微积分英文课件PPT (4).ppt

Chapter 2 Limits and Derivatives 2.1 The Tangent and Velocity Problems 2.2 The Limit of a Function 2.3 Calculating Limits Using the Limit Laws 2.4 The Precise Definition of a Limit 2.5 Continuity 2.6 Limits at Infinity: Horizontal Asymptotes 2.7 Tangents, Velocities, and Other Rates of Change ;2.8 Derivatives 2.9 The Derivative as a function;The Area Probrem;2.1 The Tangent and Velocity Problems;0;;The instantaneous velocity at t=a ;;-1;Definition We write and say “the limit of f(x), as x approaches a, equals L” If we can make the values of f(x) arbitrarily close to L(as close to L as we like) by taking x to be sufficientlyclose to a( on either side of a) but not equal to a.;Example Guess the value of ;;A;y;A;;Example Investigate ;ɡes] ;;'?sileit] ;One-Sided Limits;Definition We write and say the right-hand limit of f(x) as x approaches a [ or the limit of f(x) as x approaches a from the right ] is equal to L if we can make the values of f(x) arbitrarily close to L by taking x to be sufficiently close to a and x be greeter than a.;if and only if;1;(b) Since the left and right limits are different, thus ;;Infinite Limits;The expression is often read as;Definition Let f be a function define on both sides ofa ,except possibly at a itself .Then means that the values of f(x) can be made arbitrarily large negative by taking x sufficiently close to a ,but not equal to a.;;;;;The line x=0 is a vertical asymptotes;The line x=0 is a vertical asymptotes;The line is a vertical asymptotes;Find the vertical asymptotes of ;;;Exercises2.2;2.3 Calculating Limits Using the Limit Laws;;;Example;Example;Direct Substitution Property If f is a polynomial or a rational function and a is in the domain of f,then ;Example;Example;;Example;Example;Example;Solution;Solution;Theorem ;;;;;113/59;Exercises2.3;If we know that the limit;whenever;2.4 The Precise Definition o