托马斯微积分英文版课件-第二章.pptxVIP

? 2017 Pearson Education, Inc. All rights reserved Copyright ? 2010 Pearson Education, Inc. Publishing as Pearson Addison-WesleyChapter 2 Limits and Continuity Copyright ? 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley2.1Rates of Change and Tangents to Curves(Why study limit?)Other examples(Software/Hardware)? 1. IncrementsFor any variable u, we consider its increment △u. y = f(x): 2. (Dynamic Systems) Average rate of change and Instantaneous rate of change (1) Let y = f(x): (2) Instantaneous rate of change at x1? Instantaneous speed at x1? 3. Slopes of secant line and tangent line(1) Secant slope(2) Tangent slopeHow to get the tangent line? Copyright ? 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley2.2Limit of a Functionand Limit Laws Contents:1. Intuitive meaning of the limit of a function2. The limit operation rules: (1)+, (2) -, (3) *, (4) /, (5) ?. 3. The Sandwish Theorem 1. Intuitive meaning of the limit of a function: Suppose f(x) is defined on an open interval about x0, except possibly at x0 itself. If f(x) is arbitrarily close to L(as close to L as we wish) for all x sufficiently close to x0, we say that f(x) approaches to the limit L as x approaches x0, and we write 2. The Limit LawsLaws: Methods, RulesThe limit operation rules: (1)+, (2) -, (3) *, (4) /, (5) ?. How to prove (See Section 2.3)? Proof (Example 6 at P78)CautiousP80, 49: The limit Law of Composite Functions:(A?) Suppose that y = f(u) is defined on some interval containing u0, but not necessarily at u0, and and u = g(x) is defined on some interval containing x0 and g(x) ? u0, but not necessarily at x0, and Then Proof ?Other Cases? The limit Law of Composite Functions:(B) Suppose that y = f(u) is continuous at u0, and and u = g(x) is defined on some interval containing x0 but not necessarily at x0, and Then (See P94) Proof ? Some techniques:(1) (2) 3. The Sandw