微積分第二章課件 Limits and Derivatives

1、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

2、 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

3、 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 ex

4、cept 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

5、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. I

6、n 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 gu

7、ess 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 O

8、ne-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 tak

9、ing 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

10、 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 insta

11、nce, 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

12、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

13、 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 l

14、arge 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) i

15、f 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 pos

16、itive 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

17、 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 Usi

18、ng the Limit LawsLimit Laws If limf(x) and limg(x) both exist, then1 limf(x)+g(x)=limf(x)+limg(x), limf(x)-g(x)=limf(x)-limg(x), 2 limf(x) g(x)=limf(x) limg(x),3 limcf(x)=climf(x), for any number c,4 limf(x) /g(x)=limf(x)/limg(x), provided limg(x)0.Example 1 Find Solution 19Example 2 If f is a polyn

19、omial function and a is a real number, then Solution Since f is a polynomial function, we may assume that for real numbers bn,bn-1, b0 and some positive integer n. Applying limit laws for a number of times yields that 20A rational function f is a ratio of two polynomials:where P and Q are polynomial

20、s. The domain consists of all values of x such that . Direct Substitution Property If f is a rational function and a is in the domain of f, then 21Example 3 Find each limit.Solution (1)(2) Since the limit laws can not be applied to the quotient However, 22Theorem 1 If when x is near a (except possib

21、ly at a) and the limits of f and g both exist as x approaches a, then Theorem 2 (The Squeeze Theorem) Suppose f(x)h(x)g(x) for every x in an open interval containing a, except possibly at a.(Sandwich Theorem or Pinching Theorem) 23Example 4 Show thatSolution First note that we cannot use because doe

22、s not exist. However, since We have and By the Squeeze Theorem, we know that Figure 1242.4 The Precise Definition of a LimitDefinition 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 approach

23、es a is L, and write if for every number 0, there is a number 0 such that 25Geometric interpretation of the limitFigure 126Definition 1 can be stated as follows:1 means that the distance between f(x) and L can be made arbitrarily small by taking the distance from x to a sufficiently small (but not 0

24、).2 means that the values of f(x) can be made as close as we please to and L by taking x close enough to a (but not equal to a).3 means that for every (no matter how small is) we can find such that if x lies in the open interval and , then f(x) lies in the open interval 27Example 1 Use Definition 1

25、to prove that(1) Where c is a real number.(2) Solution (1) Let f(x)=c. Since and since 0 is less than any 0, it follows from Definition 1 that f(x) has the limit c as x approaches a. Thus (2) Let f(x)=x. Since it is obvious that if chooseas ，then we obtain whenever It follows that 28Example 2 Prove

26、that Solution For every number 0, since to make it is sufficient to make Hence let =/2，then we obtain that This implies that29Definition 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) a

27、s x approaches a from the right is L, and write if for every number0, there is a number 0 such that 30Definition 3 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. We say that the left-hand limit of f(x) as x approaches a from the

28、 left is L, and write if for every number0, there is a number 0 such that 31Example 3 Prove that Solution 1. Guessing a value for . Let be a given positive number. We want to find a number such thatthat is,This suggests that we should choose2. Showing that this works. Given 0, letIf , then This show

29、s that 32Example 4 Prove that Solution1. Guessing a value for . Let 0 be given. We have tofind a number 0 such thatthat is,Notice that if we can find a positive constant C such that then and we can make by taking 33We can find such a number C if we restrict x to lie in some interval centered at 3. I

30、n fact, since we are interested only in values of x that are close to 3, it is reasonable to assume that x is within a distance 1 from 3, that is,Then so Thus, we haveand so C = 7 is a suitable choice for the constant. But now there are two restrictions on namely 34To make sure that both of these in

31、equalities are satisfied, wetake to be the smaller of the two numbers 1 and /7. The notation for this is 2. Showing that this works. Given 0, letIf , then We also have so This shows that 35Infinite LimitsDefinition 4 Let f be a function defined on some open interval that contains the number a, excep

32、t possibly at a itself. Thenmeans that for every positive number M there is a positive number such that 36Example 5 Prove thatSolution 1. Guessing a value for . Given M 0, we want to find 0 such thatThat is,This suggests that we should take2. Showing that this works. If M 0 is given, letIf , thenThe

33、refore, by Definition 4, 372.5 ContinuityDefinition 1 A function f is continuous at a number a if38 The kind of discontinuity illustrated in part (3) of Figure 1 is called a removable discontinuity because we could remove the discontinuity by redefining at c. The discontinuity in part (2) is called

34、a jump discontinuity because the function “jumps” from one value to another. If f approaches + or - as x approaches c from either side, as, for example, in part (1), we say that f has an infinite discontinuity at c.Figure 139 In general, if a function f is not continuous at c, then it has a removabl

35、e discontinuity at a number c if the right-hand and the left-hand limits exist at c and are equal; a jump discontinuity at c if the two one-sided limits are not equal. Example 1 Where are each of the following functions discontinuous?40Solution (a) Notice that f(2) is not defined, so f is discontinu

36、ous at 2. (b) We have So f is discontinuous at 0.(c) Here f(2)=1 is defined and But , so f is not continuous at 2. (d) The greatest integer function has discontinuities at all of the integers because does not exist if n is an integer.?41Figure 2 shows the graphs of the functions in Example 1.Figure

37、2(a)(c)(d)(b)。42Definition 2 A function f is continuous from the right at a number a if and f is continuous from the left at a if Example 2 At each integer n, the function is continuous from the right but discontinuous from the left because 43 Definition 3 A function f is continuous on an interval i

38、f it is continuous at every point in the interval. (If f is defined only on one side of an endpoint of the interval, we understand continuous at the endpoint to mean continuous from the right or continuous from the left.)Example 3 Show that the function is continuous on the interval -1, 1.Solution 4

39、4Thus, by Definition 1, f is continuous at a if -1 a 0 and y = arctanx is continuous on R. Thus, y = lnx+arctanx is continuous on (0, ). The denominator, y = x2-1, is a polynomial, so it is continuous everywhere. Therefore, f is continuous at all positive numbers x except where x2-1=0. So f is conti

40、nuous on the intervals (0, 1) and (1, ).48Theorem 7 If f is continuous at b and then In other words,Example 6 Evaluate Solution Since arcsin is a continuous function, we have 49Theorem 8 If g is continuous at a and f is continuous at g(a), then the composite function f(g(x) is continuous at a.Exampl

41、e 7 Where is the function continuous? Solution We know from the Theorem 6 that f(x) = lnx is continuous and g(x) = 1+cosx is continuous (because both y =1 and y = cosx are continuous). Therefore, by Theorem 8, F(x) = f(g(x) is continuous wherever it is defined. Now ln(1+cosx) is defined when 1+cosx

42、0. So it is undefined when x = , 3 ,. Thus, F has discontinuities when x is an odd multiple of and is continuous on the intervals between these values (see Figure 4).50Figure 4Theorem 9 If f(x) is one-to-one continuous function defined on interval, then its inverse function f -1 is also continuous.5

43、1The Intermediate Value Theorem 10 Suppose that f is continuous on the closed interval a, b and let N be any number between f(a) and f(b), where f(a) f(b). Then there is at least one number c in (a, b) such that f(c) = N.Some properties of continuous functions on a closed intervalTheorem 11 If a fun

44、ction f is continuous on a closed interval a, b, then there exists a positive number M such that f(x)M for every x in a, b.Theorem 12 If f is continuous on a closed interval a, b, then f attains a maximum value f() and a minimum value f() at some numbers and in a, b. 52The Intermediate Value Theorem

45、 states that a continuous function takes on every intermediate value between the function values f(a) and f(b). It is illustrated by Figure 5. Note that the value N can be taken on once or more than once.Figure 553Example 8 Show that there is a root of the equationbetween 1 and 2.Solution Let We are

46、 looking for a solution of the given equation, that is, a number c between 1 and 2 such that f(c) = 0. Therefore, we take a = 1, b = 2, and N = 0 in Theorem 10. We have f(1) = 4-6+3-2 = -1 0 Thus, f(1) 0 0 is a rational number, then If r 0 is a rational number such that xr is defined for all x, then

47、61Example 2 Find SolutionA similar calculation shows that the limit asis also 3/5. 62Example 3 Find the horizontal and vertical asymptotes of the graph of the function Solution Dividing both numerator and denominator by x and using the properties of limits, we haveTherefore, the line is a horizontal

48、 asymptote of the graph of f. 63In computing the limit as , we must remember that for x 0, there is a number m 0, we may assume that 0 1, so the domain of f(x) is (1, ). This is smaller than the domain of f(x), which is 1, ).Figure 278Other Notations Definition 2 A function f is differentiable at a

49、if f(a) exists. It is differentiable on an open interval (a, b) or (a, ) or (- , a) or (- , ) if it is differentiable at every point in the interval.79Example 3 Where is the function is differentiable?Solution If x 0, then and we can choose h small enough that x+h 0 and hence . Therefore, for x 0 we have and so f is differentiable for any x 0. Similarly, If x 0, we have and h can