微積分英文版教學(xué)課件：Chapter1 Limits and Continuity

1、Chapter1 Limits and Continuity1.1 Rates of Change and Limits The Tangent Problem Let f be a function and let P(a, f(a) be a point on the graph of f. To find the slope m of the tangent line l at P(a, f(a) on the graph of f, we first choose another nearby point Q(x, f(x) on the graph (see Figure 1) an

2、d then compute the slope mPQ of the secant line PQ.0P(a,f(a)Q(x,f(x)f(x)0P(a,f(a)Q(x,f(x)f(x)lLet Q get closer to P and QP. The slope m of the tangent line l is the limit of the slopes of the secant lines, i.eThe velocity problemSuppose an object moves along a straight line according to an equation

3、of motion is called the position function of the objectosAverage velocity The instantaneous velocity at t=a The Limit of a Function-110.5Definition We write and say “the limit of f(x), as x approaches a, equals L” meaning: If we can make the values of f(x) arbitrarily close to L(as close to L as we

4、like) by taking x to be sufficiently close to a( on either side of a) but not equal to a.An alternative notation foriswhich is usually read “f(x) approaches L as x approaches a” Example Guess the value of XSinx/x1.050.010.0050.0010.841470980.958851080.973545860.985067360.993346650.

5、998334170.999983390.999983330.999995830.99999983AxCaution: Notice the phrase “but xa” in the definition of limit. This means that in finding the limit of f(x) as x approaches a, we never consider x=a. In fact, f(x) need not even be defined when x=a. the only thing that matters is how f is defined ne

6、ar a.yAxThe limit value does not depend on how the function is defined at X0 !AxyLimits may fail to existThe Unit step function3) The Precise Definition of a LimitDefinition Let f be a function defined on some openinterval that contains that number a ,except possiblyat a itself. Then we say that the

7、 limit of f(x) as x approaches a is L, and we writeif for every number there is a number such thatwheneverifthenorifthensuch thatGeometric interpretation of limitLExampleProve that Proof1.Guessing a value for Let be a given positive number.wheneverThis suggests that we should choose2.Showing that th

8、is works.givenLetIf then Therefore , by the definition of a limit,ExampleProve that Proof1.Guessing a value for Let be a given positive number.IfthensoTherefor,there are two restrictions on namely and Letwhenever2.Showing that this works.givenLetIf then Therefore , by the definition of a limit,1.2 F

9、inding Limits and One-Sided LimitsCalculating Limits Using the Limit LawsLimit Laws Suppose that c is a constant and limits andexist. thenIf ProofLet be a given positive number.Sinceandboth exist, thenthere exists a number such thatwheneverSimilarly, since there exists a number such thatwheneverLetN

10、otice thatif thenTherefore , by the definition of a limit,Direct Substitution Property If f is a polynomial or a rational function and a is in the domain of f, then “Continuous” conceptExampleSolutionExampleSolutionExample: Eliminating zero denominators algebraicallySolutionIn general, if when xa, t

11、henExampleSolutionExampleSolutionTheorem when x is near a ( except possibly at a )and the limits of f and g both exist as x approachesa ,thenThe Squeeze Theorem (Sandwich Theorem)when x is near a (except possibly at a )andthen Example:exists? If so, find the value of and the value of the limit.Is th

12、ere a number such thatSolutionBecauseIf we know that the limitdoes not exist,it follows that the value of must be 0,namely, so One-Sided LimitsDefinition We write and say the left-hand limit of f(x) as x approaches a or the limit of f(x) as x approaches a from the left is equal to L if we can make t

13、he values of f(x) arbitrarily close to L by taking x to be sufficientlyclose to a and x less than a.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

14、by taking x to be sufficiently close to a and x be greeter than a.if and only ifandTheorem: Relation between one-sided and two-sided limits 1-111lim)(lim11lim)(lim0000-=-=+=-+）（）（Solution:xxfxxfxxxx(a)(b) Since the left and right limits are different, thus does not exist.(c)1-11SolutionExample Prove

15、 that does not exist.Since the right- and left- hand limits are different,It follows that does not exist.Solution:the largest integer that is less than or equal to xExample The greatest integer function is defined byShow that does not exist.Definition of Left-Hand Limitif for every number there is a

16、 number such thatwheneverDefinition of Right-Hand Limitif for every number there is a number such thatwheneverExampleProve that Proof1.Guessing a value for Let be a given positive number.wheneverThis suggests that we should choose2.Showing that this works.givenLetIf then Therefore , by the definitio

17、n of a limit, Proof First suppose that 0 x/2. Figure shows a sector with center 0, central angle x and radius 1. Notice that MP=sinx and AQ=tanx.From the figure, we see that Area of AOPArea of sector AOP Area of AOQ.Hence the preceding inequality may be rewritten as sinx/2x/20, we obtain cosxsinx/x1

18、. oPMAQxIf -/2x0, we also get cos(-x)sin(-x)/(-x)1 andHence cosxsinx/x0 is a rational number, thenIf r0 is a rational number such that xr defined for all x, thenExampleSolutionDividing both numerator and denominator by x3Definition The line y=L is called a horizontal asymptoteof the curve if eithero

19、rThe line y=0 is a horizontal asymptoteThe lines are horizontal asymptotesPrecise DefinitionsDefinition Let f be a function defined on (a, ). Thenmeans that for every there is correspondingnumber N such that wheneverDefinition Let f be a function defined on (-, a). Thenmeans that for every there is

20、correspondingnumber N such that wheneverExampleProve that Proof1.Guessing a value for Let be a given positive number.wheneverThis suggests that we should choose2.Showing that this works.givenLetIf then Therefore , by the definition of a limit,Definition Let f be a function define on both sides of a

21、,except possibly at a itself .Then means that the values of f(x) can be made arbitrarily large ( as large as we please) by taking x sufficiently close to a ,but not equal to a.An alternative notation forisInfinite LimitsThe expression is often read asf(x) becomes infinite as x approaches aor The lim

22、it of f(x), as x approaches a, is infinityor f(x) increases without bound as x approaches aDefinition 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 equa

23、l to a.One-Sided Infinite LimitsDefinition 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:Caution: Asymptotes need not be two-sided.The line x=0 is a vertical asymptotesThe line x=0 is a vertical asymptotesThe line is a vertical as

24、ymptotesFind the vertical asymptotes of ExampleSolutionSince Thus The line is a vertical asymptotesPrecise definition Infinite LimitsDefinition Let f be a function defined on some openinterval that contains that number a ,except possiblyat a itself. Thenmeans that for every positive number M there i

25、s a positive number such that ExampleProve that Proof1.Guessing a value for Let be a given positive number.wheneverThis suggests that we should choose2.Showing that this works.givenLetIf then Therefore , by the definition of a limit,Definition Let f be a function defined on some openinterval that co

26、ntains that number a ,except possiblyat a itself. Thenmeans that for every negative number N there is a positive number such that Infinite Limits at InfinityDefinition Let f be a function defined on (a, ). Thenmeans that for every there is correspondingnumber N such that wheneverInfinite Limits at I

27、nfinityEnd Behavior Model The function g is a right end behavior model for f if and only if 2) a left end behavior model for f if and only if Finding an Oblique (slanted) asymptoteis an asymptote of the graph of f1.4 ContinuityDefinition A function f is continuous at a number aiff is continuous at a

28、, then f(a) is defined (that is , a is in the domain of f) exists If one (or more ) of the three conditions in Definition is not satisfied, we say that f is discontinuous at a, or that f has a discontinuity at a.Definition A function f is continuous from the rightat a number a if and f is continuous

29、 from the left at a if The kind of discontinuity illustrated in part (1) is called a removable discontinuity.a(1) In general, if a function f is not continuous at a, then it has a removable discontinuity at a number a if the right-hand and the left-hand limits exist at a and are equal; aThe disconti

30、nuity in part (2) is called a jump discontinuity (2) a jump discontinuity at a if the two one-sided limits are not equal. aIf f approaches + or - as x approaches a from either side, as, for example, in part (3), we say that f has an infinite discontinuity at a.(3)(1)(2) 機(jī)動 目錄 上頁 下頁 返回 結(jié)束 f has a rem

31、ovable discontinuity atf has a jump discontinuity atf has a infinite discontinuity at x=0Definition A function f is continuous on an intervalif it is continuous at every number in the interval. Note: (If f is defined only on one side of an endpoint ofthe interval, we understand continuous at the end

32、point to mean continuous from the right or continuous from the left.)Theorem If f and g are continuous at a and c is a constant, then the following functions are also contiuous at a:Proof Since f and g are continuous at a, we haveThis shows thatis continuous at a.Theorem Any polynomial is continuous

33、 everywhere; that is , it is continuous on R=(-,+).(b) Any rational function is continuous wherever it isdefined; that is , it is continuous on its domain.Theorem The following types of function arecontinuous at every number in their domains: polynomials rational functions root functions trigonometr

34、ic functions inverse trigonometric functions exponential functions logarithmic functionsTheorem If f is continuous at b and then In other words,Example EvaluateExample Prove thatTheorem If g is continuous at a and f is continuous at g(a), then composite functiongiven byis continuous at a.Proof Since

35、 g is continuous at a, we haveSince f is continuous at b=g(a), thenThis shows thatis continuous at a.Example Where are the following functions continuous?Solution(1) f is continuous on its domain ,that is since f is a rational function.(2) f is continuous on since f is a polynomial. f is continuous

36、on since f is a polynomial. f is continuous on since f is a exponential function.The Intermediate Value Theorem Suppose that f iscontinuous 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 exists a number c in (a, b)such that f(c) =NNabf(a)f(b)f(

37、c1) = f(c2) = f(c3) = NoExampleShow that there is a root of the equationbetween 1 and 2.SolutionLetWe haveThus N=0 is a number between f(1) and f(2).Since f is continuous on 1,2,so the IntermediateValue Theorem says there is a number c between 1 and 2 such that f(c)=0.In other words, the equation ha

38、s at least one root c in(1,2).Exampleshow that there is a number c between 0 and 2 IfSuch thatSolutionLetWe haveThus N=0 is a number between g(0) and g(2).Since g is continuous on 0,2,so the IntermediateValue Theorem says there is a number c between 0 and 2 such that g(c)=0.That is 1.5 Tangent Lines

39、 The Tangent Problem Let f be a function and let P(a, f(a) be a point on the graph of f. To find the slope m of the tangent line l at P(a, f(a) on the graph of f, we first choose another nearby point Q(x, f(x) on the graph (see Figure 1) and then compute the slope mPQ of the secant line PQ.0P(a,f(a)

40、Q(x,f(x)f(x)0P(a,f(a)Q(x,f(x)f(x)lLet Q get closer to P and QP. The slope m of the tangent line l is the limit of the slopes of the secant lines,i.eLet Then So the slope of the second line PQ is The slope m of the tangent line l isThe velocity problemSuppose an object moves along a straight line acc

41、ording to an equation of motion is called the position function of the objectosAverage velocity Rates of changeIf x change from to ,then then change , The corresponding change in y isThe instantaneous velocity at t=a then the change in (increment of )isThe average rate of change of y with respect to

42、 xisThe instantaneous rate of change of y with respect to x at isDefinition of DerivativeDefinition Let y=f(x) be a function defined on an open interval containing a number a.The derivative of f(x) at number a, denoted by f(a) , is if this limit exists. If we write x=a+h, then h=x-a and h approaches

43、 0 if and only if x approaches a.thenThe right-hand derivativeof f at a ,is denoted by f +(a)The left-hand derivative of f at a , is denoted by f -(a).f (a) exists if and only if both the right-hand derivative f +(a) and the left-hand derivative f -(a) exist and are equal. If exists, we say that f(x

44、) is differentiable at a or that f(x) has a derivative at a. Interpretation of the Derivative as the Slope of a TangentThe tangent line to at is the line through whose slope is equal to ,the derivative of at a.0P(a, f(a)Q(x, f(x) f(x)lAn equation of the tangent line to the curve at the point : The geometric in