同济大学：《高等数学 Advanced Mathematics》课程教学资源（高数D，PPT课件）函数

Part One ●Introduction

Part One ⚫Introduction

The Aim of this Course Learn Advanced Mathematics OWhat is advanced mathematics OWhat is advanced mathematics doing OHow is mathematical thinking ●Learn English OWhat is saying in English about advanced mathematics

The Aim of this Course ⚫Learn Advanced Mathematics What is advanced mathematics What is advanced mathematics doing How is mathematical thinking ⚫Learn English What is saying in English about advanced mathematics

Calculus -Advanced Mathematics Different from High School Mathematics OCalculate v.s Analysis OStatic v.s Dynamic Different from Social Science OPhilosophy v.s Arts OLogical v.s Image

Calculus – Advanced Mathematics ⚫Different from High School Mathematics Calculate v.s Analysis Static v.s Dynamic ⚫Different from Social Science Philosophy v.s Arts Logical v.s Image

Method of Studying Advanced Mathematics o Understanding deeply ●Deduce step by step ● Exercise

Method of Studying Advanced Mathematics ⚫Understanding deeply ⚫Deduce step by step ⚫Exercise

Part Two ●Functions ●Limites ●Derivatives ●Integration o Differential Equations

Part Two ⚫Functions ⚫Limites ⚫Derivatives ⚫Integration ⚫Differential Equations

Functions ●Vhat is a function? OHow does a function play in Calculus? oWhat is any difference between the function in Calculus and the Mathematics in the Middle School?

Functions ⚫What is a function? ⚫How does a function play in Calculus? ⚫What is any difference between the function in Calculus and the Mathematics in the Middle School?

Basic Concepts ●Variables:x,y ●Value or image oRelation ●Determination ●Region

Basic Concepts ⚫Variables: x,y ⚫Value or image ⚫Relation ⚫Determination ⚫Region

Definition of Function If a variable y depends on a variablex in a such way that each value of x determines exactly one value of y,then we say thaty is a function ofx. OA function fis a rule that associates a unique output with each input.If the input is denoted by x,then the output is denoted byfx)(read“fofx")

Definition of Function ⚫If a variable y depends on a variable x in a such way that each value of x determines exactly one value of y, then we say that y is a function of x. ⚫A function f is a rule that associates a unique output with each input. If the input is denoted by x, then the output is denoted by f(x) (read “f of x”)

Represent of Function Numerically by tables Geometrically by Graphs oAlgebraically by formulas ●Verbally

Represent of Function ⚫Numerically by tables ⚫Geometrically by Graphs ⚫Algebraically by formulas ⚫Verbally

Example of Function 1 0 1 2 3 f(x) 3 4 -1 3 f(0)=3,f(1)=4, f(2)=-1,f(3)=3

Example of Function 1 x 0 1 2 3 f (x) 3 4 -1 3 (0) 3, (1) 4, (2) 1, (3) 3 f f f f = = = − =