微积分第二章ppt课件.ppt

Chapter 2 Limits and Derivatives2.1The tangent and velocity problems2.1.1 The tangent problemExample 1 Find an equation of the tangent line to the parabola at the point P(1,1).SOLUTION We will be able to find an equation of the tangent line t as soon as we know its slope m.The difficulty is that we know only one point,P,on t,whereas we need two points to compute the slope.But observe that we can compute an approximation to m by choosing a nearby Q(x,x2)on the parabola and computing the slope mPQ of the secant line PQ.1 We choose ,then .For instance,for the point Q(1.5,2.25)we have The closer Q is to P,the closer x is to 1 and the closer mPQ is to 2.This suggests that the slope of the tangent line t should be m=2.Figure 1PQ22.2 Limits of Functions2.2.1 Limit of a Function f(x)as x Approaches a Definition 1 Let f be a function defined on some open interval containing a except possibly at a itself,and let L be a real number.We say that the limit of f(x)as x approaches a is L,and writeand 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 by taking x to be sufficiently close to a(on either side of a)but not equal to a.3 In order to understand the precise meaning of a function in Definition,let us begin to consider the behavior of a function as x approaches 1.From the graph of f shown in Figure 2,we can intuitively see that as x gets closer to 1 from both sides but x1,f(x)gets closer to 3/2.In this case,we use the notation and say that the limit of f(x),as x approaches 1,is 3/2,or that f(x)Figure 2 /2 as x approaches 1.approaches 3。

4Example 2 Guess the value of .SOLUTION The function f(x)=sinx/x is not defined at x=0.From the table and the graph in Figure 3 we guess thatThis guess is in fact correct,as will be proved in Chapter 3.Figure 35Example 3 The Heaviside function H is defined byAs t approaches 0 from the left,H(t)approaches 0.As t approaches 0 from the right,H(t)approaches 1.There is no single number that H(t)approaches as t approaches 0.Figure 462.2.2 One-Sided LimitsDefinition 2 Let f be a function defined on an open interval of the form(a,c)for some real number c,and let L be a real number.We say that the right-hand limit of f(x)as x approaches a from the right is L,and write if we can make the values of f(x)arbitrarily close to L by taking x to be sufficiently close to a and x greater than a.7Similarly,we get definition of the right-hand limit of f(x)as x approaches a.Let f be a function defined on an open interval of the form(c,a)for some real number c,and let L be a real number.Wesay that the left-hand limit of f(x)as x approaches a from the left is L,and write if we can make the values of f(x)arbitrarily close to L by taking x to be sufficiently close to a and x less than a.8For example,Example 4 Use the graph of y=g(x)to find the following limits,if they exist.Solution This graph shows thatFigure 59For instance,since ,therefore does not exist.Example 5 Suppose that (1)Find and(2)Discuss Solution (1),.(2)Because ,so does not exist.10Example 6 Show that Solution Recall that We have Therefore,112.2.3 Infinite LimitsDefinition 3 Let f be a function defined on both sides of a,except possibly at a itself.Then means that the value of f(x)can be made arbitrarily large by taking x sufficiently close to a,but not equal to a.Example 6 Find if it exists.SOLUTION As x becomes close to 0,x2 also becomes close to 0,and 1/x2 becomes very large.(See the table on the next page.)12It appears from the graph of the function f(x)shown in Figure that the value of the f(x)can be made arbitrarily xlarge by taking x close enough to 0.ThusFigure 613Definition 4 Let f be a function defined on both sides of a,except possibly at a itself.Then means that the value of f(x)can be made arbitrarily large negative by taking x sufficiently close to a,but not equal to a.As an example we have Similar definitions can be given for the one-sided infinite limits14Examples of these four cases are given in Figure 7.Figure 715Definition 5 The line x=a is called a vertical asymptote of the curve y=f(x)if at least one of the following statements is true:For instance,the y-axis is a vertical asymptote of the curve y=1/x2 because 16Example 7 FindSolution If x is close to 3 but larger than 3,then the denominator x-3 is a positive number and 2x is close to 6.So the quotient 2x/(x-3)is a large positive number.Thus,we see that Likewise,if x is close to 3 but smaller than 3,then x-3 is a small negative number but 2x is still a positive number(close to 6).So 2x/(x-3)is a numerically large negative number.Thus 17The line x=3 is a vertical asymptote.Example 8 Find the vertical asymptotes of f(x)=tanx.Solution Because tanx=sinx/cosx,there are potential vertical asymptotes where cosx=0.In fact,we haveThis shows that the line is a vertical asymptote.Similar reasoning shows that the lines ,where n is an integer,are all vertical asymptotes of f(x)=tanx.182.3 Calculating Limits Using the Limit LawsLimit L。