1 The How, When, and Why of Mathematics Spotlight: George Polya Tips on Doing Homework 2 Logically Speaking 3 Introducing the Contrapositive and Converse 4 Set Notation and Quantifiers Tips on Quantification 5 Proof Techniques Tips on Definitions …… 10 Relations Tips on Reading Mathematics 11 Partitions Tips on Putting It All Together 12 Order in the Reals 13 Consequences of the Completeness of R Tips: You Solved ItNow What? 14 Functions, Domain, and Range Spotlight: The Definition of Function 15 Functions, One-to-One, and Onto 16 Inverses 17 Images and Inverse Images Spotlight: Minimum or Infimum? 18 Mathematical Induction 19 Sequences 20 Convergence of Sequences of Real Numbers 21 Equivalent Sets 22 Finite Sets and an Infinite Set 23 Countable and Uncountable Sets 24 The Cantor–Schroder–Bernstein Theorem Spotlight: The Continuum Hypothesis ……

❑ Review of CLT principles ❑ Reducing cognitive load in mathematics instruction ❑ Learner prior knowledge and instructional guidance ❑ Responding to alternative approaches ❑ Implications

Part 1: Fitting Curves to Data Part 2: Symbolic Mathematics Part 3: Calculus

Recursion-breaking an object down into smaller objects of the same typeis a ma- jor theme in mathematics and computer science. For example, in an induction proof we establish the truth of a statement()from the truth of the statement P(n-1). In pro- gramming, a recursive algorithm solves a problem by applying itself to smaller instances

THE MATHEMATICS OF NMR EX ons Trigonometric Functions Differentials and Integral Coordinate Transformations Convolutions Imaginary Numbers The Fourier transform tial Functions

一.符号对象的创建(Creating symbolic object) 1.创建符号变量和表达式(Creating symbolic variable and expression)创建符号变量和表达式的两个基本函数:sym,symsx=sym(x)创建一个符号变量x,可以是字符、字符串、表达式或字符表达式

实验3的基本内容 插值的基本原理 Experiments in Mathematics 三种插值方法:拉格朗日插值,分段线性 插值,三次样条插值

This 90 minutes lecture to be delivered via telelink addresses a class of mit undergraduates unrolled in Engineering Design and Rapid prototyping Course. The students have been introduced to the basic optimization concepts of design variables design space, objective function, constraints and numerical search methods. Building on this elementary basis, the purpose of the lecture is to make the students aware of a reaud range of theoretical and practical issues that arise when optimization is applied in real-life engineering. To that end a gamut of topics is presented with emphasis on the qualitative exposition rather than in-depth mathematics. The topics include

Chapter one Introduction: 1.1 Defining Microeconomics 1.2 Economic Theories and Models（经济理论与模型） 1.3 The Mathematics of Optimization（数学最优化） 1.4 Demand-Supply Model Chapter 2 Consumer Behavior and Demand Theory 2.1 Preference and Utility 2.2 Utility Maximization and Choice 2.3 Income and Substitution Effects 2.4 Market Demand and Elasticity

1 Case analysis The proof of a statement can sometimes be broken down into can be tackled individually 1.1 The method In order to prove a proposition P using case analysis Write, We use case analysis Identify a sequence of conditions, at least one of which must hold. (If this is not obvious, you must prove it