNED AREA
  • $f'>0$ 增，$f'<0$ 减
  • $f''>0$ 凹向上，$f''<0$ 凹向下
  • 二阶判别：$f'(c)=0$ 且 $f''(c)>0$ 为极小；$f''(c)<0$ 为极大

常见陷阱

• $f'(c)=0$ 只是必要不充分（例如 $x^3$ 在0）。[21]Source: asksia-cheatsheet-math1061.pdfcosh x cosh x sinh x cosh2x - sinh2x = 1. Combine with the chain rule for any composite, e. g. d/dx In(f(x)) = f'(x)/f(x). 5 . L'Hôpital's Rule INDETERMINATE 0/0 OR 00/00 Limx-a f/g = limx-a f' /g' (when the RHS exists) Check the form first. Other forms: 0. 00 -> rewrite as 0/0 or co/co; for 100, 00, 00° take logs, then apply. Worked. limx->0 (e-1-x)/x2 is 0/0 -> (ex-1)/2x still 0/0 > ex/2 -> 1/2. 0. 00 worked. limx->0+ x In x = lim (In x)/(1/x) = (00/00) -> (1/x)/(-1/x2) =- x -> 0. Warning: never apply L'Hôpital to a determinate form (e. g. 3/0 or 5/2) - re-check the form after every step. 6 . Extrema & Curve Sketching WK 5- 6 FIRST DERIVATIVE TEST · f'>0 = increasing; f'<0 = decreasing · Critical point f'(c)=0: necessary, not sufficient (x3 at 0) · f' :- >+ at c = local min; +>- = > local max SECOND DERIVATIVE TEST · f">0 => concave up; f"<0 => concave down · f'(c)=0, f"(c)>0=min; f"(c)<0=max Sketch checklist: domain . intercepts . sign of f' (incr/decr, crit pts) . sign of f" (concavity, inflections) . end behaviour x-> +00, 6b · Optimisation · Worked METHOD MARKS Method: write the quantity & its domain -> f'(x)=0 for critical points -> classify (1st/2nd test) -> compare with endpoints for the global optimum. Eg. max area of a rectangle of perimeter 20: A=x(10-x), A'=10-2x=0=>x=5; A" =- 2<0=> max. A=25 (a square). Trap: f''(c)=0 does not force an inflection; never forget endpoint values on a closed interval. 7 . Riemann Integral DEFINITE INTEGRAL = SIGNED AREA
• 忘了端点比较（全局最值题常丢分）。[21]Source: asksia-cheatsheet-math1061.pdfcosh x cosh x sinh x cosh2x - sinh2x = 1. Combine with the chain rule for any composite, e. g. d/dx In(f(x)) = f'(x)/f(x). 5 . L'Hôpital's Rule INDETERMINATE 0/0 OR 00/00 Limx-a f/g = limx-a f' /g' (when the RHS exists) Check the form first. Other forms: 0. 00 -> rewrite as 0/0 or co/co; for 100, 00, 00° take logs, then apply. Worked. limx->0 (e-1-x)/x2 is 0/0 -> (ex-1)/2x still 0/0 > ex/2 -> 1/2. 0. 00 worked. limx->0+ x In x = lim (In x)/(1/x) = (00/00) -> (1/x)/(-1/x2) =- x -> 0. Warning: never apply L'Hôpital to a determinate form (e. g. 3/0 or 5/2) - re-check the form after every step. 6 . Extrema & Curve Sketching WK 5- 6 FIRST DERIVATIVE TEST · f'>0 = increasing; f'<0 = decreasing · Critical point f'(c)=0: necessary, not sufficient (x3 at 0) · f' :- >+ at c = local min; +>- = > local max SECOND DERIVATIVE TEST · f">0 => concave up; f"<0 => concave down · f'(c)=0, f"(c)>0=min; f"(c)<0=max Sketch checklist: domain . intercepts . sign of f' (incr/decr, crit pts) . sign of f" (concavity, inflections) . end behaviour x-> +00, 6b · Optimisation · Worked METHOD MARKS Method: write the quantity & its domain -> f'(x)=0 for critical points -> classify (1st/2nd test) -> compare with endpoints for the global optimum. Eg. max area of a rectangle of perimeter 20: A=x(10-x), A'=10-2x=0=>x=5; A" =- 2<0=> max. A=25 (a square). Trap: f''(c)=0 does not force an inflection; never forget endpoint values on a closed interval. 7 . Riemann Integral DEFINITE INTEGRAL = SIGNED AREA

A3. 积分（Integration）：后半学期微积分的大头

一个“大思想”（用一句话背）

• 积分两件事本质上同一件事：
  • 反过来做求导（原函数）
  • 计算曲线下的带符号面积（定积分 = 黎曼和极限）
  • 把它们连起来的桥是 微积分基本定理 FTC。[6]Source: asksia-bible-math1061-bilingual.pdf1 Relation (Pythagoras). x2 + y2 = 132 = 169. At x = 5: y = 1169 - 25 = 12. 关系(勾股定理)。在该时刻: . 0 2 Differentiate in t. 2xdZ + 2y dy = 0. 对 t 求导。 3 Substitute x = 5, y = 12, d = 2. 2(5)(2) + 2(12). dy It = 0. 代入数值。 4 Solve. dy dt 246 5 m/s - the minus sign says the top is descending. = − 求解。负号说明顶端正在下降。 MATH1061 . Mathematics 1A 4 . INTEGRATION EXAM CORE Accumulation & area - differentiation run backwards 累积与面积 -- 把求导反过来运行 Antiderivatives . Riemann sums . the FTC . techniques . areas, volumes & improper integrals 原函数 · 黎曼和 · FTC · 技巧 · 面积、体积与反常积分 One big idea: integration does two things that turn out to be the same thing. It reverses differentiation (antiderivatives), and it measures the signed area under a graph (the definite integral, built as a limit of Riemann sums). The bridge between them - the most important theorem in the course - is the Fundamental Theorem of Calculus. 一个大思想:积分做两件事,而结果证明它们是同一件事。它反转求导(原函数),并度量图像下方的带符号面积(定积分,作为黎曼和 的极限构造)。二者之间的桥梁 -- 本课程最重要的定理 -- 就是微积分基本定理。 ★ What the exam asks here 考试在这里会问什么 Integration carries the back half of the calculus marks. Expect: (1) a definite integral by the FTC; (2) an integral by substitution (remember to change the limits!); (3) integration by parts (LIATE); (4) a partial-fractions integral; (5) area between two curves; (6) a volume of revolution (disc or shell); (7) an improper integral - converge or diverge? 积分占据微积分部分后半段的分数。预期会考:(1)用 FTC 计算定积分;(2)用换元法积分(记得换积分限!);(3)分部积分 (LIATE);(4)部分分式积分;(5)两曲线间的面积;(6)旋转体的体积(圆盘法或柱壳法);(7)反常积分 -- 收敛还是发散? 4. 1 Antiderivatives - the standard table 4. 1 原函数 ––标准表 If F' = f then [ f(x) dx = F(x) + C. The arbitrary constant C is non-negotiable - dropping it is the most common one- mark loss in the paper. 若则。任意常数不可省略 -- 丢掉它是整张试卷中最常见的一分损失。 f(x) § fdx 2ºn (n = - 1) 2n+1 -+ C n+1 2-1 In |x|+ C

考试会问什么（材料直接列了清单）

• 你需要预期：[6]Source: asksia-bible-math1061-bilingual.pdf1 Relation (Pythagoras). x2 + y2 = 132 = 169. At x = 5: y = 1169 - 25 = 12. 关系(勾股定理)。在该时刻: . 0 2 Differentiate in t. 2xdZ + 2y dy = 0. 对 t 求导。 3 Substitute x = 5, y = 12, d = 2. 2(5)(2) + 2(12). dy It = 0. 代入数值。 4 Solve. dy dt 246 5 m/s - the minus sign says the top is descending. = − 求解。负号说明顶端正在下降。 MATH1061 . Mathematics 1A 4 . INTEGRATION EXAM CORE Accumulation & area - differentiation run backwards 累积与面积 -- 把求导反过来运行 Antiderivatives . Riemann sums . the FTC . techniques . areas, volumes & improper integrals 原函数 · 黎曼和 · FTC · 技巧 · 面积、体积与反常积分 One big idea: integration does two things that turn out to be the same thing. It reverses differentiation (antiderivatives), and it measures the signed area under a graph (the definite integral, built as a limit of Riemann sums). The bridge between them - the most important theorem in the course - is the Fundamental Theorem of Calculus. 一个大思想:积分做两件事,而结果证明它们是同一件事。它反转求导(原函数),并度量图像下方的带符号面积(定积分,作为黎曼和 的极限构造)。二者之间的桥梁 -- 本课程最重要的定理 -- 就是微积分基本定理。 ★ What the exam asks here 考试在这里会问什么 Integration carries the back half of the calculus marks. Expect: (1) a definite integral by the FTC; (2) an integral by substitution (remember to change the limits!); (3) integration by parts (LIATE); (4) a partial-fractions integral; (5) area between two curves; (6) a volume of revolution (disc or shell); (7) an improper integral - converge or diverge? 积分占据微积分部分后半段的分数。预期会考:(1)用 FTC 计算定积分;(2)用换元法积分(记得换积分限!);(3)分部积分 (LIATE);(4)部分分式积分;(5)两曲线间的面积;(6)旋转体的体积(圆盘法或柱壳法);(7)反常积分 -- 收敛还是发散? 4. 1 Antiderivatives - the standard table 4. 1 原函数 ––标准表 If F' = f then [ f(x) dx = F(x) + C. The arbitrary constant C is non-negotiable - dropping it is the most common one- mark loss in the paper. 若则。任意常数不可省略 -- 丢掉它是整张试卷中最常见的一分损失。 f(x) § fdx 2ºn (n = - 1) 2n+1 -+ C n+1 2-1 In |x|+ C
  • FTC算定积分
  • 换元（定积分要换积分限）
  • 分部积分（LIATE）
  • 部分分式
  • 两曲线间面积
  • 旋转体体积（圆盘/柱壳）
  • 反常积分：收敛/发散

你必须背的公式/定理

• 不定积分定义：若 $F'=f$，则
  $$\int f(x),dx = F(x)+C$$
  其中 $+C$ 不能省（材料明确说这是最常见的一分损失）。[6]Source: asksia-bible-math1061-bilingual.pdf1 Relation (Pythagoras). x2 + y2 = 132 = 169. At x = 5: y = 1169 - 25 = 12. 关系(勾股定理)。在该时刻: . 0 2 Differentiate in t. 2xdZ + 2y dy = 0. 对 t 求导。 3 Substitute x = 5, y = 12, d = 2. 2(5)(2) + 2(12). dy It = 0. 代入数值。 4 Solve. dy dt 246 5 m/s - the minus sign says the top is descending. = − 求解。负号说明顶端正在下降。 MATH1061 . Mathematics 1A 4 . INTEGRATION EXAM CORE Accumulation & area - differentiation run backwards 累积与面积 -- 把求导反过来运行 Antiderivatives . Riemann sums . the FTC . techniques . areas, volumes & improper integrals 原函数 · 黎曼和 · FTC · 技巧 · 面积、体积与反常积分 One big idea: integration does two things that turn out to be the same thing. It reverses differentiation (antiderivatives), and it measures the signed area under a graph (the definite integral, built as a limit of Riemann sums). The bridge between them - the most important theorem in the course - is the Fundamental Theorem of Calculus. 一个大思想:积分做两件事,而结果证明它们是同一件事。它反转求导(原函数),并度量图像下方的带符号面积(定积分,作为黎曼和 的极限构造)。二者之间的桥梁 -- 本课程最重要的定理 -- 就是微积分基本定理。 ★ What the exam asks here 考试在这里会问什么 Integration carries the back half of the calculus marks. Expect: (1) a definite integral by the FTC; (2) an integral by substitution (remember to change the limits!); (3) integration by parts (LIATE); (4) a partial-fractions integral; (5) area between two curves; (6) a volume of revolution (disc or shell); (7) an improper integral - converge or diverge? 积分占据微积分部分后半段的分数。预期会考:(1)用 FTC 计算定积分;(2)用换元法积分(记得换积分限!);(3)分部积分 (LIATE);(4)部分分式积分;(5)两曲线间的面积;(6)旋转体的体积(圆盘法或柱壳法);(7)反常积分 -- 收敛还是发散? 4. 1 Antiderivatives - the standard table 4. 1 原函数 ––标准表 If F' = f then [ f(x) dx = F(x) + C. The arbitrary constant C is non-negotiable - dropping it is the most common one- mark loss in the paper. 若则。任意常数不可省略 -- 丢掉它是整张试卷中最常见的一分损失。 f(x) § fdx 2ºn (n = - 1) 2n+1 -+ C n+1 2-1 In |x|+ C[17]Source: asksia-cheatsheet-math1061.pdf· Invert A: [A||] -> RREF -> [I|A-1] 26 . High-Yield Traps DON'T LOSE MARKS · Integration: always +C; convert limits in substitution; reduce improper fractions before partial fractions · Projection uses ||u||2; cross product is R3-only & order- sensitive · (AB)-1, (AB)" reverse order; AB#BA; no matrix division Algebra Formula Belt SIDE 2 e10=cos0+i sine . (rei8)n=pleine n-th roots: R1/exp(i(+2km)/n) proju(v)=(u . v/llull2)u . Iluxvll=llulllvlsine 2×2-1 = (1/(ad-bc)) [d -b; - c a] det(A-AI)=0 . A=PDP-1 - Ak=pDKp-1 Σλ=tr Α . Πλ=det A SIDE 2/2 LINEAR ALGEBRA . Complex numbers . Polar / Euler / de Moivre . Vectors . Dot / cross / projection . Lines & planes . Matrices . Determinants · Eigenvalues . Diagonalisation REVISION SHEET . ALL TOPICS Compiled by AskSia . mapped to the MATH1061 syllabus . asksia. ai/cheatsheet/usyd-math1061 - WK 7
• FTC（两条）：[22]Source: asksia-cheatsheet-math1061.pdfMETHOD MARKS Method: write the quantity & its domain -> f'(x)=0 for critical points -> classify (1st/2nd test) -> compare with endpoints for the global optimum. Eg. max area of a rectangle of perimeter 20: A=x(10-x), A'=10-2x=0=>x=5; A" =- 2<0=> max. A=25 (a square). Trap: f''(c)=0 does not force an inflection; never forget endpoint values on a closed interval. 7 . Riemann Integral DEFINITE INTEGRAL = SIGNED AREA Sab f dx = LimN=oo ΣK f (Xk*) ΔΧ Ax = (b-a)/N Exists for continuous f. Lower/upper sums LN $ J ≤ UN bound it (handy for monotonic f). Sub-intervals partition a=x0<>> <. . . << N=b; the sample point xk* € k BASIC PROPERTIES Saª f = 0 . Jab f = - 5. ª f Ja (af+ßg) = aff + B[g (linearity) Ja C + + JCP f = Ja f Signed area: regions below the x-axis count negative - split at the zeros if you want true geometric area. 7b · Curve Sketch . Worked f'=3×2-3=0=>x=+1; f"=6x. x =- 1: f"<0 => local max f=2; x=1: f">0=> local min f =- 2. Inflection at x=0. Odd, roots 0,±/3, ends to0. The sign charts of f' and f" give the whole shape. On a closed interval, also test endpoints for the global extremum; a critical point alone need not be one. 8 . Fundamental Theorem FTC . WK 10 FTC I & II I: d/dx f. x f(t) dt = f(x) II: [ b F' (x) dx = F(b) - F(a) VARIABLE LIMITS (LEIBNIZ) d/dx [ g(x) f(t) dt = f(g(x) ) . g' (x) d/dx [xº f(t) dt = - f(x) Don't drop g'(x) on a variable upper limit; the sign flips for a variable lower limit. FTC worked. J. T/2 cos x dx = [sin x]]™/2 = 1 - 0 = 1. And d/dx f"" sin t dt = sin(x2)-2x. Part I > II. Part I says J " f is an antiderivative of f; Part II evaluates any antiderivative at the endpoints. Continuity of f on [a,b] is the hypothesis both need; FTC is what turns "antiderivative" into a number. 9 . Antiderivatives S . * TABLE F(X
  • I：$\dfrac{d}{dx}\int_a^x f(t),dt=f(x)$
  • II：$\displaystyle \int_a^b F'(x),dx=F(b)-F(a)$
• 带变量上限/下限（Leibniz）：
  • 上限 $g(x)$：$\dfrac{d}{dx}\int_a^{g(x)} f(t),dt=f(g(x))g'(x)$
  • 下限是 $x$：会出现负号（材料提醒“符号会翻”）。[22]Source: asksia-cheatsheet-math1061.pdfMETHOD MARKS Method: write the quantity & its domain -> f'(x)=0 for critical points -> classify (1st/2nd test) -> compare with endpoints for the global optimum. Eg. max area of a rectangle of perimeter 20: A=x(10-x), A'=10-2x=0=>x=5; A" =- 2<0=> max. A=25 (a square). Trap: f''(c)=0 does not force an inflection; never forget endpoint values on a closed interval. 7 . Riemann Integral DEFINITE INTEGRAL = SIGNED AREA Sab f dx = LimN=oo ΣK f (Xk*) ΔΧ Ax = (b-a)/N Exists for continuous f. Lower/upper sums LN $ J ≤ UN bound it (handy for monotonic f). Sub-intervals partition a=x0<>> <. . . << N=b; the sample point xk* € k BASIC PROPERTIES Saª f = 0 . Jab f = - 5. ª f Ja (af+ßg) = aff + B[g (linearity) Ja C + + JCP f = Ja f Signed area: regions below the x-axis count negative - split at the zeros if you want true geometric area. 7b · Curve Sketch . Worked f'=3×2-3=0=>x=+1; f"=6x. x =- 1: f"<0 => local max f=2; x=1: f">0=> local min f =- 2. Inflection at x=0. Odd, roots 0,±/3, ends to0. The sign charts of f' and f" give the whole shape. On a closed interval, also test endpoints for the global extremum; a critical point alone need not be one. 8 . Fundamental Theorem FTC . WK 10 FTC I & II I: d/dx f. x f(t) dt = f(x) II: [ b F' (x) dx = F(b) - F(a) VARIABLE LIMITS (LEIBNIZ) d/dx [ g(x) f(t) dt = f(g(x) ) . g' (x) d/dx [xº f(t) dt = - f(x) Don't drop g'(x) on a variable upper limit; the sign flips for a variable lower limit. FTC worked. J. T/2 cos x dx = [sin x]]™/2 = 1 - 0 = 1. And d/dx f"" sin t dt = sin(x2)-2x. Part I > II. Part I says J " f is an antiderivative of f; Part II evaluates any antiderivative at the endpoints. Continuity of f on [a,b] is the hypothesis both need; FTC is what turns "antiderivative" into a number. 9 . Antiderivatives S . * TABLE F(X

三个积分高频陷阱（必背）

• 不定积分忘写 $+C$。[6]Source: asksia-bible-math1061-bilingual.pdf1 Relation (Pythagoras). x2 + y2 = 132 = 169. At x = 5: y = 1169 - 25 = 12. 关系(勾股定理)。在该时刻: . 0 2 Differentiate in t. 2xdZ + 2y dy = 0. 对 t 求导。 3 Substitute x = 5, y = 12, d = 2. 2(5)(2) + 2(12). dy It = 0. 代入数值。 4 Solve. dy dt 246 5 m/s - the minus sign says the top is descending. = − 求解。负号说明顶端正在下降。 MATH1061 . Mathematics 1A 4 . INTEGRATION EXAM CORE Accumulation & area - differentiation run backwards 累积与面积 -- 把求导反过来运行 Antiderivatives . Riemann sums . the FTC . techniques . areas, volumes & improper integrals 原函数 · 黎曼和 · FTC · 技巧 · 面积、体积与反常积分 One big idea: integration does two things that turn out to be the same thing. It reverses differentiation (antiderivatives), and it measures the signed area under a graph (the definite integral, built as a limit of Riemann sums). The bridge between them - the most important theorem in the course - is the Fundamental Theorem of Calculus. 一个大思想:积分做两件事,而结果证明它们是同一件事。它反转求导(原函数),并度量图像下方的带符号面积(定积分,作为黎曼和 的极限构造)。二者之间的桥梁 -- 本课程最重要的定理 -- 就是微积分基本定理。 ★ What the exam asks here 考试在这里会问什么 Integration carries the back half of the calculus marks. Expect: (1) a definite integral by the FTC; (2) an integral by substitution (remember to change the limits!); (3) integration by parts (LIATE); (4) a partial-fractions integral; (5) area between two curves; (6) a volume of revolution (disc or shell); (7) an improper integral - converge or diverge? 积分占据微积分部分后半段的分数。预期会考:(1)用 FTC 计算定积分;(2)用换元法积分(记得换积分限!);(3)分部积分 (LIATE);(4)部分分式积分;(5)两曲线间的面积;(6)旋转体的体积(圆盘法或柱壳法);(7)反常积分 -- 收敛还是发散? 4. 1 Antiderivatives - the standard table 4. 1 原函数 ––标准表 If F' = f then [ f(x) dx = F(x) + C. The arbitrary constant C is non-negotiable - dropping it is the most common one- mark loss in the paper. 若则。任意常数不可省略 -- 丢掉它是整张试卷中最常见的一分损失。 f(x) § fdx 2ºn (n = - 1) 2n+1 -+ C n+1 2-1 In |x|+ C[17]Source: asksia-cheatsheet-math1061.pdf· Invert A: [A||] -> RREF -> [I|A-1] 26 . High-Yield Traps DON'T LOSE MARKS · Integration: always +C; convert limits in substitution; reduce improper fractions before partial fractions · Projection uses ||u||2; cross product is R3-only & order- sensitive · (AB)-1, (AB)" reverse order; AB#BA; no matrix division Algebra Formula Belt SIDE 2 e10=cos0+i sine . (rei8)n=pleine n-th roots: R1/exp(i(+2km)/n) proju(v)=(u . v/llull2)u . Iluxvll=llulllvlsine 2×2-1 = (1/(ad-bc)) [d -b; - c a] det(A-AI)=0 . A=PDP-1 - Ak=pDKp-1 Σλ=tr Α . Πλ=det A SIDE 2/2 LINEAR ALGEBRA . Complex numbers . Polar / Euler / de Moivre . Vectors . Dot / cross / projection . Lines & planes . Matrices . Determinants · Eigenvalues . Diagonalisation REVISION SHEET . ALL TOPICS Compiled by AskSia . mapped to the MATH1061 syllabus . asksia. ai/cheatsheet/usyd-math1061 - WK 7
• 换元做定积分却不换积分限（材料多次强调）。[6]Source: asksia-bible-math1061-bilingual.pdf1 Relation (Pythagoras). x2 + y2 = 132 = 169. At x = 5: y = 1169 - 25 = 12. 关系(勾股定理)。在该时刻: . 0 2 Differentiate in t. 2xdZ + 2y dy = 0. 对 t 求导。 3 Substitute x = 5, y = 12, d = 2. 2(5)(2) + 2(12). dy It = 0. 代入数值。 4 Solve. dy dt 246 5 m/s - the minus sign says the top is descending. = − 求解。负号说明顶端正在下降。 MATH1061 . Mathematics 1A 4 . INTEGRATION EXAM CORE Accumulation & area - differentiation run backwards 累积与面积 -- 把求导反过来运行 Antiderivatives . Riemann sums . the FTC . techniques . areas, volumes & improper integrals 原函数 · 黎曼和 · FTC · 技巧 · 面积、体积与反常积分 One big idea: integration does two things that turn out to be the same thing. It reverses differentiation (antiderivatives), and it measures the signed area under a graph (the definite integral, built as a limit of Riemann sums). The bridge between them - the most important theorem in the course - is the Fundamental Theorem of Calculus. 一个大思想:积分做两件事,而结果证明它们是同一件事。它反转求导(原函数),并度量图像下方的带符号面积(定积分,作为黎曼和 的极限构造)。二者之间的桥梁 -- 本课程最重要的定理 -- 就是微积分基本定理。 ★ What the exam asks here 考试在这里会问什么 Integration carries the back half of the calculus marks. Expect: (1) a definite integral by the FTC; (2) an integral by substitution (remember to change the limits!); (3) integration by parts (LIATE); (4) a partial-fractions integral; (5) area between two curves; (6) a volume of revolution (disc or shell); (7) an improper integral - converge or diverge? 积分占据微积分部分后半段的分数。预期会考:(1)用 FTC 计算定积分;(2)用换元法积分(记得换积分限!);(3)分部积分 (LIATE);(4)部分分式积分;(5)两曲线间的面积;(6)旋转体的体积(圆盘法或柱壳法);(7)反常积分 -- 收敛还是发散? 4. 1 Antiderivatives - the standard table 4. 1 原函数 ––标准表 If F' = f then [ f(x) dx = F(x) + C. The arbitrary constant C is non-negotiable - dropping it is the most common one- mark loss in the paper. 若则。任意常数不可省略 -- 丢掉它是整张试卷中最常见的一分损失。 f(x) § fdx 2ºn (n = - 1) 2n+1 -+ C n+1 2-1 In |x|+ C[17]Source: asksia-cheatsheet-math1061.pdf· Invert A: [A||] -> RREF -> [I|A-1] 26 . High-Yield Traps DON'T LOSE MARKS · Integration: always +C; convert limits in substitution; reduce improper fractions before partial fractions · Projection uses ||u||2; cross product is R3-only & order- sensitive · (AB)-1, (AB)" reverse order; AB#BA; no matrix division Algebra Formula Belt SIDE 2 e10=cos0+i sine . (rei8)n=pleine n-th roots: R1/exp(i(+2km)/n) proju(v)=(u . v/llull2)u . Iluxvll=llulllvlsine 2×2-1 = (1/(ad-bc)) [d -b; - c a] det(A-AI)=0 . A=PDP-1 - Ak=pDKp-1 Σλ=tr Α . Πλ=det A SIDE 2/2 LINEAR ALGEBRA . Complex numbers . Polar / Euler / de Moivre . Vectors . Dot / cross / projection . Lines & planes . Matrices . Determinants · Eigenvalues . Diagonalisation REVISION SHEET . ALL TOPICS Compiled by AskSia . mapped to the MATH1061 syllabus . asksia. ai/cheatsheet/usyd-math1061 - WK 7
• 面积题把“带符号面积”当几何面积：在 $x$ 轴下方的部分是负的；要真面积就要按零点分段。[22]Source: asksia-cheatsheet-math1061.pdfMETHOD MARKS Method: write the quantity & its domain -> f'(x)=0 for critical points -> classify (1st/2nd test) -> compare with endpoints for the global optimum. Eg. max area of a rectangle of perimeter 20: A=x(10-x), A'=10-2x=0=>x=5; A" =- 2<0=> max. A=25 (a square). Trap: f''(c)=0 does not force an inflection; never forget endpoint values on a closed interval. 7 . Riemann Integral DEFINITE INTEGRAL = SIGNED AREA Sab f dx = LimN=oo ΣK f (Xk*) ΔΧ Ax = (b-a)/N Exists for continuous f. Lower/upper sums LN $ J ≤ UN bound it (handy for monotonic f). Sub-intervals partition a=x0<>> <. . . << N=b; the sample point xk* € k BASIC PROPERTIES Saª f = 0 . Jab f = - 5. ª f Ja (af+ßg) = aff + B[g (linearity) Ja C + + JCP f = Ja f Signed area: regions below the x-axis count negative - split at the zeros if you want true geometric area. 7b · Curve Sketch . Worked f'=3×2-3=0=>x=+1; f"=6x. x =- 1: f"<0 => local max f=2; x=1: f">0=> local min f =- 2. Inflection at x=0. Odd, roots 0,±/3, ends to0. The sign charts of f' and f" give the whole shape. On a closed interval, also test endpoints for the global extremum; a critical point alone need not be one. 8 . Fundamental Theorem FTC . WK 10 FTC I & II I: d/dx f. x f(t) dt = f(x) II: [ b F' (x) dx = F(b) - F(a) VARIABLE LIMITS (LEIBNIZ) d/dx [ g(x) f(t) dt = f(g(x) ) . g' (x) d/dx [xº f(t) dt = - f(x) Don't drop g'(x) on a variable upper limit; the sign flips for a variable lower limit. FTC worked. J. T/2 cos x dx = [sin x]]™/2 = 1 - 0 = 1. And d/dx f"" sin t dt = sin(x2)-2x. Part I > II. Part I says J " f is an antiderivative of f; Part II evaluates any antiderivative at the endpoints. Continuity of f on [a,b] is the hypothesis both need; FTC is what turns "antiderivative" into a number. 9 . Antiderivatives S . * TABLE F(X

反常积分（Improper integrals）你要抓住的“关卡”

• 反常积分是用极限定义的；在奇点处要分段，每一段都必须收敛，不能靠“相消”救命。[11]Source: asksia-bible-math1061-bilingual.pdfx = lim 6-700 . b 1 x -2 z = Jim [ - ]]] 6-700 000 dx = lim b+00/1 1. ab 2-2 dr = lim -] 11b 3 Evaluate. = lima-+% (1 - }) = 1 - finite, so it converges. Check the test. Here p = 2 > 1, so the p-test predicts convergence V. 检验判别法。此处 p>1,故 p 判别法预言收敛 √。 - MATH1061 . Mathematics 1A 6->00 求值。 有限,故收敛。 dx converges p>1 ! Both pieces must converge at a singularity 在奇点处两部分都必须收敛 If the integrand blows up inside the interval (e. g. at x = 0 for f_ dz), split at the singularity and require each piece to converge separately. One divergent half makes the whole integral diverge - you cannot let cancellation rescue it. 若被积函数在区间内部发散(例如在某点处),要在奇点处分段,并要求每段各自收敛。任一段发散都会使整个积分发散 -- 不能 靠相消来补救。 i Chapter recap - the convergence gates 本章回顾 ––收敛的关卡 Two gates run this chapter. Series: the geometric Lar" converges iff |r| < 1; a Taylor series equals f iff Rn -> 0, checked via Lagrange. Integrals: the power test p > 1 (at ) or p < 1 (at O). Master those two gates and the whole topic is procedural. 本章由两道“关卡”主导。级数:几何级数当且仅当Irl<1时收敛;Taylor 级数当且仅当 Rn→0(用 Lagrange 余项检验)时等于函 数。积分:幂判别法(在 ∞处或在0处)。掌握这两道关卡,整个主题便是程式化的。 MATH1061 . Mathematics 1A ARITHMETIC WEEK 1 - COMPLEX ARITHMETIC One new symbol, then ordinary algebra 一个新符号,其余都是普通代数 Set i2 = - 1 and treat a + bi like a binomial - add, multiply, conjugate, divide 令 i2 =- 1,把 a+ bi 当作二项式处理––加、乘、取共轭、除
• 幂$p$检验作为关键关卡（材料强调这是 MATH1061 的常用判定法）：掌握它，题就变程序化。[9]Source: asksia-bible-math1061-bilingual.pdfWEEKS 11-12 Inverses, and a p-test bookend 反函数,以及一个 p-检验的收尾 enx and In x mirror across y = x; improper integrals converge by a clean power test e^x 与 In x 关于 y = x 互为镜像;反常积分由一个简洁的幂检验判定收敛 Two loose ends close the chapter. First the e? / In x pair - the functions whose series we just built are inverses, each the other reflected in y = x. Second, the MATH1061 way to decide convergence of an improper integral: a clean power (p) test, the integral analogue of the geometric gate. 两个收尾结束本章。第一是这一对 -- 我们刚构造其级数的那些函数互为反函数,彼此关于对称。第二是 MATH1061 判定反常积分收 敛的方法:一个简洁的幂(p)检验,是几何关卡的积分类比。 y y = X enx ln x x e^x and ln x are inverse functions: each is the other reflected in the dashed line y = x. e^x passes through (0,1); ln x through (1,0); ln undoes exp. ex 与 In x 互为反函数:彼此关于虚线 y=x 对称。ex 过(0,1),In x 过(1,0); In 抵消 exp。 Inverse functions, mirrored 反函数,互为镜像 e passes through (0, 1) and grows without bound; In x passes through (1, 0) and is its reflection across y = x. Formally In(e) = x and elnx = x. The series for In(1 + x) is what you get by integrating the geometric series for 1 term by term. 过且无界增长;过且是它关于的反射。形式上与。的级数正是把 的几何级数逐项积分所得。 KEY EQUATION . P-TEST FOR IMPROPER INTEGRALS ∞ ∫ 1 ∫ 0 p dx converges > p<1 EX 6 Improper integral by the definition standard Evaluate . 00 dx and confirm the p-test. 求并验证 p-检验。 1 Replace o by a limit. 把 ∞ 替换为极限。 dx 2[11]Source: asksia-bible-math1061-bilingual.pdfx = lim 6-700 . b 1 x -2 z = Jim [ - ]]] 6-700 000 dx = lim b+00/1 1. ab 2-2 dr = lim -] 11b 3 Evaluate. = lima-+% (1 - }) = 1 - finite, so it converges. Check the test. Here p = 2 > 1, so the p-test predicts convergence V. 检验判别法。此处 p>1,故 p 判别法预言收敛 √。 - MATH1061 . Mathematics 1A 6->00 求值。 有限,故收敛。 dx converges p>1 ! Both pieces must converge at a singularity 在奇点处两部分都必须收敛 If the integrand blows up inside the interval (e. g. at x = 0 for f_ dz), split at the singularity and require each piece to converge separately. One divergent half makes the whole integral diverge - you cannot let cancellation rescue it. 若被积函数在区间内部发散(例如在某点处),要在奇点处分段,并要求每段各自收敛。任一段发散都会使整个积分发散 -- 不能 靠相消来补救。 i Chapter recap - the convergence gates 本章回顾 ––收敛的关卡 Two gates run this chapter. Series: the geometric Lar" converges iff |r| < 1; a Taylor series equals f iff Rn -> 0, checked via Lagrange. Integrals: the power test p > 1 (at ) or p < 1 (at O). Master those two gates and the whole topic is procedural. 本章由两道“关卡”主导。级数:几何级数当且仅当Irl<1时收敛;Taylor 级数当且仅当 Rn→0(用 Lagrange 余项检验)时等于函 数。积分:幂判别法(在 ∞处或在0处)。掌握这两道关卡,整个主题便是程式化的。 MATH1061 . Mathematics 1A ARITHMETIC WEEK 1 - COMPLEX ARITHMETIC One new symbol, then ordinary algebra 一个新符号,其余都是普通代数 Set i2 = - 1 and treat a + bi like a binomial - add, multiply, conjugate, divide 令 i2 =- 1,把 a+ bi 当作二项式处理––加、乘、取共轭、除

A4. 泰勒 / 麦克劳林（Taylor/Maclaurin）与“级数使用”

你复习时要抓的三个核心

• 会写一个已知标准展开式并用它做极限（材料给了例子：$\cos x$ 的麦克劳林展开到 $x^4$，用来秒杀 $\lim_{x\to 0}\frac{1-\cos x}{x^2}$）。[18]Source: asksia-cheatsheet-math1061.pdf13c . Taylor . Worked USE A KNOWN SERIES Maclaurin of cos x to x4: cos x = 1 - x2/2 + x4/24. So limx->0 (1-cos x)/x2 = 1/2 (the x2/2 term dominates) - faster than two L'Hôpitals. And ex" = 1 +x2 +x4/2+ . . . by substituting x2 into ex - no new derivatives needed. Integrate term-by-term => [ ex- dx series (no elementary closed form). The first non-zero term controls the small-x behaviour. About a0. In x about a=1: f(1)=0, f'=1/x->1, f" =- 1/x2->-1 =Inx= (x-1) - (x-1)2/2 + (x-1)3/3 - . . . (matches In(1+u), u=x-1). Build coefficients f(k) (a)/k! one derivative at a time. Radius / convergence. e", sin, cos converge for all x; In(1+x) and (1+x)" only for |x|<1; geometric for |x|<1. Always state the radius before equating a series to f. Off- by-one in the factorial/power is the standard slip. Calculus Formula Belt SIDE 1 (gof)'=g'(f). f' . (f/g)'=(f'g-fg')/g2 S u dv = uv - [ v du . LIATE FTC: [ bF' =F(b)-F(a) . d/dxf. Xf=f(x) L'Hôpital: 0/0,c/c = lim f' /g' Σxk =1/ (1-x) (|x|<1) SIA - When a limit/integral looks ugly, ask "is there a standard series or a substitution that turns it into something on this sheet?" - usually yes. State the rule, show the substitution, then the answer - that is where the method marks live. asksia. ai/cheatsheet/ usyd-math1061 . side 1/2 AskSia CHEATSHEET SERIES Revision aid . check the current unit outline for exam conditions . @ 2026 flip - for side 2 . linear algebra k x x F=X3-3X REVISION SHEET . ALL TOPICS Compiled by AskSia . mapped to the MATH1061 syllabus . asksia. ai/cheatsheet/usyd-math1061 WK 7-8
• 会做“代入替换”生成新级数：例如把 $x^2$ 代进 $e^x$ 得到 $e^{x^2}$（材料强调“不用重新求导”）。[18]Source: asksia-cheatsheet-math1061.pdf13c . Taylor . Worked USE A KNOWN SERIES Maclaurin of cos x to x4: cos x = 1 - x2/2 + x4/24. So limx->0 (1-cos x)/x2 = 1/2 (the x2/2 term dominates) - faster than two L'Hôpitals. And ex" = 1 +x2 +x4/2+ . . . by substituting x2 into ex - no new derivatives needed. Integrate term-by-term => [ ex- dx series (no elementary closed form). The first non-zero term controls the small-x behaviour. About a0. In x about a=1: f(1)=0, f'=1/x->1, f" =- 1/x2->-1 =Inx= (x-1) - (x-1)2/2 + (x-1)3/3 - . . . (matches In(1+u), u=x-1). Build coefficients f(k) (a)/k! one derivative at a time. Radius / convergence. e", sin, cos converge for all x; In(1+x) and (1+x)" only for |x|<1; geometric for |x|<1. Always state the radius before equating a series to f. Off- by-one in the factorial/power is the standard slip. Calculus Formula Belt SIDE 1 (gof)'=g'(f). f' . (f/g)'=(f'g-fg')/g2 S u dv = uv - [ v du . LIATE FTC: [ bF' =F(b)-F(a) . d/dxf. Xf=f(x) L'Hôpital: 0/0,c/c = lim f' /g' Σxk =1/ (1-x) (|x|<1) SIA - When a limit/integral looks ugly, ask "is there a standard series or a substitution that turns it into something on this sheet?" - usually yes. State the rule, show the substitution, then the answer - that is where the method marks live. asksia. ai/cheatsheet/ usyd-math1061 . side 1/2 AskSia CHEATSHEET SERIES Revision aid . check the current unit outline for exam conditions . @ 2026 flip - for side 2 . linear algebra k x x F=X3-3X REVISION SHEET . ALL TOPICS Compiled by AskSia . mapped to the MATH1061 syllabus . asksia. ai/cheatsheet/usyd-math1061 WK 7-8
• 一定要说收敛半径/范围再把级数当函数：材料明确提醒：$e^x,\sin x,\cos x$ 对所有 $x$ 收敛；$\ln(1+x)$、$(1+x)^\alpha$ 只在 $|x|<1$。[18]Source: asksia-cheatsheet-math1061.pdf13c . Taylor . Worked USE A KNOWN SERIES Maclaurin of cos x to x4: cos x = 1 - x2/2 + x4/24. So limx->0 (1-cos x)/x2 = 1/2 (the x2/2 term dominates) - faster than two L'Hôpitals. And ex" = 1 +x2 +x4/2+ . . . by substituting x2 into ex - no new derivatives needed. Integrate term-by-term => [ ex- dx series (no elementary closed form). The first non-zero term controls the small-x behaviour. About a0. In x about a=1: f(1)=0, f'=1/x->1, f" =- 1/x2->-1 =Inx= (x-1) - (x-1)2/2 + (x-1)3/3 - . . . (matches In(1+u), u=x-1). Build coefficients f(k) (a)/k! one derivative at a time. Radius / convergence. e", sin, cos converge for all x; In(1+x) and (1+x)" only for |x|<1; geometric for |x|<1. Always state the radius before equating a series to f. Off- by-one in the factorial/power is the standard slip. Calculus Formula Belt SIDE 1 (gof)'=g'(f). f' . (f/g)'=(f'g-fg')/g2 S u dv = uv - [ v du . LIATE FTC: [ bF' =F(b)-F(a) . d/dxf. Xf=f(x) L'Hôpital: 0/0,c/c = lim f' /g' Σxk =1/ (1-x) (|x|<1) SIA - When a limit/integral looks ugly, ask "is there a standard series or a substitution that turns it into something on this sheet?" - usually yes. State the rule, show the substitution, then the answer - that is where the method marks live. asksia. ai/cheatsheet/ usyd-math1061 . side 1/2 AskSia CHEATSHEET SERIES Revision aid . check the current unit outline for exam conditions . @ 2026 flip - for side 2 . linear algebra k x x F=X3-3X REVISION SHEET . ALL TOPICS Compiled by AskSia . mapped to the MATH1061 syllabus . asksia. ai/cheatsheet/usyd-math1061 WK 7-8

级数高频陷阱

• 阶乘/幂次“差一位”（off-by-one）。[18]Source: asksia-cheatsheet-math1061.pdf13c . Taylor . Worked USE A KNOWN SERIES Maclaurin of cos x to x4: cos x = 1 - x2/2 + x4/24. So limx->0 (1-cos x)/x2 = 1/2 (the x2/2 term dominates) - faster than two L'Hôpitals. And ex" = 1 +x2 +x4/2+ . . . by substituting x2 into ex - no new derivatives needed. Integrate term-by-term => [ ex- dx series (no elementary closed form). The first non-zero term controls the small-x behaviour. About a0. In x about a=1: f(1)=0, f'=1/x->1, f" =- 1/x2->-1 =Inx= (x-1) - (x-1)2/2 + (x-1)3/3 - . . . (matches In(1+u), u=x-1). Build coefficients f(k) (a)/k! one derivative at a time. Radius / convergence. e", sin, cos converge for all x; In(1+x) and (1+x)" only for |x|<1; geometric for |x|<1. Always state the radius before equating a series to f. Off- by-one in the factorial/power is the standard slip. Calculus Formula Belt SIDE 1 (gof)'=g'(f). f' . (f/g)'=(f'g-fg')/g2 S u dv = uv - [ v du . LIATE FTC: [ bF' =F(b)-F(a) . d/dxf. Xf=f(x) L'Hôpital: 0/0,c/c = lim f' /g' Σxk =1/ (1-x) (|x|<1) SIA - When a limit/integral looks ugly, ask "is there a standard series or a substitution that turns it into something on this sheet?" - usually yes. State the rule, show the substitution, then the answer - that is where the method marks live. asksia. ai/cheatsheet/ usyd-math1061 . side 1/2 AskSia CHEATSHEET SERIES Revision aid . check the current unit outline for exam conditions . @ 2026 flip - for side 2 . linear algebra k x x F=X3-3X REVISION SHEET . ALL TOPICS Compiled by AskSia . mapped to the MATH1061 syllabus . asksia. ai/cheatsheet/usyd-math1061 WK 7-8
• 没写收敛半径就直接“等于函数”。[18]Source: asksia-cheatsheet-math1061.pdf13c . Taylor . Worked USE A KNOWN SERIES Maclaurin of cos x to x4: cos x = 1 - x2/2 + x4/24. So limx->0 (1-cos x)/x2 = 1/2 (the x2/2 term dominates) - faster than two L'Hôpitals. And ex" = 1 +x2 +x4/2+ . . . by substituting x2 into ex - no new derivatives needed. Integrate term-by-term => [ ex- dx series (no elementary closed form). The first non-zero term controls the small-x behaviour. About a0. In x about a=1: f(1)=0, f'=1/x->1, f" =- 1/x2->-1 =Inx= (x-1) - (x-1)2/2 + (x-1)3/3 - . . . (matches In(1+u), u=x-1). Build coefficients f(k) (a)/k! one derivative at a time. Radius / convergence. e", sin, cos converge for all x; In(1+x) and (1+x)" only for |x|<1; geometric for |x|<1. Always state the radius before equating a series to f. Off- by-one in the factorial/power is the standard slip. Calculus Formula Belt SIDE 1 (gof)'=g'(f). f' . (f/g)'=(f'g-fg')/g2 S u dv = uv - [ v du . LIATE FTC: [ bF' =F(b)-F(a) . d/dxf. Xf=f(x) L'Hôpital: 0/0,c/c = lim f' /g' Σxk =1/ (1-x) (|x|<1) SIA - When a limit/integral looks ugly, ask "is there a standard series or a substitution that turns it into something on this sheet?" - usually yes. State the rule, show the substitution, then the answer - that is where the method marks live. asksia. ai/cheatsheet/ usyd-math1061 . side 1/2 AskSia CHEATSHEET SERIES Revision aid . check the current unit outline for exam conditions . @ 2026 flip - for side 2 . linear algebra k x x F=X3-3X REVISION SHEET . ALL TOPICS Compiled by AskSia . mapped to the MATH1061 syllabus . asksia. ai/cheatsheet/usyd-math1061 WK 7-8

B）线性代数（Linear Algebra）复习重点（按高频→低频）

B1. 复数（Complex Numbers）

你必须会的基础定义/操作

• $i^2=-1$，$z=a+ib$，$\operatorname{Re}(z)=a$，$\operatorname{Im}(z)=b$。[23]Source: asksia-cheatsheet-math1061.pdfCALCULUS . Limits & squeeze . Derivative + rules . Standard derivatives . L'Hopital . Optimisation . FTC . Integration techniques WK 5 WK 9 SQUEEZE (SANDWICH) x MATH1061 Mathematics 1A UNIVERSITY OF SYDNEY . SCHOOL OF MATHEMATICS & STATISTICS EXAM REVISION Sem 1 2026 . SIDE 2 OF 2 Linear algebra · C -> eigenvalues 14 . Complex Numbers WK 1-2 ¡2 =- 1; z=a+ib, Re(z)=a, Im(z)=b (both real). RcC. ARITHMETIC (a+ib)+(c+id) = (a+c)+i(b+d) (a+ib) (c+id) = (ac-bd)+i(ad+bc) Conjugate ż=a-ib: z ż = a2+b2 E R , and zw = z. w, z+w = ż+ẅ. DIVISION - REALISE THE DENOMINATOR w/z = w ż / (z ż) = w ż / | z | 2 Equality: a+ib = c+id = a=c and b=d. Powers of i cycle: i, -1, -i, 1, i, . . . (period 4) - reduce the exponent mod 4. Division worked. (3+i)/(1-2i) . (1+2i)/(1+2i) = (3+6i+i-2)/(1+4) = (1+7i)/5 = 1/5 + (7/5)i . Quadratic over C. z2+z+1=0=z=(-1±/(-3))/2 =- 1/2 + (/3/2)i - a conjugate pair, the primitive cube roots of unity (#1). The quadratic formula works over C with / of a negative; the discriminant being negative is what forces complex roots. 15 . Modulus, Polar & Euler ARGAND PLANE Modulus |z|=/(a2+b2)=/(z z) = distance from 0; |z-w| = distance z+> w. |zw|=|z||w|, |z/w|=|z|/|w|, triangle ineq |z+w|≤|z|+|w|. Argument arg z=0 with z=|z|(cos0+i sin0); principal Arg ZE(-TI,Tt]. Read the quadrant from the diagram - not just tan-1(b/a). POLAR / EXPONENTIAL (EULER) e18 = cos0 + i sin8 = z = r e18 Z1Z2 = rir2 e1(81+82) Z1/Z2 = (r1/r2) ei(01-02) Euler's identity: e'" + 1 = 0. Multiplying = multiply moduli, add arguments. Useful identities: Re(z)=(z+z)/2, Im(z)=(z-z)/2i, z"1 = Z/|z|2 (|z|=1 = z-1=), and cos0=(e" +e-19)/2, sin0= (ei0-e-19)/2i. 15b . de Moivre & Roots WK 2 DE MOIVRE (cose + i sine)" = cos ne + i sin ne (r e18)n = pn eine N-TH ROOTS OF W = R E'+ ZK = R1/n exp(i(ų+2km)/n) k = 0,1, . . , n-1 (spaced 2m/n apart) Roots of unity (z"=1): e2Tik/n _ n points equally spaced on the unit circle. List all n roots (run k=0 . . . n-1).
• 共轭：$\bar z=a-ib$，且 $z\bar z=a^2+b^2=|z|^2$。[23]Source: asksia-cheatsheet-math1061.pdfCALCULUS . Limits & squeeze . Derivative + rules . Standard derivatives . L'Hopital . Optimisation . FTC . Integration techniques WK 5 WK 9 SQUEEZE (SANDWICH) x MATH1061 Mathematics 1A UNIVERSITY OF SYDNEY . SCHOOL OF MATHEMATICS & STATISTICS EXAM REVISION Sem 1 2026 . SIDE 2 OF 2 Linear algebra · C -> eigenvalues 14 . Complex Numbers WK 1-2 ¡2 =- 1; z=a+ib, Re(z)=a, Im(z)=b (both real). RcC. ARITHMETIC (a+ib)+(c+id) = (a+c)+i(b+d) (a+ib) (c+id) = (ac-bd)+i(ad+bc) Conjugate ż=a-ib: z ż = a2+b2 E R , and zw = z. w, z+w = ż+ẅ. DIVISION - REALISE THE DENOMINATOR w/z = w ż / (z ż) = w ż / | z | 2 Equality: a+ib = c+id = a=c and b=d. Powers of i cycle: i, -1, -i, 1, i, . . . (period 4) - reduce the exponent mod 4. Division worked. (3+i)/(1-2i) . (1+2i)/(1+2i) = (3+6i+i-2)/(1+4) = (1+7i)/5 = 1/5 + (7/5)i . Quadratic over C. z2+z+1=0=z=(-1±/(-3))/2 =- 1/2 + (/3/2)i - a conjugate pair, the primitive cube roots of unity (#1). The quadratic formula works over C with / of a negative; the discriminant being negative is what forces complex roots. 15 . Modulus, Polar & Euler ARGAND PLANE Modulus |z|=/(a2+b2)=/(z z) = distance from 0; |z-w| = distance z+> w. |zw|=|z||w|, |z/w|=|z|/|w|, triangle ineq |z+w|≤|z|+|w|. Argument arg z=0 with z=|z|(cos0+i sin0); principal Arg ZE(-TI,Tt]. Read the quadrant from the diagram - not just tan-1(b/a). POLAR / EXPONENTIAL (EULER) e18 = cos0 + i sin8 = z = r e18 Z1Z2 = rir2 e1(81+82) Z1/Z2 = (r1/r2) ei(01-02) Euler's identity: e'" + 1 = 0. Multiplying = multiply moduli, add arguments. Useful identities: Re(z)=(z+z)/2, Im(z)=(z-z)/2i, z"1 = Z/|z|2 (|z|=1 = z-1=), and cos0=(e" +e-19)/2, sin0= (ei0-e-19)/2i. 15b . de Moivre & Roots WK 2 DE MOIVRE (cose + i sine)" = cos ne + i sin ne (r e18)n = pn eine N-TH ROOTS OF W = R E'+ ZK = R1/n exp(i(ų+2km)/n) k = 0,1, . . , n-1 (spaced 2m/n apart) Roots of unity (z"=1): e2Tik/n _ n points equally spaced on the unit circle. List all n roots (run k=0 . . . n-1).
• 加减乘：按二项式展开就行。[23]Source: asksia-cheatsheet-math1061.pdfCALCULUS . Limits & squeeze . Derivative + rules . Standard derivatives . L'Hopital . Optimisation . FTC . Integration techniques WK 5 WK 9 SQUEEZE (SANDWICH) x MATH1061 Mathematics 1A UNIVERSITY OF SYDNEY . SCHOOL OF MATHEMATICS & STATISTICS EXAM REVISION Sem 1 2026 . SIDE 2 OF 2 Linear algebra · C -> eigenvalues 14 . Complex Numbers WK 1-2 ¡2 =- 1; z=a+ib, Re(z)=a, Im(z)=b (both real). RcC. ARITHMETIC (a+ib)+(c+id) = (a+c)+i(b+d) (a+ib) (c+id) = (ac-bd)+i(ad+bc) Conjugate ż=a-ib: z ż = a2+b2 E R , and zw = z. w, z+w = ż+ẅ. DIVISION - REALISE THE DENOMINATOR w/z = w ż / (z ż) = w ż / | z | 2 Equality: a+ib = c+id = a=c and b=d. Powers of i cycle: i, -1, -i, 1, i, . . . (period 4) - reduce the exponent mod 4. Division worked. (3+i)/(1-2i) . (1+2i)/(1+2i) = (3+6i+i-2)/(1+4) = (1+7i)/5 = 1/5 + (7/5)i . Quadratic over C. z2+z+1=0=z=(-1±/(-3))/2 =- 1/2 + (/3/2)i - a conjugate pair, the primitive cube roots of unity (#1). The quadratic formula works over C with / of a negative; the discriminant being negative is what forces complex roots. 15 . Modulus, Polar & Euler ARGAND PLANE Modulus |z|=/(a2+b2)=/(z z) = distance from 0; |z-w| = distance z+> w. |zw|=|z||w|, |z/w|=|z|/|w|, triangle ineq |z+w|≤|z|+|w|. Argument arg z=0 with z=|z|(cos0+i sin0); principal Arg ZE(-TI,Tt]. Read the quadrant from the diagram - not just tan-1(b/a). POLAR / EXPONENTIAL (EULER) e18 = cos0 + i sin8 = z = r e18 Z1Z2 = rir2 e1(81+82) Z1/Z2 = (r1/r2) ei(01-02) Euler's identity: e'" + 1 = 0. Multiplying = multiply moduli, add arguments. Useful identities: Re(z)=(z+z)/2, Im(z)=(z-z)/2i, z"1 = Z/|z|2 (|z|=1 = z-1=), and cos0=(e" +e-19)/2, sin0= (ei0-e-19)/2i. 15b . de Moivre & Roots WK 2 DE MOIVRE (cose + i sine)" = cos ne + i sin ne (r e18)n = pn eine N-TH ROOTS OF W = R E'+ ZK = R1/n exp(i(ų+2km)/n) k = 0,1, . . , n-1 (spaced 2m/n apart) Roots of unity (z"=1): e2Tik/n _ n points equally spaced on the unit circle. List all n roots (run k=0 . . . n-1).

考试最爱考的“题链”

• 除法：分母实数化（realise the denominator）：
  $$\frac{w}{z}=\frac{w\bar z}{z\bar z}=\frac{w\bar z}{|z|^2}$$
  这是宝典点名的 recurring chain。[3]Source: asksia-bible-math1061-bilingual.pdfThe strategy this dictates 由此决定的策略 MATH1061 . Mathematics 1A ★ Quiz A - the closed-book MCQ in Week 8 Quiz A ––第8周的闭卷选择题 Held in the Week-8 Linear-Algebra tutorial: 40 minutes, 12 MCQ, 1 mark each, NO calculators, no notes, no extra paper, one correct answer per question. It samples the first half of BOTH streams - sets & functions, limits, differentiation, complex numbers, vectors, lines & planes, cross product. Exact-value arithmetic by hand under time is the whole test. There is no Special Consideration for the online quizzes (a better-mark rule covers them) - but Quiz A and the final are the marks you sit once. 在第 8周的线性代数习题课进行:40 分钟,12 道选择题, 每题1分,不许用计算器、不许带笔记、不许用额外草稿 纸,每题只有一个正确答案。它对两条主线的前半抽样 -- 集合与函数、极限、求导、复数、向量、直线与平 面、叉积。限时下手算精确值是全部考点。在线小测没有特 殊考量(由取较好成绩的规则覆盖) -- 但 Quiz A 和期末 是你只考一次的分数。 ✓ Drill the recurring chains, both streams 通刷反复出现的题链,两条主线 Every question is procedural: take a function, limit, matrix or complex number, apply the right technique, give the exact value. The chains that recur - Calculus: factor/cancel - limit; squeeze-bound - limit; differentiate + f'=0 - classify; Linear Algebra: realise denominator - complex division; dot product - angle/orthogonality; row- reduce - back-substitute; det(A-MI)=O - eigen. Show every line - method marks are real. Drill the chains and fresh numbers can't surprise you. 每道题都是程序化的:拿一个函数、极限、矩阵或复数,运 用正确的技巧,给出精确值。反复出现的题链 -- 微积 分:因式分解/约分→极限;夹逼定界→极限;求导→ f'=0→分类;线性代数:分母实数化→复数除法;点积 →夹角/正交;行化简→回代;det(A-入)=0→ 特征。写 出每一行 -- 步骤分是实打实的。把题链练熟,全新数字 便无法让你措手不及。 MATH1061 . Mathematics 1A CONTENTS - CONTENTS Both streams, one exam-ready book 两条主线,一本应考之书 Calculus first, then Linear Algebra - the order you actually meet them 先微积分,再线性代数 -- 按你实际遇到它们的顺序 Ch Topic Core methods Stream 1 . Calculus (Monday lectures) 1 Functions & limits function families . limit laws . squeeze . continuity . IVT → 2 Differentiation difference quotient . rules . chain . implicit . L'Hôpital → 3 Applications & curve sketching extrema . concavity · optimisation . MVT → 4 Taylor polynomials T_n about a . remainder . standard series → 5 Integration Riemann sum . FTC . substitution . parts . applications[23]Source: asksia-cheatsheet-math1061.pdfCALCULUS . Limits & squeeze . Derivative + rules . Standard derivatives . L'Hopital . Optimisation . FTC . Integration techniques WK 5 WK 9 SQUEEZE (SANDWICH) x MATH1061 Mathematics 1A UNIVERSITY OF SYDNEY . SCHOOL OF MATHEMATICS & STATISTICS EXAM REVISION Sem 1 2026 . SIDE 2 OF 2 Linear algebra · C -> eigenvalues 14 . Complex Numbers WK 1-2 ¡2 =- 1; z=a+ib, Re(z)=a, Im(z)=b (both real). RcC. ARITHMETIC (a+ib)+(c+id) = (a+c)+i(b+d) (a+ib) (c+id) = (ac-bd)+i(ad+bc) Conjugate ż=a-ib: z ż = a2+b2 E R , and zw = z. w, z+w = ż+ẅ. DIVISION - REALISE THE DENOMINATOR w/z = w ż / (z ż) = w ż / | z | 2 Equality: a+ib = c+id = a=c and b=d. Powers of i cycle: i, -1, -i, 1, i, . . . (period 4) - reduce the exponent mod 4. Division worked. (3+i)/(1-2i) . (1+2i)/(1+2i) = (3+6i+i-2)/(1+4) = (1+7i)/5 = 1/5 + (7/5)i . Quadratic over C. z2+z+1=0=z=(-1±/(-3))/2 =- 1/2 + (/3/2)i - a conjugate pair, the primitive cube roots of unity (#1). The quadratic formula works over C with / of a negative; the discriminant being negative is what forces complex roots. 15 . Modulus, Polar & Euler ARGAND PLANE Modulus |z|=/(a2+b2)=/(z z) = distance from 0; |z-w| = distance z+> w. |zw|=|z||w|, |z/w|=|z|/|w|, triangle ineq |z+w|≤|z|+|w|. Argument arg z=0 with z=|z|(cos0+i sin0); principal Arg ZE(-TI,Tt]. Read the quadrant from the diagram - not just tan-1(b/a). POLAR / EXPONENTIAL (EULER) e18 = cos0 + i sin8 = z = r e18 Z1Z2 = rir2 e1(81+82) Z1/Z2 = (r1/r2) ei(01-02) Euler's identity: e'" + 1 = 0. Multiplying = multiply moduli, add arguments. Useful identities: Re(z)=(z+z)/2, Im(z)=(z-z)/2i, z"1 = Z/|z|2 (|z|=1 = z-1=), and cos0=(e" +e-19)/2, sin0= (ei0-e-19)/2i. 15b . de Moivre & Roots WK 2 DE MOIVRE (cose + i sine)" = cos ne + i sin ne (r e18)n = pn eine N-TH ROOTS OF W = R E'+ ZK = R1/n exp(i(ų+2km)/n) k = 0,1, . . , n-1 (spaced 2m/n apart) Roots of unity (z"=1): e2Tik/n _ n points equally spaced on the unit circle. List all n roots (run k=0 . . . n-1).
• 模长与幅角（极坐标/Euler）：
  $$z=r(\cos\theta+i\sin\theta)=re^{i\theta},\quad e^{i\theta}=\cos\theta+i\sin\theta$$ [23]Source: asksia-cheatsheet-math1061.pdfCALCULUS . Limits & squeeze . Derivative + rules . Standard derivatives . L'Hopital . Optimisation . FTC . Integration techniques WK 5 WK 9 SQUEEZE (SANDWICH) x MATH1061 Mathematics 1A UNIVERSITY OF SYDNEY . SCHOOL OF MATHEMATICS & STATISTICS EXAM REVISION Sem 1 2026 . SIDE 2 OF 2 Linear algebra · C -> eigenvalues 14 . Complex Numbers WK 1-2 ¡2 =- 1; z=a+ib, Re(z)=a, Im(z)=b (both real). RcC. ARITHMETIC (a+ib)+(c+id) = (a+c)+i(b+d) (a+ib) (c+id) = (ac-bd)+i(ad+bc) Conjugate ż=a-ib: z ż = a2+b2 E R , and zw = z. w, z+w = ż+ẅ. DIVISION - REALISE THE DENOMINATOR w/z = w ż / (z ż) = w ż / | z | 2 Equality: a+ib = c+id = a=c and b=d. Powers of i cycle: i, -1, -i, 1, i, . . . (period 4) - reduce the exponent mod 4. Division worked. (3+i)/(1-2i) . (1+2i)/(1+2i) = (3+6i+i-2)/(1+4) = (1+7i)/5 = 1/5 + (7/5)i . Quadratic over C. z2+z+1=0=z=(-1±/(-3))/2 =- 1/2 + (/3/2)i - a conjugate pair, the primitive cube roots of unity (#1). The quadratic formula works over C with / of a negative; the discriminant being negative is what forces complex roots. 15 . Modulus, Polar & Euler ARGAND PLANE Modulus |z|=/(a2+b2)=/(z z) = distance from 0; |z-w| = distance z+> w. |zw|=|z||w|, |z/w|=|z|/|w|, triangle ineq |z+w|≤|z|+|w|. Argument arg z=0 with z=|z|(cos0+i sin0); principal Arg ZE(-TI,Tt]. Read the quadrant from the diagram - not just tan-1(b/a). POLAR / EXPONENTIAL (EULER) e18 = cos0 + i sin8 = z = r e18 Z1Z2 = rir2 e1(81+82) Z1/Z2 = (r1/r2) ei(01-02) Euler's identity: e'" + 1 = 0. Multiplying = multiply moduli, add arguments. Useful identities: Re(z)=(z+z)/2, Im(z)=(z-z)/2i, z"1 = Z/|z|2 (|z|=1 = z-1=), and cos0=(e" +e-19)/2, sin0= (ei0-e-19)/2i. 15b . de Moivre & Roots WK 2 DE MOIVRE (cose + i sine)" = cos ne + i sin ne (r e18)n = pn eine N-TH ROOTS OF W = R E'+ ZK = R1/n exp(i(ų+2km)/n) k = 0,1, . . , n-1 (spaced 2m/n apart) Roots of unity (z"=1): e2Tik/n _ n points equally spaced on the unit circle. List all n roots (run k=0 . . . n-1).
• 棣莫弗（de Moivre）与$n$次根：
  $$(re^{i\theta})^n=r^n e^{in\theta}$$
  $$\text{$w$ 的 $n$ 次根： }R^{1/n}\exp\left(i\frac{\theta+2k\pi}{n}\right),\ k=0,\dots,n-1$$ [23]Source: asksia-cheatsheet-math1061.pdfCALCULUS . Limits & squeeze . Derivative + rules . Standard derivatives . L'Hopital . Optimisation . FTC . Integration techniques WK 5 WK 9 SQUEEZE (SANDWICH) x MATH1061 Mathematics 1A UNIVERSITY OF SYDNEY . SCHOOL OF MATHEMATICS & STATISTICS EXAM REVISION Sem 1 2026 . SIDE 2 OF 2 Linear algebra · C -> eigenvalues 14 . Complex Numbers WK 1-2 ¡2 =- 1; z=a+ib, Re(z)=a, Im(z)=b (both real). RcC. ARITHMETIC (a+ib)+(c+id) = (a+c)+i(b+d) (a+ib) (c+id) = (ac-bd)+i(ad+bc) Conjugate ż=a-ib: z ż = a2+b2 E R , and zw = z. w, z+w = ż+ẅ. DIVISION - REALISE THE DENOMINATOR w/z = w ż / (z ż) = w ż / | z | 2 Equality: a+ib = c+id = a=c and b=d. Powers of i cycle: i, -1, -i, 1, i, . . . (period 4) - reduce the exponent mod 4. Division worked. (3+i)/(1-2i) . (1+2i)/(1+2i) = (3+6i+i-2)/(1+4) = (1+7i)/5 = 1/5 + (7/5)i . Quadratic over C. z2+z+1=0=z=(-1±/(-3))/2 =- 1/2 + (/3/2)i - a conjugate pair, the primitive cube roots of unity (#1). The quadratic formula works over C with / of a negative; the discriminant being negative is what forces complex roots. 15 . Modulus, Polar & Euler ARGAND PLANE Modulus |z|=/(a2+b2)=/(z z) = distance from 0; |z-w| = distance z+> w. |zw|=|z||w|, |z/w|=|z|/|w|, triangle ineq |z+w|≤|z|+|w|. Argument arg z=0 with z=|z|(cos0+i sin0); principal Arg ZE(-TI,Tt]. Read the quadrant from the diagram - not just tan-1(b/a). POLAR / EXPONENTIAL (EULER) e18 = cos0 + i sin8 = z = r e18 Z1Z2 = rir2 e1(81+82) Z1/Z2 = (r1/r2) ei(01-02) Euler's identity: e'" + 1 = 0. Multiplying = multiply moduli, add arguments. Useful identities: Re(z)=(z+z)/2, Im(z)=(z-z)/2i, z"1 = Z/|z|2 (|z|=1 = z-1=), and cos0=(e" +e-19)/2, sin0= (ei0-e-19)/2i. 15b . de Moivre & Roots WK 2 DE MOIVRE (cose + i sine)" = cos ne + i sin ne (r e18)n = pn eine N-TH ROOTS OF W = R E'+ ZK = R1/n exp(i(ų+2km)/n) k = 0,1, . . , n-1 (spaced 2m/n apart) Roots of unity (z"=1): e2Tik/n _ n points equally spaced on the unit circle. List all n roots (run k=0 . . . n-1).

陷阱

• 幅角要看象限，不是只算 $\tan^{-1}(b/a)$（材料提醒要读象限）。[23]Source: asksia-cheatsheet-math1061.pdfCALCULUS . Limits & squeeze . Derivative + rules . Standard derivatives . L'Hopital . Optimisation . FTC . Integration techniques WK 5 WK 9 SQUEEZE (SANDWICH) x MATH1061 Mathematics 1A UNIVERSITY OF SYDNEY . SCHOOL OF MATHEMATICS & STATISTICS EXAM REVISION Sem 1 2026 . SIDE 2 OF 2 Linear algebra · C -> eigenvalues 14 . Complex Numbers WK 1-2 ¡2 =- 1; z=a+ib, Re(z)=a, Im(z)=b (both real). RcC. ARITHMETIC (a+ib)+(c+id) = (a+c)+i(b+d) (a+ib) (c+id) = (ac-bd)+i(ad+bc) Conjugate ż=a-ib: z ż = a2+b2 E R , and zw = z. w, z+w = ż+ẅ. DIVISION - REALISE THE DENOMINATOR w/z = w ż / (z ż) = w ż / | z | 2 Equality: a+ib = c+id = a=c and b=d. Powers of i cycle: i, -1, -i, 1, i, . . . (period 4) - reduce the exponent mod 4. Division worked. (3+i)/(1-2i) . (1+2i)/(1+2i) = (3+6i+i-2)/(1+4) = (1+7i)/5 = 1/5 + (7/5)i . Quadratic over C. z2+z+1=0=z=(-1±/(-3))/2 =- 1/2 + (/3/2)i - a conjugate pair, the primitive cube roots of unity (#1). The quadratic formula works over C with / of a negative; the discriminant being negative is what forces complex roots. 15 . Modulus, Polar & Euler ARGAND PLANE Modulus |z|=/(a2+b2)=/(z z) = distance from 0; |z-w| = distance z+> w. |zw|=|z||w|, |z/w|=|z|/|w|, triangle ineq |z+w|≤|z|+|w|. Argument arg z=0 with z=|z|(cos0+i sin0); principal Arg ZE(-TI,Tt]. Read the quadrant from the diagram - not just tan-1(b/a). POLAR / EXPONENTIAL (EULER) e18 = cos0 + i sin8 = z = r e18 Z1Z2 = rir2 e1(81+82) Z1/Z2 = (r1/r2) ei(01-02) Euler's identity: e'" + 1 = 0. Multiplying = multiply moduli, add arguments. Useful identities: Re(z)=(z+z)/2, Im(z)=(z-z)/2i, z"1 = Z/|z|2 (|z|=1 = z-1=), and cos0=(e" +e-19)/2, sin0= (ei0-e-19)/2i. 15b . de Moivre & Roots WK 2 DE MOIVRE (cose + i sine)" = cos ne + i sin ne (r e18)n = pn eine N-TH ROOTS OF W = R E'+ ZK = R1/n exp(i(ų+2km)/n) k = 0,1, . . , n-1 (spaced 2m/n apart) Roots of unity (z"=1): e2Tik/n _ n points equally spaced on the unit circle. List all n roots (run k=0 . . . n-1).

B2. 向量：点积/投影/叉积（Vectors）

反复出现题链

• 点积 → 夹角/正交（材料点名）：[3]Source: asksia-bible-math1061-bilingual.pdfThe strategy this dictates 由此决定的策略 MATH1061 . Mathematics 1A ★ Quiz A - the closed-book MCQ in Week 8 Quiz A ––第8周的闭卷选择题 Held in the Week-8 Linear-Algebra tutorial: 40 minutes, 12 MCQ, 1 mark each, NO calculators, no notes, no extra paper, one correct answer per question. It samples the first half of BOTH streams - sets & functions, limits, differentiation, complex numbers, vectors, lines & planes, cross product. Exact-value arithmetic by hand under time is the whole test. There is no Special Consideration for the online quizzes (a better-mark rule covers them) - but Quiz A and the final are the marks you sit once. 在第 8周的线性代数习题课进行:40 分钟,12 道选择题, 每题1分,不许用计算器、不许带笔记、不许用额外草稿 纸,每题只有一个正确答案。它对两条主线的前半抽样 -- 集合与函数、极限、求导、复数、向量、直线与平 面、叉积。限时下手算精确值是全部考点。在线小测没有特 殊考量(由取较好成绩的规则覆盖) -- 但 Quiz A 和期末 是你只考一次的分数。 ✓ Drill the recurring chains, both streams 通刷反复出现的题链,两条主线 Every question is procedural: take a function, limit, matrix or complex number, apply the right technique, give the exact value. The chains that recur - Calculus: factor/cancel - limit; squeeze-bound - limit; differentiate + f'=0 - classify; Linear Algebra: realise denominator - complex division; dot product - angle/orthogonality; row- reduce - back-substitute; det(A-MI)=O - eigen. Show every line - method marks are real. Drill the chains and fresh numbers can't surprise you. 每道题都是程序化的:拿一个函数、极限、矩阵或复数,运 用正确的技巧,给出精确值。反复出现的题链 -- 微积 分:因式分解/约分→极限;夹逼定界→极限;求导→ f'=0→分类;线性代数:分母实数化→复数除法;点积 →夹角/正交;行化简→回代;det(A-入)=0→ 特征。写 出每一行 -- 步骤分是实打实的。把题链练熟,全新数字 便无法让你措手不及。 MATH1061 . Mathematics 1A CONTENTS - CONTENTS Both streams, one exam-ready book 两条主线,一本应考之书 Calculus first, then Linear Algebra - the order you actually meet them 先微积分,再线性代数 -- 按你实际遇到它们的顺序 Ch Topic Core methods Stream 1 . Calculus (Monday lectures) 1 Functions & limits function families . limit laws . squeeze . continuity . IVT → 2 Differentiation difference quotient . rules . chain . implicit . L'Hôpital → 3 Applications & curve sketching extrema . concavity · optimisation . MVT → 4 Taylor polynomials T_n about a . remainder . standard series → 5 Integration Riemann sum . FTC . substitution . parts . applications
• 投影公式（cheatsheet 直接写了）：
  $$\operatorname{proj}_u(v)=\left(\frac{u\cdot v}{|u|^2}\right)u$$ [16]Source: asksia-cheatsheet-math1061.pdfDO THIS IN ORDER · Eigen/diag: det(A-NI)=0 > > > eigenspaces > if any geom#alg, stop (not diag. ) -> else assemble P (columns), D (matching order) Solve Ax=b: row-reduce [A|b] > REF -> free vars > . back-sub · Invert A: [A||] -> RREF -> [I|A-1] 26 . High-Yield Traps DON'T LOSE MARKS · Integration: always +C; convert limits in substitution; reduce improper fractions before partial fractions · Projection uses ||u||2; cross product is R3-only & order- sensitive · (AB)-1, (AB)" reverse order; AB#BA; no matrix division Algebra Formula Belt SIDE 2 e10=cos0+i sine . (rei8)n=pleine n-th roots: R1/exp(i(+2km)/n) proju(v)=(u . v/llull2)u . Iluxvll=llulllvlsine 2×2-1 = (1/(ad-bc)) [d -b; - c a] det(A-AI)=0 . A=PDP-1 - Ak=pDKp-1 Σλ=tr Α . Πλ=det A SIDE 2/2 LINEAR ALGEBRA . Complex numbers . Polar / Euler / de Moivre . Vectors . Dot / cross / projection . Lines & planes . Matrices . Determinants · Eigenvalues . Diagonalisation REVISION SHEET . ALL TOPICS Compiled by AskSia . mapped to the MATH1061 syllabus . asksia. ai/cheatsheet/usyd-math1061 - WK 7 MATH1061 Mathematics 1A UNIVERSITY OF SYDNEY . SCHOOL OF MATHEMATICS & STATISTICS EXAM REVISION Sem 1 2026 . SIDE 1 OF 2 Calculus · limits -> series

两个高频陷阱（别白送分）

• 投影分母是 $|u|^2$，不是 $|u|$。[16]Source: asksia-cheatsheet-math1061.pdfDO THIS IN ORDER · Eigen/diag: det(A-NI)=0 > > > eigenspaces > if any geom#alg, stop (not diag. ) -> else assemble P (columns), D (matching order) Solve Ax=b: row-reduce [A|b] > REF -> free vars > . back-sub · Invert A: [A||] -> RREF -> [I|A-1] 26 . High-Yield Traps DON'T LOSE MARKS · Integration: always +C; convert limits in substitution; reduce improper fractions before partial fractions · Projection uses ||u||2; cross product is R3-only & order- sensitive · (AB)-1, (AB)" reverse order; AB#BA; no matrix division Algebra Formula Belt SIDE 2 e10=cos0+i sine . (rei8)n=pleine n-th roots: R1/exp(i(+2km)/n) proju(v)=(u . v/llull2)u . Iluxvll=llulllvlsine 2×2-1 = (1/(ad-bc)) [d -b; - c a] det(A-AI)=0 . A=PDP-1 - Ak=pDKp-1 Σλ=tr Α . Πλ=det A SIDE 2/2 LINEAR ALGEBRA . Complex numbers . Polar / Euler / de Moivre . Vectors . Dot / cross / projection . Lines & planes . Matrices . Determinants · Eigenvalues . Diagonalisation REVISION SHEET . ALL TOPICS Compiled by AskSia . mapped to the MATH1061 syllabus . asksia. ai/cheatsheet/usyd-math1061 - WK 7 MATH1061 Mathematics 1A UNIVERSITY OF SYDNEY . SCHOOL OF MATHEMATICS & STATISTICS EXAM REVISION Sem 1 2026 . SIDE 1 OF 2 Calculus · limits -> series
• 叉积只在 $\mathbb{R}^3$，而且有顺序（order-sensitive）。[16]Source: asksia-cheatsheet-math1061.pdfDO THIS IN ORDER · Eigen/diag: det(A-NI)=0 > > > eigenspaces > if any geom#alg, stop (not diag. ) -> else assemble P (columns), D (matching order) Solve Ax=b: row-reduce [A|b] > REF -> free vars > . back-sub · Invert A: [A||] -> RREF -> [I|A-1] 26 . High-Yield Traps DON'T LOSE MARKS · Integration: always +C; convert limits in substitution; reduce improper fractions before partial fractions · Projection uses ||u||2; cross product is R3-only & order- sensitive · (AB)-1, (AB)" reverse order; AB#BA; no matrix division Algebra Formula Belt SIDE 2 e10=cos0+i sine . (rei8)n=pleine n-th roots: R1/exp(i(+2km)/n) proju(v)=(u . v/llull2)u . Iluxvll=llulllvlsine 2×2-1 = (1/(ad-bc)) [d -b; - c a] det(A-AI)=0 . A=PDP-1 - Ak=pDKp-1 Σλ=tr Α . Πλ=det A SIDE 2/2 LINEAR ALGEBRA . Complex numbers . Polar / Euler / de Moivre . Vectors . Dot / cross / projection . Lines & planes . Matrices . Determinants · Eigenvalues . Diagonalisation REVISION SHEET . ALL TOPICS Compiled by AskSia . mapped to the MATH1061 syllabus . asksia. ai/cheatsheet/usyd-math1061 - WK 7 MATH1061 Mathematics 1A UNIVERSITY OF SYDNEY . SCHOOL OF MATHEMATICS & STATISTICS EXAM REVISION Sem 1 2026 . SIDE 1 OF 2 Calculus · limits -> series

B3. 方程组与矩阵：高斯消元 / RREF / 求逆

你必须能背出的流程（直接写在cheatsheet上）

• 解 $Ax=b$：行化简增广矩阵 $[A|b]$ → 到 REF → 识别自由变量 → 回代。[16]Source: asksia-cheatsheet-math1061.pdfDO THIS IN ORDER · Eigen/diag: det(A-NI)=0 > > > eigenspaces > if any geom#alg, stop (not diag. ) -> else assemble P (columns), D (matching order) Solve Ax=b: row-reduce [A|b] > REF -> free vars > . back-sub · Invert A: [A||] -> RREF -> [I|A-1] 26 . High-Yield Traps DON'T LOSE MARKS · Integration: always +C; convert limits in substitution; reduce improper fractions before partial fractions · Projection uses ||u||2; cross product is R3-only & order- sensitive · (AB)-1, (AB)" reverse order; AB#BA; no matrix division Algebra Formula Belt SIDE 2 e10=cos0+i sine . (rei8)n=pleine n-th roots: R1/exp(i(+2km)/n) proju(v)=(u . v/llull2)u . Iluxvll=llulllvlsine 2×2-1 = (1/(ad-bc)) [d -b; - c a] det(A-AI)=0 . A=PDP-1 - Ak=pDKp-1 Σλ=tr Α . Πλ=det A SIDE 2/2 LINEAR ALGEBRA . Complex numbers . Polar / Euler / de Moivre . Vectors . Dot / cross / projection . Lines & planes . Matrices . Determinants · Eigenvalues . Diagonalisation REVISION SHEET . ALL TOPICS Compiled by AskSia . mapped to the MATH1061 syllabus . asksia. ai/cheatsheet/usyd-math1061 - WK 7 MATH1061 Mathematics 1A UNIVERSITY OF SYDNEY . SCHOOL OF MATHEMATICS & STATISTICS EXAM REVISION Sem 1 2026 . SIDE 1 OF 2 Calculus · limits -> series
• 求逆 $A^{-1}$：对 $[A|I]$ 做 RREF → 得到 $[I|A^{-1}]$。[16]Source: asksia-cheatsheet-math1061.pdfDO THIS IN ORDER · Eigen/diag: det(A-NI)=0 > > > eigenspaces > if any geom#alg, stop (not diag. ) -> else assemble P (columns), D (matching order) Solve Ax=b: row-reduce [A|b] > REF -> free vars > . back-sub · Invert A: [A||] -> RREF -> [I|A-1] 26 . High-Yield Traps DON'T LOSE MARKS · Integration: always +C; convert limits in substitution; reduce improper fractions before partial fractions · Projection uses ||u||2; cross product is R3-only & order- sensitive · (AB)-1, (AB)" reverse order; AB#BA; no matrix division Algebra Formula Belt SIDE 2 e10=cos0+i sine . (rei8)n=pleine n-th roots: R1/exp(i(+2km)/n) proju(v)=(u . v/llull2)u . Iluxvll=llulllvlsine 2×2-1 = (1/(ad-bc)) [d -b; - c a] det(A-AI)=0 . A=PDP-1 - Ak=pDKp-1 Σλ=tr Α . Πλ=det A SIDE 2/2 LINEAR ALGEBRA . Complex numbers . Polar / Euler / de Moivre . Vectors . Dot / cross / projection . Lines & planes . Matrices . Determinants · Eigenvalues . Diagonalisation REVISION SHEET . ALL TOPICS Compiled by AskSia . mapped to the MATH1061 syllabus . asksia. ai/cheatsheet/usyd-math1061 - WK 7 MATH1061 Mathematics 1A UNIVERSITY OF SYDNEY . SCHOOL OF MATHEMATICS & STATISTICS EXAM REVISION Sem 1 2026 . SIDE 1 OF 2 Calculus · limits -> series

高频陷阱

• 牢记“陷阱配对”：REF vs RREF（材料点名是 trap pairs）。[4]Source: asksia-bible-math1061-bilingual.pdfDiagonalisation A = PDP-1 — P = eigenvectors, D = eigenvalues; possible iff geom = alg mult for every ). Matrix power via diagonalisation — Ak = PDk P-1, Dk = diag(}}) - the payoff for fast high powers. i How to drill this glossary 如何通刷这份术语表 Cover the meaning column and recite from the term; then flip and recall the English term from the +X. Pay special attention to the trap pairs - REF vs RREF, algebraic vs geometric multiplicity, dot vs cross product, and | u|| vs | u||2 in the projection. 盖住释义列,从术语回忆其含义;再反过来,从中文回忆英文术语。要特别留意陷阱配对 -- REF 与 RREF、代数重数与几何重 数、点积与叉积,以及投影中 |lull 与 llull2 的区别。 MATH1061 . Mathematics 1A — — — — — — — — — — — PRACTICE . Q1-Q4 - CHAPTER . PRACTICE BANK & WORKED SOLUTIONS DRILL TO EXAM STANDARD Drill the whole paper, both streams 通刷整张试卷,两条主线 Fourteen exam-style problems - calculus and linear algebra - each worked end to end on fresh numbers 十四道考试风格题目 -- 微积分与线性代数 -- 每题用全新数字从头算到尾 One-line takeaway. The MATH1061 final is computation-focused and samples both streams: a limit, a derivative/optimisation, an integral, a Taylor series and a complex-number question from calculus; a vector/plane, a Gaussian-elimination system, a determinant and an eigenvalue/eigenvector question from linear algebra. This bank gives one fresh problem per slot, fully worked. Cover the solution, do it by hand, then check.
• 没有矩阵除法；一般 $AB\neq BA$；$(AB)^{-1}$ 要反序。[16]Source: asksia-cheatsheet-math1061.pdfDO THIS IN ORDER · Eigen/diag: det(A-NI)=0 > > > eigenspaces > if any geom#alg, stop (not diag. ) -> else assemble P (columns), D (matching order) Solve Ax=b: row-reduce [A|b] > REF -> free vars > . back-sub · Invert A: [A||] -> RREF -> [I|A-1] 26 . High-Yield Traps DON'T LOSE MARKS · Integration: always +C; convert limits in substitution; reduce improper fractions before partial fractions · Projection uses ||u||2; cross product is R3-only & order- sensitive · (AB)-1, (AB)" reverse order; AB#BA; no matrix division Algebra Formula Belt SIDE 2 e10=cos0+i sine . (rei8)n=pleine n-th roots: R1/exp(i(+2km)/n) proju(v)=(u . v/llull2)u . Iluxvll=llulllvlsine 2×2-1 = (1/(ad-bc)) [d -b; - c a] det(A-AI)=0 . A=PDP-1 - Ak=pDKp-1 Σλ=tr Α . Πλ=det A SIDE 2/2 LINEAR ALGEBRA . Complex numbers . Polar / Euler / de Moivre . Vectors . Dot / cross / projection . Lines & planes . Matrices . Determinants · Eigenvalues . Diagonalisation REVISION SHEET . ALL TOPICS Compiled by AskSia . mapped to the MATH1061 syllabus . asksia. ai/cheatsheet/usyd-math1061 - WK 7 MATH1061 Mathematics 1A UNIVERSITY OF SYDNEY . SCHOOL OF MATHEMATICS & STATISTICS EXAM REVISION Sem 1 2026 . SIDE 1 OF 2 Calculus · limits -> series

B4. 行列式（Determinants）

必背性质与考试用法

• 用途：判断可逆（$\det A\neq 0$ 才可逆）。
• 材料给了高频性质提醒：$\det(cA)=c^n\det A$（$n\times n$）。[5]Source: asksia-bible-math1061-bilingual.pdf4 1 Let A = 2 . Find A-1 and use it to solve Ax = 1 2 ] Do CT ] ★ Exam-morning reminders - the recurring chains 考前提醒 -- 反复出现的题链 Calculus: factor/cancel for & limits; LIATE for parts; change limits in substitution; check Rn -> 0 before claiming a series. Linear algebra: projection divides by | u|2; det (cA) = c" det A; eigenvectors are nonzero; match P and D order. Always show working - method marks survive an arithmetic slip. 微积分:极限要因式分解约分;分部用 LIATE;换元要换积分限;断言级数收敛前先检验 lim an=0。线性代数:投影要除以 lull2; det(AB)=det A·det B; 特征向量非零;P与 D 的次序要对应。务必写出过程 -- 即便算术出错,步骤分仍能保住。 MATH1061 . Mathematics 1A AskSia Library VISUAL STUDY BIBLE . ASKSIA SCHOOL OF MATHEMATICS & STATISTICS SEMESTER 1 . 2026 0 3 r THE COMPLETE FIRST -YEAR BIBLE Mathematics 1A 数学 1A TWO STREAMS, ONE MARK - SINGLE-VARIABLE CALCULUS AND LINEAR ALGEBRA, DRILLED BY HAND ON FRESH NUMBERS. 悉尼大学 MATH1061 · 双语视觉精读 · LaTeX 公式排版 · 微积分 + 线性代数 · 期末 60% MATH1061 . THE UNIVERSITY OF SYDNEY 中英双语版 · BILINGUAL EDITION 英文主讲,中文随行 一 考试要点与术语保留英文原词 The final exam is 60% of your mark - the single biggest lever. This book covers both parallel streams: the Calculus half (functions - limits - differentiation - integration & Taylor) and the Linear Algebra half (complex numbers - vectors - systems - matrices - eigenvalues). Every definition is stated plainly, every method shown on a worked example with real arithmetic. Independent study companion. Not affiliated with or endorsed by The University of Sydney. Corrections: takedowns@asksia. ai PREFACE - HOW TO USE THIS BOOK Two streams, by hand, on fresh numbers 两条主线,纯手算,全新数字 Calculus and Linear Algebra run in parallel all semester - this book follows both 微积分与线性代数整学期并行推进 -- 本书两条线都跟 This is not a transcript of the lecture slides. It is a self-contained course in every technique MATH1061 examines, organised by the two streams you sit each week - Calculus (functions, limits, differentiation, integration, Taylor) and Linear Algebra (complex numbers, vectors, lines/planes, systems, matrices, eigenvalues). Each topic is built the same way so you always know where you are. 这并不是讲义幻灯片的誊录。它是一门自成体系的课程,涵盖 MATH1061 所考查的每一项技巧,按你每周分别上的两条主线组织 -- 微积分(函数、极限、求导、积分、泰勒)与线性代数(复数、向量、直线/平面、方程组、矩阵、特征值)。每个主题都以相同方式搭 建,让你始终清楚自己身在何处。 A 1 . LEARN
• 还提醒了：$\det(AB)=\det A\cdot \det B$。[5]Source: asksia-bible-math1061-bilingual.pdf4 1 Let A = 2 . Find A-1 and use it to solve Ax = 1 2 ] Do CT ] ★ Exam-morning reminders - the recurring chains 考前提醒 -- 反复出现的题链 Calculus: factor/cancel for & limits; LIATE for parts; change limits in substitution; check Rn -> 0 before claiming a series. Linear algebra: projection divides by | u|2; det (cA) = c" det A; eigenvectors are nonzero; match P and D order. Always show working - method marks survive an arithmetic slip. 微积分:极限要因式分解约分;分部用 LIATE;换元要换积分限;断言级数收敛前先检验 lim an=0。线性代数:投影要除以 lull2; det(AB)=det A·det B; 特征向量非零;P与 D 的次序要对应。务必写出过程 -- 即便算术出错,步骤分仍能保住。 MATH1061 . Mathematics 1A AskSia Library VISUAL STUDY BIBLE . ASKSIA SCHOOL OF MATHEMATICS & STATISTICS SEMESTER 1 . 2026 0 3 r THE COMPLETE FIRST -YEAR BIBLE Mathematics 1A 数学 1A TWO STREAMS, ONE MARK - SINGLE-VARIABLE CALCULUS AND LINEAR ALGEBRA, DRILLED BY HAND ON FRESH NUMBERS. 悉尼大学 MATH1061 · 双语视觉精读 · LaTeX 公式排版 · 微积分 + 线性代数 · 期末 60% MATH1061 . THE UNIVERSITY OF SYDNEY 中英双语版 · BILINGUAL EDITION 英文主讲,中文随行 一 考试要点与术语保留英文原词 The final exam is 60% of your mark - the single biggest lever. This book covers both parallel streams: the Calculus half (functions - limits - differentiation - integration & Taylor) and the Linear Algebra half (complex numbers - vectors - systems - matrices - eigenvalues). Every definition is stated plainly, every method shown on a worked example with real arithmetic. Independent study companion. Not affiliated with or endorsed by The University of Sydney. Corrections: takedowns@asksia. ai PREFACE - HOW TO USE THIS BOOK Two streams, by hand, on fresh numbers 两条主线,纯手算,全新数字 Calculus and Linear Algebra run in parallel all semester - this book follows both 微积分与线性代数整学期并行推进 -- 本书两条线都跟 This is not a transcript of the lecture slides. It is a self-contained course in every technique MATH1061 examines, organised by the two streams you sit each week - Calculus (functions, limits, differentiation, integration, Taylor) and Linear Algebra (complex numbers, vectors, lines/planes, systems, matrices, eigenvalues). Each topic is built the same way so you always know where you are. 这并不是讲义幻灯片的誊录。它是一门自成体系的课程,涵盖 MATH1061 所考查的每一项技巧,按你每周分别上的两条主线组织 -- 微积分(函数、极限、求导、积分、泰勒)与线性代数(复数、向量、直线/平面、方程组、矩阵、特征值)。每个主题都以相同方式搭 建,让你始终清楚自己身在何处。 A 1 . LEARN

B5. 特征值/特征向量/对角化（Eigen & Diagonalisation）

你必须会的“题链”（cheatsheet写得非常明确）

• 求特征值：
  $$\det(A-\lambda I)=0$$
  先解 $\lambda$，再求 eigenspaces（特征子空间）；若出现某个特征值的“几何重数 < 代数重数”，就不可对角化，直接停。[16]Source: asksia-cheatsheet-math1061.pdfDO THIS IN ORDER · Eigen/diag: det(A-NI)=0 > > > eigenspaces > if any geom#alg, stop (not diag. ) -> else assemble P (columns), D (matching order) Solve Ax=b: row-reduce [A|b] > REF -> free vars > . back-sub · Invert A: [A||] -> RREF -> [I|A-1] 26 . High-Yield Traps DON'T LOSE MARKS · Integration: always +C; convert limits in substitution; reduce improper fractions before partial fractions · Projection uses ||u||2; cross product is R3-only & order- sensitive · (AB)-1, (AB)" reverse order; AB#BA; no matrix division Algebra Formula Belt SIDE 2 e10=cos0+i sine . (rei8)n=pleine n-th roots: R1/exp(i(+2km)/n) proju(v)=(u . v/llull2)u . Iluxvll=llulllvlsine 2×2-1 = (1/(ad-bc)) [d -b; - c a] det(A-AI)=0 . A=PDP-1 - Ak=pDKp-1 Σλ=tr Α . Πλ=det A SIDE 2/2 LINEAR ALGEBRA . Complex numbers . Polar / Euler / de Moivre . Vectors . Dot / cross / projection . Lines & planes . Matrices . Determinants · Eigenvalues . Diagonalisation REVISION SHEET . ALL TOPICS Compiled by AskSia . mapped to the MATH1061 syllabus . asksia. ai/cheatsheet/usyd-math1061 - WK 7 MATH1061 Mathematics 1A UNIVERSITY OF SYDNEY . SCHOOL OF MATHEMATICS & STATISTICS EXAM REVISION Sem 1 2026 . SIDE 1 OF 2 Calculus · limits -> series
• 可对角化时组装：
  $$A=PDP^{-1}$$
  • $P$ 的列是特征向量
  • $D$ 的对角线是对应特征值
  • 顺序必须匹配（材料反复提醒）。[10]Source: asksia-bible-math1061-bilingual.pdfP-1 = 1 [3k + 1 3k - 17 5 4 2 3k -1 3k + 1 4 5 幂运算的回报:A^n=PD^nP-1 -- 例如 . . . 。 ! Eigenvectors are nonzero; match P and D order 特征向量非零;让 P 与 D 顺序匹配 An eigenvector must be nonzero (0 is never one). When building P and D, the k-th column of P must be an eigenvector for the k-th diagonal entry of D. A mismatch breaks A = PDP-1. Also: diagonalisable <> geometric mult = algebraic mult for every eigenvalue. 特征向量必须非零(0永远不是特征向量)。构造P与 D时,P的第 i列必须是 D第 i个对角元所对应的特征向量。错配会破坏 A=PDP-1。另外:可对角化 ⇒)每个特征值的几何重数=代数重数。 MATH1061 . Mathematics 1A - e. g. A2 = V. GLOSSARY . CALCULUS I - CHAPTER . GLOSSARY & KEY TERMS EN + 中文 Every examinable term, one line each 每个可考术语,每条一行 English term . X . crisp meaning - grouped by stream and topic 英文术语 · 中文 · 精炼释义 -- 按主线与主题分组 A fast reference for the vocabulary MATH1061 actually tests, across both streams - calculus and linear algebra. About 55 terms, each with a one-line meaning and small formula where it helps. Cover the right column and recite from the term; the +X column is filled during the bilingual pass. 一份针对 MATH1061 实际考查词汇的快速参考,横跨两条主线 -- 微积分与线性代数。约55个术语,每个配一行释义,有帮助处附 小公式。遮住右栏,对着术语背诵;中文一栏在双语整理阶段填入。 Term (EN) 中文 One-line meaning Part A - Calculus: functions, limits, derivatives Function f: A-+B — Rule giving each input x exactly one output f(x); has a domain, codomain and range. Natural domain All x for which the formula makes sense (no /O, no V of negatives, etc. ). Injective / surjective / bijective — One-to-one / onto / both; a bijection is invertible (Horizontal Line Test 'exactly once'). Inverse function f-1[16]Source: asksia-cheatsheet-math1061.pdfDO THIS IN ORDER · Eigen/diag: det(A-NI)=0 > > > eigenspaces > if any geom#alg, stop (not diag. ) -> else assemble P (columns), D (matching order) Solve Ax=b: row-reduce [A|b] > REF -> free vars > . back-sub · Invert A: [A||] -> RREF -> [I|A-1] 26 . High-Yield Traps DON'T LOSE MARKS · Integration: always +C; convert limits in substitution; reduce improper fractions before partial fractions · Projection uses ||u||2; cross product is R3-only & order- sensitive · (AB)-1, (AB)" reverse order; AB#BA; no matrix division Algebra Formula Belt SIDE 2 e10=cos0+i sine . (rei8)n=pleine n-th roots: R1/exp(i(+2km)/n) proju(v)=(u . v/llull2)u . Iluxvll=llulllvlsine 2×2-1 = (1/(ad-bc)) [d -b; - c a] det(A-AI)=0 . A=PDP-1 - Ak=pDKp-1 Σλ=tr Α . Πλ=det A SIDE 2/2 LINEAR ALGEBRA . Complex numbers . Polar / Euler / de Moivre . Vectors . Dot / cross / projection . Lines & planes . Matrices . Determinants · Eigenvalues . Diagonalisation REVISION SHEET . ALL TOPICS Compiled by AskSia . mapped to the MATH1061 syllabus . asksia. ai/cheatsheet/usyd-math1061 - WK 7 MATH1061 Mathematics 1A UNIVERSITY OF SYDNEY . SCHOOL OF MATHEMATICS & STATISTICS EXAM REVISION Sem 1 2026 . SIDE 1 OF 2 Calculus · limits -> series
• 矩阵高次幂（对角化收益点）：
  $$A^k=PD^kP^{-1}$$ [4]Source: asksia-bible-math1061-bilingual.pdfDiagonalisation A = PDP-1 — P = eigenvectors, D = eigenvalues; possible iff geom = alg mult for every ). Matrix power via diagonalisation — Ak = PDk P-1, Dk = diag(}}) - the payoff for fast high powers. i How to drill this glossary 如何通刷这份术语表 Cover the meaning column and recite from the term; then flip and recall the English term from the +X. Pay special attention to the trap pairs - REF vs RREF, algebraic vs geometric multiplicity, dot vs cross product, and | u|| vs | u||2 in the projection. 盖住释义列,从术语回忆其含义;再反过来,从中文回忆英文术语。要特别留意陷阱配对 -- REF 与 RREF、代数重数与几何重 数、点积与叉积,以及投影中 |lull 与 llull2 的区别。 MATH1061 . Mathematics 1A — — — — — — — — — — — PRACTICE . Q1-Q4 - CHAPTER . PRACTICE BANK & WORKED SOLUTIONS DRILL TO EXAM STANDARD Drill the whole paper, both streams 通刷整张试卷,两条主线 Fourteen exam-style problems - calculus and linear algebra - each worked end to end on fresh numbers 十四道考试风格题目 -- 微积分与线性代数 -- 每题用全新数字从头算到尾 One-line takeaway. The MATH1061 final is computation-focused and samples both streams: a limit, a derivative/optimisation, an integral, a Taylor series and a complex-number question from calculus; a vector/plane, a Gaussian-elimination system, a determinant and an eigenvalue/eigenvector question from linear algebra. This bank gives one fresh problem per slot, fully worked. Cover the solution, do it by hand, then check.

两个大陷阱（考试很爱扣）

• 特征向量必须非零（$0$ 永远不是特征向量）。[10]Source: asksia-bible-math1061-bilingual.pdfP-1 = 1 [3k + 1 3k - 17 5 4 2 3k -1 3k + 1 4 5 幂运算的回报:A^n=PD^nP-1 -- 例如 . . . 。 ! Eigenvectors are nonzero; match P and D order 特征向量非零;让 P 与 D 顺序匹配 An eigenvector must be nonzero (0 is never one). When building P and D, the k-th column of P must be an eigenvector for the k-th diagonal entry of D. A mismatch breaks A = PDP-1. Also: diagonalisable <> geometric mult = algebraic mult for every eigenvalue. 特征向量必须非零(0永远不是特征向量)。构造P与 D时,P的第 i列必须是 D第 i个对角元所对应的特征向量。错配会破坏 A=PDP-1。另外:可对角化 ⇒)每个特征值的几何重数=代数重数。 MATH1061 . Mathematics 1A - e. g. A2 = V. GLOSSARY . CALCULUS I - CHAPTER . GLOSSARY & KEY TERMS EN + 中文 Every examinable term, one line each 每个可考术语,每条一行 English term . X . crisp meaning - grouped by stream and topic 英文术语 · 中文 · 精炼释义 -- 按主线与主题分组 A fast reference for the vocabulary MATH1061 actually tests, across both streams - calculus and linear algebra. About 55 terms, each with a one-line meaning and small formula where it helps. Cover the right column and recite from the term; the +X column is filled during the bilingual pass. 一份针对 MATH1061 实际考查词汇的快速参考,横跨两条主线 -- 微积分与线性代数。约55个术语,每个配一行释义,有帮助处附 小公式。遮住右栏,对着术语背诵;中文一栏在双语整理阶段填入。 Term (EN) 中文 One-line meaning Part A - Calculus: functions, limits, derivatives Function f: A-+B — Rule giving each input x exactly one output f(x); has a domain, codomain and range. Natural domain All x for which the formula makes sense (no /O, no V of negatives, etc. ). Injective / surjective / bijective — One-to-one / onto / both; a bijection is invertible (Horizontal Line Test 'exactly once'). Inverse function f-1
• $P$ 的第 $k$ 列必须对应 $D$ 的第 $k$ 个对角元，错配会让 $A=PDP^{-1}$ 直接崩。[10]Source: asksia-bible-math1061-bilingual.pdfP-1 = 1 [3k + 1 3k - 17 5 4 2 3k -1 3k + 1 4 5 幂运算的回报:A^n=PD^nP-1 -- 例如 . . . 。 ! Eigenvectors are nonzero; match P and D order 特征向量非零;让 P 与 D 顺序匹配 An eigenvector must be nonzero (0 is never one). When building P and D, the k-th column of P must be an eigenvector for the k-th diagonal entry of D. A mismatch breaks A = PDP-1. Also: diagonalisable <> geometric mult = algebraic mult for every eigenvalue. 特征向量必须非零(0永远不是特征向量)。构造P与 D时,P的第 i列必须是 D第 i个对角元所对应的特征向量。错配会破坏 A=PDP-1。另外:可对角化 ⇒)每个特征值的几何重数=代数重数。 MATH1061 . Mathematics 1A - e. g. A2 = V. GLOSSARY . CALCULUS I - CHAPTER . GLOSSARY & KEY TERMS EN + 中文 Every examinable term, one line each 每个可考术语,每条一行 English term . X . crisp meaning - grouped by stream and topic 英文术语 · 中文 · 精炼释义 -- 按主线与主题分组 A fast reference for the vocabulary MATH1061 actually tests, across both streams - calculus and linear algebra. About 55 terms, each with a one-line meaning and small formula where it helps. Cover the right column and recite from the term; the +X column is filled during the bilingual pass. 一份针对 MATH1061 实际考查词汇的快速参考,横跨两条主线 -- 微积分与线性代数。约55个术语,每个配一行释义,有帮助处附 小公式。遮住右栏,对着术语背诵;中文一栏在双语整理阶段填入。 Term (EN) 中文 One-line meaning Part A - Calculus: functions, limits, derivatives Function f: A-+B — Rule giving each input x exactly one output f(x); has a domain, codomain and range. Natural domain All x for which the formula makes sense (no /O, no V of negatives, etc. ). Injective / surjective / bijective — One-to-one / onto / both; a bijection is invertible (Horizontal Line Test 'exactly once'). Inverse function f-1

3）整张卷“最容易白送丢分”的高收益陷阱清单（请逐条打钩）

• 极限：看到 $0/0$ 先因式分解约分；分母极限为0别用商法则。[13]Source: asksia-bible-math1061-bilingual.pdfEvery limit, derivative and integral this semester is a question about a function near a point or over an interval. The natural domain (where the formula makes sense) tells you where you are even allowed to ask the limit question - so pin it down before you compute anything. 本学期的每个极限、导数和积分都是关于某点附近或某区间上的函数的问题。自然定义域(公式有意义之处)告诉你在哪里才被允 许提出极限问题 -- 所以在计算任何东西之前先把它确定下来。 MATH1061 . Mathematics 1A C1 . LIMITS - THE LIMIT What f (x) approaches - not what it equals 趋近的是什么 -- 而非它等于什么 The single most important idea in the calculus stream 微积分主线中最重要的一个概念 lim f(x) = L means: we can force f(x) as close to L as we like, just by taking a close enough to a - but not equal to a. The value f (a) itself is irrelevant; it can even be undefined. 意思是:只要取得足够靠近,我们就能迫使任意接近 -- 但不等于。函数值本身无关紧要;它甚至可以无定义。 i The & idea, informally (full E-d is the Advanced course) 非正式的概念阐述(完整版在 Advanced 课程) Pick any tolerance & > 0 on the output. The limit is L if there is a window 8 > 0 on the input so that staying within o of a (but off a) keeps f(x) within & of L: 在输出端任取一个容差。极限是,如果在输入端存在一个窗口,使得保持在距内(但不等于)就能让保持在距内: 0 < |x -a| < 8 -> |f(x) - LE. 1. 3 The limit laws - build big limits from small ones 1. 3 极限定律–––由小极限搭建大极限 If limx-+a f and limx +a g both exist, limits pass straight through the arithmetic: 若与都存在,极限可直接穿过算术运算: lim (kf) = k lim f, lim (f + g) = lim f + lim g, 2-a lim(fg) =(lim f) (lim g), lim & = 17 limetaf f lim g + 0). 1 Try direct substitution first. If f is built from continuous families and a is in the domain, limx-+a f(x) = f(a) - just plug in. 先尝试直接代入。若函数由连续族构成且该点在定义域内,则极 限就等于函数值 -- 直接代入即可。 2 If you hit &, that is a signal, not an answer. Factor and cancel the common factor causing the zero, then substitute. 若得到 0/0,那是一个信号,而非答案。把导致零的公因式因式 分解并约去,然后再代入。 3 Never use the quotient law when the bottom limit is 0. Fix the form first. 当分母极限为 0时,绝不可使用商的极限法则。先把形式整理 好。 EX 1 A & rational limit no calculator
• 求导：链式法则别漏内层导数；商法则别写反号。[12]Source: asksia-bible-math1061-bilingual.pdf考试钟爱的两个法则陷阱 (1) Chain rule: after differentiating the outer function, you must multiply by the derivative of the inside. ₫ sin(x2) = cos(x2) . 2x, not just cos(x2). (2) Quotient rule sign & order: the numerator is f'g - fg' (top-derivative first), not fg' - f'g. Swapping them flips the whole sign. (1)链式法则:对外层函数求导后,你必须乘以内部的导数。,而不只是。(2)商法则的符号与顺序:分子是(分子导数在前),而 非。把它们互换会翻转整个符号。 EX 3. 2 Chain x product x quotient together composite Differentiate y = x2 e 32 COS x . 求导。 1 Top is a product. Let u = x2e3x. Then u' = 2x e3x + x2 . 3e3x = xe3x (2 + 3x) (product rule; the 3 is the chain factor from e3x). 分子是一个乘积。令 . . ,则由乘积法则(其中含因子来自对内层求导的链式因子)。 2 Bottom. v = cos x, v' = - sin x. 分母。 3 Quotient rule. y' = u'v - uv' 02 xe3x (2 + 3x) cos x + x2e3ª sin x cos2 x = 商的法则。 4 Tidy. Factor xe32: y' xe3x (2+ 3x) cos x + x sin x] . cos2 x 整理。提取公因子: MATH1061 . Mathematics 1A tangent EX 3. 3 Tangent line Find the tangent line to f(x) = x3 - 4x at x = 1. 求在处的切线。 1 Point. f(1) = 1-4 =- 3, so the point is (1, -3). 点。故该点为 …. . 。 Slope. f'(x) =3x2-4, so f'(1) =3-4 =- 1. 斜率。故 . . . 。 Assemble. y = f(1) + f'(1)(x-1) = - 3- (x-1), i. e. y =- 2-2. 组装。即得切线方程。 MATH1061 . Mathematics 1A 3 . DERIVATIVE 3. 3 The standard-derivative table - memorise cold
• 积分：不定积分永远写 $+C$；换元定积分要换上下限；分部积分按 LIATE 选 $u$。[6]Source: asksia-bible-math1061-bilingual.pdf1 Relation (Pythagoras). x2 + y2 = 132 = 169. At x = 5: y = 1169 - 25 = 12. 关系(勾股定理)。在该时刻: . 0 2 Differentiate in t. 2xdZ + 2y dy = 0. 对 t 求导。 3 Substitute x = 5, y = 12, d = 2. 2(5)(2) + 2(12). dy It = 0. 代入数值。 4 Solve. dy dt 246 5 m/s - the minus sign says the top is descending. = − 求解。负号说明顶端正在下降。 MATH1061 . Mathematics 1A 4 . INTEGRATION EXAM CORE Accumulation & area - differentiation run backwards 累积与面积 -- 把求导反过来运行 Antiderivatives . Riemann sums . the FTC . techniques . areas, volumes & improper integrals 原函数 · 黎曼和 · FTC · 技巧 · 面积、体积与反常积分 One big idea: integration does two things that turn out to be the same thing. It reverses differentiation (antiderivatives), and it measures the signed area under a graph (the definite integral, built as a limit of Riemann sums). The bridge between them - the most important theorem in the course - is the Fundamental Theorem of Calculus. 一个大思想:积分做两件事,而结果证明它们是同一件事。它反转求导(原函数),并度量图像下方的带符号面积(定积分,作为黎曼和 的极限构造)。二者之间的桥梁 -- 本课程最重要的定理 -- 就是微积分基本定理。 ★ What the exam asks here 考试在这里会问什么 Integration carries the back half of the calculus marks. Expect: (1) a definite integral by the FTC; (2) an integral by substitution (remember to change the limits!); (3) integration by parts (LIATE); (4) a partial-fractions integral; (5) area between two curves; (6) a volume of revolution (disc or shell); (7) an improper integral - converge or diverge? 积分占据微积分部分后半段的分数。预期会考:(1)用 FTC 计算定积分;(2)用换元法积分(记得换积分限!);(3)分部积分 (LIATE);(4)部分分式积分;(5)两曲线间的面积;(6)旋转体的体积(圆盘法或柱壳法);(7)反常积分 -- 收敛还是发散? 4. 1 Antiderivatives - the standard table 4. 1 原函数 ––标准表 If F' = f then [ f(x) dx = F(x) + C. The arbitrary constant C is non-negotiable - dropping it is the most common one- mark loss in the paper. 若则。任意常数不可省略 -- 丢掉它是整张试卷中最常见的一分损失。 f(x) § fdx 2ºn (n = - 1) 2n+1 -+ C n+1 2-1 In |x|+ C[17]Source: asksia-cheatsheet-math1061.pdf· Invert A: [A||] -> RREF -> [I|A-1] 26 . High-Yield Traps DON'T LOSE MARKS · Integration: always +C; convert limits in substitution; reduce improper fractions before partial fractions · Projection uses ||u||2; cross product is R3-only & order- sensitive · (AB)-1, (AB)" reverse order; AB#BA; no matrix division Algebra Formula Belt SIDE 2 e10=cos0+i sine . (rei8)n=pleine n-th roots: R1/exp(i(+2km)/n) proju(v)=(u . v/llull2)u . Iluxvll=llulllvlsine 2×2-1 = (1/(ad-bc)) [d -b; - c a] det(A-AI)=0 . A=PDP-1 - Ak=pDKp-1 Σλ=tr Α . Πλ=det A SIDE 2/2 LINEAR ALGEBRA . Complex numbers . Polar / Euler / de Moivre . Vectors . Dot / cross / projection . Lines & planes . Matrices . Determinants · Eigenvalues . Diagonalisation REVISION SHEET . ALL TOPICS Compiled by AskSia . mapped to the MATH1061 syllabus . asksia. ai/cheatsheet/usyd-math1061 - WK 7
• 反常积分：奇点处分段，两边都要收敛；不能靠相消。[11]Source: asksia-bible-math1061-bilingual.pdfx = lim 6-700 . b 1 x -2 z = Jim [ - ]]] 6-700 000 dx = lim b+00/1 1. ab 2-2 dr = lim -] 11b 3 Evaluate. = lima-+% (1 - }) = 1 - finite, so it converges. Check the test. Here p = 2 > 1, so the p-test predicts convergence V. 检验判别法。此处 p>1,故 p 判别法预言收敛 √。 - MATH1061 . Mathematics 1A 6->00 求值。 有限,故收敛。 dx converges p>1 ! Both pieces must converge at a singularity 在奇点处两部分都必须收敛 If the integrand blows up inside the interval (e. g. at x = 0 for f_ dz), split at the singularity and require each piece to converge separately. One divergent half makes the whole integral diverge - you cannot let cancellation rescue it. 若被积函数在区间内部发散(例如在某点处),要在奇点处分段,并要求每段各自收敛。任一段发散都会使整个积分发散 -- 不能 靠相消来补救。 i Chapter recap - the convergence gates 本章回顾 ––收敛的关卡 Two gates run this chapter. Series: the geometric Lar" converges iff |r| < 1; a Taylor series equals f iff Rn -> 0, checked via Lagrange. Integrals: the power test p > 1 (at ) or p < 1 (at O). Master those two gates and the whole topic is procedural. 本章由两道“关卡”主导。级数:几何级数当且仅当Irl<1时收敛;Taylor 级数当且仅当 Rn→0(用 Lagrange 余项检验)时等于函 数。积分:幂判别法(在 ∞处或在0处)。掌握这两道关卡,整个主题便是程式化的。 MATH1061 . Mathematics 1A ARITHMETIC WEEK 1 - COMPLEX ARITHMETIC One new symbol, then ordinary algebra 一个新符号,其余都是普通代数 Set i2 = - 1 and treat a + bi like a binomial - add, multiply, conjugate, divide 令 i2 =- 1,把 a+ bi 当作二项式处理––加、乘、取共轭、除
• 投影：分母是 $|u|^2$。[16]Source: asksia-cheatsheet-math1061.pdfDO THIS IN ORDER · Eigen/diag: det(A-NI)=0 > > > eigenspaces > if any geom#alg, stop (not diag. ) -> else assemble P (columns), D (matching order) Solve Ax=b: row-reduce [A|b] > REF -> free vars > . back-sub · Invert A: [A||] -> RREF -> [I|A-1] 26 . High-Yield Traps DON'T LOSE MARKS · Integration: always +C; convert limits in substitution; reduce improper fractions before partial fractions · Projection uses ||u||2; cross product is R3-only & order- sensitive · (AB)-1, (AB)" reverse order; AB#BA; no matrix division Algebra Formula Belt SIDE 2 e10=cos0+i sine . (rei8)n=pleine n-th roots: R1/exp(i(+2km)/n) proju(v)=(u . v/llull2)u . Iluxvll=llulllvlsine 2×2-1 = (1/(ad-bc)) [d -b; - c a] det(A-AI)=0 . A=PDP-1 - Ak=pDKp-1 Σλ=tr Α . Πλ=det A SIDE 2/2 LINEAR ALGEBRA . Complex numbers . Polar / Euler / de Moivre . Vectors . Dot / cross / projection . Lines & planes . Matrices . Determinants · Eigenvalues . Diagonalisation REVISION SHEET . ALL TOPICS Compiled by AskSia . mapped to the MATH1061 syllabus . asksia. ai/cheatsheet/usyd-math1061 - WK 7 MATH1061 Mathematics 1A UNIVERSITY OF SYDNEY . SCHOOL OF MATHEMATICS & STATISTICS EXAM REVISION Sem 1 2026 . SIDE 1 OF 2 Calculus · limits -> series
• 叉积：只在 $\mathbb{R}^3$ 且有顺序。[16]Source: asksia-cheatsheet-math1061.pdfDO THIS IN ORDER · Eigen/diag: det(A-NI)=0 > > > eigenspaces > if any geom#alg, stop (not diag. ) -> else assemble P (columns), D (matching order) Solve Ax=b: row-reduce [A|b] > REF -> free vars > . back-sub · Invert A: [A||] -> RREF -> [I|A-1] 26 . High-Yield Traps DON'T LOSE MARKS · Integration: always +C; convert limits in substitution; reduce improper fractions before partial fractions · Projection uses ||u||2; cross product is R3-only & order- sensitive · (AB)-1, (AB)" reverse order; AB#BA; no matrix division Algebra Formula Belt SIDE 2 e10=cos0+i sine . (rei8)n=pleine n-th roots: R1/exp(i(+2km)/n) proju(v)=(u . v/llull2)u . Iluxvll=llulllvlsine 2×2-1 = (1/(ad-bc)) [d -b; - c a] det(A-AI)=0 . A=PDP-1 - Ak=pDKp-1 Σλ=tr Α . Πλ=det A SIDE 2/2 LINEAR ALGEBRA . Complex numbers . Polar / Euler / de Moivre . Vectors . Dot / cross / projection . Lines & planes . Matrices . Determinants · Eigenvalues . Diagonalisation REVISION SHEET . ALL TOPICS Compiled by AskSia . mapped to the MATH1061 syllabus . asksia. ai/cheatsheet/usyd-math1061 - WK 7 MATH1061 Mathematics 1A UNIVERSITY OF SYDNEY . SCHOOL OF MATHEMATICS & STATISTICS EXAM REVISION Sem 1 2026 . SIDE 1 OF 2 Calculus · limits -> series
• 矩阵：$AB\neq BA$；$(AB)^{-1}$ 反序；别写“矩阵除法”。[16]Source: asksia-cheatsheet-math1061.pdfDO THIS IN ORDER · Eigen/diag: det(A-NI)=0 > > > eigenspaces > if any geom#alg, stop (not diag. ) -> else assemble P (columns), D (matching order) Solve Ax=b: row-reduce [A|b] > REF -> free vars > . back-sub · Invert A: [A||] -> RREF -> [I|A-1] 26 . High-Yield Traps DON'T LOSE MARKS · Integration: always +C; convert limits in substitution; reduce improper fractions before partial fractions · Projection uses ||u||2; cross product is R3-only & order- sensitive · (AB)-1, (AB)" reverse order; AB#BA; no matrix division Algebra Formula Belt SIDE 2 e10=cos0+i sine . (rei8)n=pleine n-th roots: R1/exp(i(+2km)/n) proju(v)=(u . v/llull2)u . Iluxvll=llulllvlsine 2×2-1 = (1/(ad-bc)) [d -b; - c a] det(A-AI)=0 . A=PDP-1 - Ak=pDKp-1 Σλ=tr Α . Πλ=det A SIDE 2/2 LINEAR ALGEBRA . Complex numbers . Polar / Euler / de Moivre . Vectors . Dot / cross / projection . Lines & planes . Matrices . Determinants · Eigenvalues . Diagonalisation REVISION SHEET . ALL TOPICS Compiled by AskSia . mapped to the MATH1061 syllabus . asksia. ai/cheatsheet/usyd-math1061 - WK 7 MATH1061 Mathematics 1A UNIVERSITY OF SYDNEY . SCHOOL OF MATHEMATICS & STATISTICS EXAM REVISION Sem 1 2026 . SIDE 1 OF 2 Calculus · limits -> series
• 特征：特征向量非零；$P$ 和 $D$ 顺序匹配。[10]Source: asksia-bible-math1061-bilingual.pdfP-1 = 1 [3k + 1 3k - 17 5 4 2 3k -1 3k + 1 4 5 幂运算的回报:A^n=PD^nP-1 -- 例如 . . . 。 ! Eigenvectors are nonzero; match P and D order 特征向量非零;让 P 与 D 顺序匹配 An eigenvector must be nonzero (0 is never one). When building P and D, the k-th column of P must be an eigenvector for the k-th diagonal entry of D. A mismatch breaks A = PDP-1. Also: diagonalisable <> geometric mult = algebraic mult for every eigenvalue. 特征向量必须非零(0永远不是特征向量)。构造P与 D时,P的第 i列必须是 D第 i个对角元所对应的特征向量。错配会破坏 A=PDP-1。另外:可对角化 ⇒)每个特征值的几何重数=代数重数。 MATH1061 . Mathematics 1A - e. g. A2 = V. GLOSSARY . CALCULUS I - CHAPTER . GLOSSARY & KEY TERMS EN + 中文 Every examinable term, one line each 每个可考术语,每条一行 English term . X . crisp meaning - grouped by stream and topic 英文术语 · 中文 · 精炼释义 -- 按主线与主题分组 A fast reference for the vocabulary MATH1061 actually tests, across both streams - calculus and linear algebra. About 55 terms, each with a one-line meaning and small formula where it helps. Cover the right column and recite from the term; the +X column is filled during the bilingual pass. 一份针对 MATH1061 实际考查词汇的快速参考,横跨两条主线 -- 微积分与线性代数。约55个术语,每个配一行释义,有帮助处附 小公式。遮住右栏,对着术语背诵;中文一栏在双语整理阶段填入。 Term (EN) 中文 One-line meaning Part A - Calculus: functions, limits, derivatives Function f: A-+B — Rule giving each input x exactly one output f(x); has a domain, codomain and range. Natural domain All x for which the formula makes sense (no /O, no V of negatives, etc. ). Injective / surjective / bijective — One-to-one / onto / both; a bijection is invertible (Horizontal Line Test 'exactly once'). Inverse function f-1

4）我建议你现在立刻做的“冲刺复习安排”（按最省命的顺序）

第1轮（把题链跑通，打底）

• 微积分：极限（代入/因式分解/夹逼）→ 求导（链式×乘积/商）→ 最优化（模板）→ 积分（FTC+换元+分部）→ 泰勒（会用已知级数）。[3]Source: asksia-bible-math1061-bilingual.pdfThe strategy this dictates 由此决定的策略 MATH1061 . Mathematics 1A ★ Quiz A - the closed-book MCQ in Week 8 Quiz A ––第8周的闭卷选择题 Held in the Week-8 Linear-Algebra tutorial: 40 minutes, 12 MCQ, 1 mark each, NO calculators, no notes, no extra paper, one correct answer per question. It samples the first half of BOTH streams - sets & functions, limits, differentiation, complex numbers, vectors, lines & planes, cross product. Exact-value arithmetic by hand under time is the whole test. There is no Special Consideration for the online quizzes (a better-mark rule covers them) - but Quiz A and the final are the marks you sit once. 在第 8周的线性代数习题课进行:40 分钟,12 道选择题, 每题1分,不许用计算器、不许带笔记、不许用额外草稿 纸,每题只有一个正确答案。它对两条主线的前半抽样 -- 集合与函数、极限、求导、复数、向量、直线与平 面、叉积。限时下手算精确值是全部考点。在线小测没有特 殊考量(由取较好成绩的规则覆盖) -- 但 Quiz A 和期末 是你只考一次的分数。 ✓ Drill the recurring chains, both streams 通刷反复出现的题链,两条主线 Every question is procedural: take a function, limit, matrix or complex number, apply the right technique, give the exact value. The chains that recur - Calculus: factor/cancel - limit; squeeze-bound - limit; differentiate + f'=0 - classify; Linear Algebra: realise denominator - complex division; dot product - angle/orthogonality; row- reduce - back-substitute; det(A-MI)=O - eigen. Show every line - method marks are real. Drill the chains and fresh numbers can't surprise you. 每道题都是程序化的:拿一个函数、极限、矩阵或复数,运 用正确的技巧,给出精确值。反复出现的题链 -- 微积 分:因式分解/约分→极限;夹逼定界→极限;求导→ f'=0→分类;线性代数:分母实数化→复数除法;点积 →夹角/正交;行化简→回代;det(A-入)=0→ 特征。写 出每一行 -- 步骤分是实打实的。把题链练熟,全新数字 便无法让你措手不及。 MATH1061 . Mathematics 1A CONTENTS - CONTENTS Both streams, one exam-ready book 两条主线,一本应考之书 Calculus first, then Linear Algebra - the order you actually meet them 先微积分,再线性代数 -- 按你实际遇到它们的顺序 Ch Topic Core methods Stream 1 . Calculus (Monday lectures) 1 Functions & limits function families . limit laws . squeeze . continuity . IVT → 2 Differentiation difference quotient . rules . chain . implicit . L'Hôpital → 3 Applications & curve sketching extrema . concavity · optimisation . MVT → 4 Taylor polynomials T_n about a . remainder . standard series → 5 Integration Riemann sum . FTC . substitution . parts . applications[6]Source: asksia-bible-math1061-bilingual.pdf1 Relation (Pythagoras). x2 + y2 = 132 = 169. At x = 5: y = 1169 - 25 = 12. 关系(勾股定理)。在该时刻: . 0 2 Differentiate in t. 2xdZ + 2y dy = 0. 对 t 求导。 3 Substitute x = 5, y = 12, d = 2. 2(5)(2) + 2(12). dy It = 0. 代入数值。 4 Solve. dy dt 246 5 m/s - the minus sign says the top is descending. = − 求解。负号说明顶端正在下降。 MATH1061 . Mathematics 1A 4 . INTEGRATION EXAM CORE Accumulation & area - differentiation run backwards 累积与面积 -- 把求导反过来运行 Antiderivatives . Riemann sums . the FTC . techniques . areas, volumes & improper integrals 原函数 · 黎曼和 · FTC · 技巧 · 面积、体积与反常积分 One big idea: integration does two things that turn out to be the same thing. It reverses differentiation (antiderivatives), and it measures the signed area under a graph (the definite integral, built as a limit of Riemann sums). The bridge between them - the most important theorem in the course - is the Fundamental Theorem of Calculus. 一个大思想:积分做两件事,而结果证明它们是同一件事。它反转求导(原函数),并度量图像下方的带符号面积(定积分,作为黎曼和 的极限构造)。二者之间的桥梁 -- 本课程最重要的定理 -- 就是微积分基本定理。 ★ What the exam asks here 考试在这里会问什么 Integration carries the back half of the calculus marks. Expect: (1) a definite integral by the FTC; (2) an integral by substitution (remember to change the limits!); (3) integration by parts (LIATE); (4) a partial-fractions integral; (5) area between two curves; (6) a volume of revolution (disc or shell); (7) an improper integral - converge or diverge? 积分占据微积分部分后半段的分数。预期会考:(1)用 FTC 计算定积分;(2)用换元法积分(记得换积分限!);(3)分部积分 (LIATE);(4)部分分式积分;(5)两曲线间的面积;(6)旋转体的体积(圆盘法或柱壳法);(7)反常积分 -- 收敛还是发散? 4. 1 Antiderivatives - the standard table 4. 1 原函数 ––标准表 If F' = f then [ f(x) dx = F(x) + C. The arbitrary constant C is non-negotiable - dropping it is the most common one- mark loss in the paper. 若则。任意常数不可省略 -- 丢掉它是整张试卷中最常见的一分损失。 f(x) § fdx 2ºn (n = - 1) 2n+1 -+ C n+1 2-1 In |x|+ C[20]Source: asksia-cheatsheet-math1061.pdfMATH1061 Mathematics 1A UNIVERSITY OF SYDNEY . SCHOOL OF MATHEMATICS & STATISTICS EXAM REVISION Sem 1 2026 . SIDE 1 OF 2 Calculus · limits -> series SIDE 1/2 Taylor / Maclaurin series 0 · Exam Blueprint READ FIRST * MATH1061 runs two parallel streams: this side is Calculus (limits > derivatives -> integrals -> series); flip for Linear Algebra . Assessment: weekly quizzes 8% · A1 5% . A2 10% · in-person Quiz A 15% . tutorials 2% . final exam 60%. Most-tested moves: 0/0 limits (factor or L'Hôpital); differentiate with chain + product/quotient; classify critical points; evaluate a definite integral via FTC + a technique (sub / parts / partial fractions); write a Maclaurin series and use it. Method marks: show the working - state the rule, then the substitution, then the answer. A dropped chain-rule factor or a missed +C is the standard mark- loss. -- SIA > Two reflexes: name the form before you compute (is it 0/0? is it a product?), and always check the hypotheses - L'Hôpital needs 0/0 or w/c, FTC needs continuity. 1 . Functions & Limits WK 1-2 Function f:A-> B, one output per input; range = image £ codomain. Injective (1-1), surjective (onto, range = codomain), bijective = both => f-1 exists. Composition (g · f)(x)=g(f(x)). Limit. limx> f(x)=L: f(x) is forced arbitrarily close to L by taking x close to (#) a. Two-sided limit exists iff both one-sided limits exist and agree. LIMIT LAWS (IF BOTH LIMITS EXIST) ALGEBRA OF LIMITS lim(kf)=k·lim f . lim(f+g)=lim f + lim g Lim(fg)=(lim f) (lim g) lim(f/g)=lim f / lim g only if lim g=0 If f is continuous at a, limx-> a f(x)=f(a) - so most limits are "plug in"; only the joints of piecewise functions need care. Never use the quotient law when the denominator limit is 0 - factor, rationalise or use L'Hôpital instead. 1b . Squeeze & Standard Limits ★ MEMORISE gsfsh near a, lim g = lim h = L = lim f = L STANDARD LIMITS Limx-@ sin x / x = 1 Limx-0 (1-cos x)/x = 0 Limx-co (1+1/x)X = e Classic squeeze: - |x| ≤ x. sin(1/x) ≤ |x| => limx>0x sin(1/x)=0. 1c . 0/0 Limits . Worked FACTOR FIRST limx=>1 (x2-1)/(x2+x-2):
• 线代：复数除法分母实数化→ 向量点积/投影→ 高斯消元→ 行列式可逆→ 特征值/对角化。[3]Source: asksia-bible-math1061-bilingual.pdfThe strategy this dictates 由此决定的策略 MATH1061 . Mathematics 1A ★ Quiz A - the closed-book MCQ in Week 8 Quiz A ––第8周的闭卷选择题 Held in the Week-8 Linear-Algebra tutorial: 40 minutes, 12 MCQ, 1 mark each, NO calculators, no notes, no extra paper, one correct answer per question. It samples the first half of BOTH streams - sets & functions, limits, differentiation, complex numbers, vectors, lines & planes, cross product. Exact-value arithmetic by hand under time is the whole test. There is no Special Consideration for the online quizzes (a better-mark rule covers them) - but Quiz A and the final are the marks you sit once. 在第 8周的线性代数习题课进行:40 分钟,12 道选择题, 每题1分,不许用计算器、不许带笔记、不许用额外草稿 纸,每题只有一个正确答案。它对两条主线的前半抽样 -- 集合与函数、极限、求导、复数、向量、直线与平 面、叉积。限时下手算精确值是全部考点。在线小测没有特 殊考量(由取较好成绩的规则覆盖) -- 但 Quiz A 和期末 是你只考一次的分数。 ✓ Drill the recurring chains, both streams 通刷反复出现的题链,两条主线 Every question is procedural: take a function, limit, matrix or complex number, apply the right technique, give the exact value. The chains that recur - Calculus: factor/cancel - limit; squeeze-bound - limit; differentiate + f'=0 - classify; Linear Algebra: realise denominator - complex division; dot product - angle/orthogonality; row- reduce - back-substitute; det(A-MI)=O - eigen. Show every line - method marks are real. Drill the chains and fresh numbers can't surprise you. 每道题都是程序化的:拿一个函数、极限、矩阵或复数,运 用正确的技巧,给出精确值。反复出现的题链 -- 微积 分:因式分解/约分→极限;夹逼定界→极限;求导→ f'=0→分类;线性代数:分母实数化→复数除法;点积 →夹角/正交;行化简→回代;det(A-入)=0→ 特征。写 出每一行 -- 步骤分是实打实的。把题链练熟,全新数字 便无法让你措手不及。 MATH1061 . Mathematics 1A CONTENTS - CONTENTS Both streams, one exam-ready book 两条主线,一本应考之书 Calculus first, then Linear Algebra - the order you actually meet them 先微积分,再线性代数 -- 按你实际遇到它们的顺序 Ch Topic Core methods Stream 1 . Calculus (Monday lectures) 1 Functions & limits function families . limit laws . squeeze . continuity . IVT → 2 Differentiation difference quotient . rules . chain . implicit . L'Hôpital → 3 Applications & curve sketching extrema . concavity · optimisation . MVT → 4 Taylor polynomials T_n about a . remainder . standard series → 5 Integration Riemann sum . FTC . substitution . parts . applications[16]Source: asksia-cheatsheet-math1061.pdfDO THIS IN ORDER · Eigen/diag: det(A-NI)=0 > > > eigenspaces > if any geom#alg, stop (not diag. ) -> else assemble P (columns), D (matching order) Solve Ax=b: row-reduce [A|b] > REF -> free vars > . back-sub · Invert A: [A||] -> RREF -> [I|A-1] 26 . High-Yield Traps DON'T LOSE MARKS · Integration: always +C; convert limits in substitution; reduce improper fractions before partial fractions · Projection uses ||u||2; cross product is R3-only & order- sensitive · (AB)-1, (AB)" reverse order; AB#BA; no matrix division Algebra Formula Belt SIDE 2 e10=cos0+i sine . (rei8)n=pleine n-th roots: R1/exp(i(+2km)/n) proju(v)=(u . v/llull2)u . Iluxvll=llulllvlsine 2×2-1 = (1/(ad-bc)) [d -b; - c a] det(A-AI)=0 . A=PDP-1 - Ak=pDKp-1 Σλ=tr Α . Πλ=det A SIDE 2/2 LINEAR ALGEBRA . Complex numbers . Polar / Euler / de Moivre . Vectors . Dot / cross / projection . Lines & planes . Matrices . Determinants · Eigenvalues . Diagonalisation REVISION SHEET . ALL TOPICS Compiled by AskSia . mapped to the MATH1061 syllabus . asksia. ai/cheatsheet/usyd-math1061 - WK 7 MATH1061 Mathematics 1A UNIVERSITY OF SYDNEY . SCHOOL OF MATHEMATICS & STATISTICS EXAM REVISION Sem 1 2026 . SIDE 1 OF 2 Calculus · limits -> series[23]Source: asksia-cheatsheet-math1061.pdfCALCULUS . Limits & squeeze . Derivative + rules . Standard derivatives . L'Hopital . Optimisation . FTC . Integration techniques WK 5 WK 9 SQUEEZE (SANDWICH) x MATH1061 Mathematics 1A UNIVERSITY OF SYDNEY . SCHOOL OF MATHEMATICS & STATISTICS EXAM REVISION Sem 1 2026 . SIDE 2 OF 2 Linear algebra · C -> eigenvalues 14 . Complex Numbers WK 1-2 ¡2 =- 1; z=a+ib, Re(z)=a, Im(z)=b (both real). RcC. ARITHMETIC (a+ib)+(c+id) = (a+c)+i(b+d) (a+ib) (c+id) = (ac-bd)+i(ad+bc) Conjugate ż=a-ib: z ż = a2+b2 E R , and zw = z. w, z+w = ż+ẅ. DIVISION - REALISE THE DENOMINATOR w/z = w ż / (z ż) = w ż / | z | 2 Equality: a+ib = c+id = a=c and b=d. Powers of i cycle: i, -1, -i, 1, i, . . . (period 4) - reduce the exponent mod 4. Division worked. (3+i)/(1-2i) . (1+2i)/(1+2i) = (3+6i+i-2)/(1+4) = (1+7i)/5 = 1/5 + (7/5)i . Quadratic over C. z2+z+1=0=z=(-1±/(-3))/2 =- 1/2 + (/3/2)i - a conjugate pair, the primitive cube roots of unity (#1). The quadratic formula works over C with / of a negative; the discriminant being negative is what forces complex roots. 15 . Modulus, Polar & Euler ARGAND PLANE Modulus |z|=/(a2+b2)=/(z z) = distance from 0; |z-w| = distance z+> w. |zw|=|z||w|, |z/w|=|z|/|w|, triangle ineq |z+w|≤|z|+|w|. Argument arg z=0 with z=|z|(cos0+i sin0); principal Arg ZE(-TI,Tt]. Read the quadrant from the diagram - not just tan-1(b/a). POLAR / EXPONENTIAL (EULER) e18 = cos0 + i sin8 = z = r e18 Z1Z2 = rir2 e1(81+82) Z1/Z2 = (r1/r2) ei(01-02) Euler's identity: e'" + 1 = 0. Multiplying = multiply moduli, add arguments. Useful identities: Re(z)=(z+z)/2, Im(z)=(z-z)/2i, z"1 = Z/|z|2 (|z|=1 = z-1=), and cos0=(e" +e-19)/2, sin0= (ei0-e-19)/2i. 15b . de Moivre & Roots WK 2 DE MOIVRE (cose + i sine)" = cos ne + i sin ne (r e18)n = pn eine N-TH ROOTS OF W = R E'+ ZK = R1/n exp(i(ų+2km)/n) k = 0,1, . . , n-1 (spaced 2m/n apart) Roots of unity (z"=1): e2Tik/n _ n points equally spaced on the unit circle. List all n roots (run k=0 . . . n-1).

第2轮（只抓“陷阱表”，能立刻涨分）

• 把上面第3部分的清单逐条做1道题并“故意不看答案手算”，练到不会漏。[5]Source: asksia-bible-math1061-bilingual.pdf4 1 Let A = 2 . Find A-1 and use it to solve Ax = 1 2 ] Do CT ] ★ Exam-morning reminders - the recurring chains 考前提醒 -- 反复出现的题链 Calculus: factor/cancel for & limits; LIATE for parts; change limits in substitution; check Rn -> 0 before claiming a series. Linear algebra: projection divides by | u|2; det (cA) = c" det A; eigenvectors are nonzero; match P and D order. Always show working - method marks survive an arithmetic slip. 微积分:极限要因式分解约分;分部用 LIATE;换元要换积分限;断言级数收敛前先检验 lim an=0。线性代数:投影要除以 lull2; det(AB)=det A·det B; 特征向量非零;P与 D 的次序要对应。务必写出过程 -- 即便算术出错,步骤分仍能保住。 MATH1061 . Mathematics 1A AskSia Library VISUAL STUDY BIBLE . ASKSIA SCHOOL OF MATHEMATICS & STATISTICS SEMESTER 1 . 2026 0 3 r THE COMPLETE FIRST -YEAR BIBLE Mathematics 1A 数学 1A TWO STREAMS, ONE MARK - SINGLE-VARIABLE CALCULUS AND LINEAR ALGEBRA, DRILLED BY HAND ON FRESH NUMBERS. 悉尼大学 MATH1061 · 双语视觉精读 · LaTeX 公式排版 · 微积分 + 线性代数 · 期末 60% MATH1061 . THE UNIVERSITY OF SYDNEY 中英双语版 · BILINGUAL EDITION 英文主讲,中文随行 一 考试要点与术语保留英文原词 The final exam is 60% of your mark - the single biggest lever. This book covers both parallel streams: the Calculus half (functions - limits - differentiation - integration & Taylor) and the Linear Algebra half (complex numbers - vectors - systems - matrices - eigenvalues). Every definition is stated plainly, every method shown on a worked example with real arithmetic. Independent study companion. Not affiliated with or endorsed by The University of Sydney. Corrections: takedowns@asksia. ai PREFACE - HOW TO USE THIS BOOK Two streams, by hand, on fresh numbers 两条主线,纯手算,全新数字 Calculus and Linear Algebra run in parallel all semester - this book follows both 微积分与线性代数整学期并行推进 -- 本书两条线都跟 This is not a transcript of the lecture slides. It is a self-contained course in every technique MATH1061 examines, organised by the two streams you sit each week - Calculus (functions, limits, differentiation, integration, Taylor) and Linear Algebra (complex numbers, vectors, lines/planes, systems, matrices, eigenvalues). Each topic is built the same way so you always know where you are. 这并不是讲义幻灯片的誊录。它是一门自成体系的课程,涵盖 MATH1061 所考查的每一项技巧,按你每周分别上的两条主线组织 -- 微积分(函数、极限、求导、积分、泰勒)与线性代数(复数、向量、直线/平面、方程组、矩阵、特征值)。每个主题都以相同方式搭 建,让你始终清楚自己身在何处。 A 1 . LEARN[16]Source: asksia-cheatsheet-math1061.pdfDO THIS IN ORDER · Eigen/diag: det(A-NI)=0 > > > eigenspaces > if any geom#alg, stop (not diag. ) -> else assemble P (columns), D (matching order) Solve Ax=b: row-reduce [A|b] > REF -> free vars > . back-sub · Invert A: [A||] -> RREF -> [I|A-1] 26 . High-Yield Traps DON'T LOSE MARKS · Integration: always +C; convert limits in substitution; reduce improper fractions before partial fractions · Projection uses ||u||2; cross product is R3-only & order- sensitive · (AB)-1, (AB)" reverse order; AB#BA; no matrix division Algebra Formula Belt SIDE 2 e10=cos0+i sine . (rei8)n=pleine n-th roots: R1/exp(i(+2km)/n) proju(v)=(u . v/llull2)u . Iluxvll=llulllvlsine 2×2-1 = (1/(ad-bc)) [d -b; - c a] det(A-AI)=0 . A=PDP-1 - Ak=pDKp-1 Σλ=tr Α . Πλ=det A SIDE 2/2 LINEAR ALGEBRA . Complex numbers . Polar / Euler / de Moivre . Vectors . Dot / cross / projection . Lines & planes . Matrices . Determinants · Eigenvalues . Diagonalisation REVISION SHEET . ALL TOPICS Compiled by AskSia . mapped to the MATH1061 syllabus . asksia. ai/cheatsheet/usyd-math1061 - WK 7 MATH1061 Mathematics 1A UNIVERSITY OF SYDNEY . SCHOOL OF MATHEMATICS & STATISTICS EXAM REVISION Sem 1 2026 . SIDE 1 OF 2 Calculus · limits -> series

第3轮（考前24小时：只做“整套跨两条主线”的题）

• 你的宝典里明确有“考试风格练习库（fresh numbers）”来对标题位：遮住解答、纯手算、再核对。[4]Source: asksia-bible-math1061-bilingual.pdfDiagonalisation A = PDP-1 — P = eigenvectors, D = eigenvalues; possible iff geom = alg mult for every ). Matrix power via diagonalisation — Ak = PDk P-1, Dk = diag(}}) - the payoff for fast high powers. i How to drill this glossary 如何通刷这份术语表 Cover the meaning column and recite from the term; then flip and recall the English term from the +X. Pay special attention to the trap pairs - REF vs RREF, algebraic vs geometric multiplicity, dot vs cross product, and | u|| vs | u||2 in the projection. 盖住释义列,从术语回忆其含义;再反过来,从中文回忆英文术语。要特别留意陷阱配对 -- REF 与 RREF、代数重数与几何重 数、点积与叉积,以及投影中 |lull 与 llull2 的区别。 MATH1061 . Mathematics 1A — — — — — — — — — — — PRACTICE . Q1-Q4 - CHAPTER . PRACTICE BANK & WORKED SOLUTIONS DRILL TO EXAM STANDARD Drill the whole paper, both streams 通刷整张试卷,两条主线 Fourteen exam-style problems - calculus and linear algebra - each worked end to end on fresh numbers 十四道考试风格题目 -- 微积分与线性代数 -- 每题用全新数字从头算到尾 One-line takeaway. The MATH1061 final is computation-focused and samples both streams: a limit, a derivative/optimisation, an integral, a Taylor series and a complex-number question from calculus; a vector/plane, a Gaussian-elimination system, a determinant and an eigenvalue/eigenvector question from linear algebra. This bank gives one fresh problem per slot, fully worked. Cover the solution, do it by hand, then check.[7]Source: asksia-bible-math1061-bilingual.pdf一句话要点。MATH1061 期末以计算为主,对两条主线都抽样:微积分出一道极限、一道导数/最优化、一道积分、一道泰勒级数和一 道复数题;线性代数出一道向量/平面、一道高斯消元方程组、一道行列式和一道特征值/特征向量题。本题库为每个题位提供一道全新 题目,完整解答。遮住解答,手算一遍,再核对。 ★ Fresh numbers, exam style - not the real stems 全新数字,考试风格 -- 并非真实题干 These are AskSia-authored questions written in the MATH1061 exam and Quiz-A style - they are not copied from any real paper. Give answers as exact values (fractions in lowest terms, exact surds/Tt) unless a decimal is asked for, and show every line - method marks are real. 这些是按 MATH1061 考试与 Quiz A 风格编写的 AskSia 原创题目 -- 并非抄自任何真实试卷。除非要求小数,答案应给出精确 值(最简分数、精确根式/π),并写出每一步 -- 步骤分是实打实的。 Q1-Q4 Calculus - limit, optimisation, parts, substitution Q1-Q4 微积分––极限、最优化、分部积分、换元 Q1 LIMIT 3 marks . Calculus Evaluate lim x2 - x-2 22 x2 - 4 . 02 OPTIMISATION 4 marks . Calculus A rectangle has perimeter 40. Maximise its area, classifying the stationary point. INTEGRATION BY PARTS Evaluate xe2ª dx. 3 marks . Calculus Q4 SUBSTITUTION Evaluate ∫ x 0 Vx2+1 =dx. 3 marks . Calculus MATH1061 . Mathematics 1A Q1-Q4 Worked solutions - calculus I 1 Q1. Factor: x2 - x -2 (x-2)(x+1) Cancel (x - 2) (the source of the g): lim2-+2 x+1 3 = = . x2 - 4 (x-2)(x+2) x+2 4 Q1. 因式分解: 约去(导致0/0的因子) :…。 2 Q2. Sides x and 20 - x (since 2x + 2y = 40). Area A(x) = x(20 - x) = 20x - x2. A'(x) = 20 -2x = 0=> x = 10. A" (x) = - 2 < 0, so it is a maximum: A = 10 . 10 = 100 (a square).

5）你告诉我两个信息，我就能把它“定制成你个人冲刺清单”

• 你 Final 还有几天？（我好给你排到每天做什么）
• 你最虚的是哪3块（从下面选）：
  • 微积分：极限 / 求导法则 / 最优化 / 积分技巧 / 反常积分 / 泰勒
  • 线代：复数极坐标与根 / 投影与夹角 / 高斯消元 / 行列式 / 特征值与对角化