Chapter0Preliminaries讲义.doc

Preliminaries Suggestion Your study likely follows such process: Listening teacher’s lecture, making notes ↓ Receiving; recalling; thoroughly thinking; reading references and notebook ↓ Questioning, inquiring ↓ Forming your own knowledge system ↓ Doing exercises ↓ Foreseeing Conceptual Structure of Mathematical Analysis Preliminaries ∣ Axioms of Real Number System ∣ Limit of Sequences Euler Constant Number e ∣ Limit and continuity of function Two important limits Functions continuous on closed interval Elementary functions’ continuity ∣ Differenciation Derivatives of functions Differential of functions ∣ Differential Mean-value Theorems Applications ∣ Converve of Derivatives or Differentials: indefinite integral ∣ Integration Definite integral of functions ∣ Newton-Leibniz theorem Preliminaries 1.Some familiar mathematical experiences 1.Length of a segment; Area of a rectangle; Length of a segment：L=|AB|=a Area of a triangle; Area of a parallelogram Rectangle(矩形) Triangle(三角形) Parallelogram(平行四边形) 2.Pythagorean Theorem In a right triangle △ABC，∠C=90°，the side AB opposite ∠C is called the hypotenuse, with length c; the sides of ∠C are called legs of the triangle, with length a and b, respectively. Pythagorean Theorem In any right triangle the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs; that is c2=a2+b2 Pf: Drawing a, perpendicular CD from C to AB. Label segments of AB as x and y, x+y=c. Since △ABC～△ACD～△CBD, ∴=,=, And cy=a? cx=b? c(x+y)= a?b?So c?a?b? . ngle Another proof. Drawing two squares both with side a+b Pf2: Drawing a perpendicular CD from C to AB, then Converse of the Pythagorean Theorem In a triangle, if the square of the length of one side equals the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle. The right angle is opposite the longest side.