高等数学英文版课件PPT 01 Limits and Rates of Change

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1、 Chapter 0ne Limits and Rates of Changeup down return end 1.4 The Precise Definition of a Limit?)(limLxfax We know that it means f(x) is moving close to L while x is moving close to a as we desire. And it can reaches L as near as we like only on condition of the x is in a neighbor.(2) DEFINITION Let

2、 f(x) be a function defined on some open interval that contains the number a, except possibly at a itself. Then we say that the limit of f(x) as x approaches a is L, and we write , if for very number 0 there is a corresponding number 0 such that |f(x) - L| whenever 0|x - a|0, there exists a 0 such t

3、hat if all x that 0|x - a| then |f(x) - L| . Another notation for is f(x) L as x a.Lxfax)(limGeometric interpretation of limits can be given in terms of the graph of the functiony=L+ y=L - y=L a a-a+oxy=f(x)yup down return end Example 1 Prove that . 7)54(lim3xxSolution Let be a given positive number

4、, we want to find a positive number such that |(4x-5)-7| whenever 0|x-3|.But |(4x-5)-7|=4|x-3|. Therefore 4|x-3| whenever 0|x-3|.That is, |x-3| /4 whenever 0|x-3|0 there is a corresponding number 0 such that |f(x) - L| whenever 0 a - x , i.e, a - x 0 there is a corresponding number 0 such that |f(x)

5、 - L| whenever 0 x - a , i.e, a x 0 there is a corresponding number 0 such that f(x)M whenever 0 |x - a| .)(limxfaxup down return end .) 1(1lim21xxExample Prove that Example 5 Prove that(6)DEFINITION Let f(x) be a function defined on some open interval that contains the number a, except possibly at

6、a itself. Then we say that the limit of f(x) as x approaches a is infinity, and we write , if for very number N0 such that f(x)N whenever 0 |x - a|0 there is a corresponding number 0 such that |f(x) - f(a) | whenever |x - a| .Note that: (1) f(a) is defined)(limxfax(2) exists.up down return end Examp

7、le is discontinuous at x=2, since f(2) is not defined.22)(2xxxxfExample is continuous at x=2.22322)(2xxxxxxfExample Prove that sinx is continuous at x=a.(2) Definition A function f(x) is continuous from the right at every number a if A function f(x) is continuous from the left at every number a if)(

8、)(limafxfax)()(limafxfaxup down return end (2) Definition A function f(x) is continuous on an interval if it is continuous at every number in the interval. (at an endpoint of the interval we understand continuous to mean continuous from the right or continuous from the left)Example At each integer n

9、, the function f(x)=x is continuous from the right and discontinuous from the left.Example Show that the function f(x)=1-(1-x2) 1/2 is continuous on the interval -1,1.(4)Theorem If functions f(x), g(x) is continuous at a and c is a constant, then the following functions are continuous at a:1. f(x)+g

10、(x) 2. f(x)-g(x) 3. f(x)g(x) 4. f(x)g(x) -1 (g(a) isnt 0.)up down return end (5) THEOREM (a) any polynomial is continuous everywhere, that is, it is continuous on R1=(). (b) any rational function is continuous wherever it is defined, that is, it is continuous on its domain. Example Find.2553lim22xxx

11、x(6) THEOREM If n is a positive even integer, then f(x)= is continuous on 0, ). If n is a positive odd integer, then f(x)= is continuous on ().nxnxExample On what intervals is each function continuous?,453)(22xxxa.11111)(22xxxxxbup down return end (8) THEOREM If g(x) is continuous at a and f(x) is c

12、ontinuous at g(a) then (fog)(x)= f(g(x) is continuous at a .(7) THE INTERMEDIATE VALUE THEOREM Suppose that f(x) is continuous on the closed interval a,b. Let N be any number strictly between f(a) and f(b). Then there exists a number c in (a,b) such that f(c)=Nyxby=Na(7) THEOREM If f(x) is continuou

13、s at b and , then bg(x)axlim).lim()()(limg(x)fbf)xf(gaxaxup down return end Example Show that there is a root of the equation 4x3- 6x2 + 3x -2 =0 between 1 and 2.up down return end 1.6 Tangent, and Other Rates of ChangeA. Tangent(1) Definition The Tangent line to the curve y=f(x) at point P( a, f(a)

14、 is the line through P with slope provided that this limit exists.axafxfmax)()(limExample Find the equation of the tangent line to the parabola y=x2 at the point P(1,1).up down return end B. Other rates of changeThe difference quotient is called the average rate change of y with respect x over the i

15、nterval x1 , x2.1212)()(xxxfxfxy(4) instantaneous rate of change=at point P(x1, f(x1) with respect to x.12120)()(limlim12xxxfxfxyxxxSuppose y is a quantity that depends on another quantity x. Thus y is a function of x and we write y=f(x). If x changes from x1 and x2, then the change in x (also calle

16、d the increment of x) is x= x2 - x1 and the corresponding change in y is x= f(x2) - f(x1) .up down return end (1) what is a tangent to a circle? Can we copy the definition of the tangent to a circle by replacing circle by curve?1.1 The tangent and velocity problemsThe tangent to a circle is a line w

17、hich intersectsthe circle once and only once. How to give the definition of tangent line to a curve?For example,up down return end Fig. (a) In Fig. (b) there are straight lines which touch the given curve, but they seem to be different from the tangent to the circle. L2Fig. (b)L1up down return end L

18、et us see the tangent to a circle as a moving line to a certain line:So we can think the tangent to a curve is the line approached by moving secant lines.PQup down return end Q x mPQ 2 3 1.5 2.5 1.1 2.1 1.01 2.01 1.001 2.001Example 1: Find the equation of the tangent line to a parabola y=x2 at point

19、 (1,1). Q is a point on the curve.Q y=x2 Pup down return end Then we can say that the slope m of the tangent line is the limit of the slopes mQP of the secants lines. And we express this symbolically by writing mmQPPQlimAnd 211lim21xxxSo we can guess that slope of the tangent to the parabola at (1,1

20、) is very closed to 2, actually it is 2. Then the equation of the tangent line to the parabola isy-1=2(x-2) i.e y=2x-3.up down return end Suppose that a ball is dropped from the upper observation deck of the Oriental Pearl Tower in Shanghai, 280m above the ground. Find the velocity of the ball after

21、 5 seconds. elapsedtimetravelleddistancevelocityaverage From physics we know that the distance fallen after t seconds is denoted by s(t) and measured in meters, so we have s(t)=4.9t2. How to find the velocity at t=5? (2) The velocity problem:Solution up down return end So we can approximate the desi

22、red quantity by computing the average velocity over the brief time interval of the n-th of a second from t=5, such as, the tenth, twenty-th and so on. Then we have the table:Time interval Average velocity(m/s) 5t6 53.9 5t5.1 49.49 5t5.05 49.245 5t5.01 49.049 5t2 or x2), f(x) is close to 4. Then we c

23、an say that: the limit of the function f(x)=x2-x+2 as x approaches 2 is equal to 4. Then we give a notation for this :4)2(lim22xxxIn general, the following notation:(1) Definition: We write Lxfax)(limGuess the value of .11lim21xxxNotice that the function is not defined at x=1, and x1 f(x)0.5 0.66666

24、7 1.5 0.4000000.9 0.526316 1.1 0.4761900.99 0.502513 1.01 0.497512 0.999 0.500250 1.001 0.4997500.999. 0.500025 1.0001 0.499975Example 1up down return end and say “the limit of f(x), as x approaches a, equals L”.SolutionIf we can make the values of f(x) arbitrarily close to L (as close to L as we li

25、ke)by taking x to be sufficiently close to a but not equal to a. Sometimes we use notation f(x) L as x a. Example 1 Find459lim24tttExample 2 Findxxxsinlim0Notice that as x a which means that x approaches a, x may a and x may a.Example 3 Discuss , where )(lim0 xHx0100)(xifxifxHThe function H(x) appro

26、aches 0 as x approaches 0 and x0.So we can not say H(x) approaches a number asx a.up down return end One -side Limits:Even though there is no single number that H(x) approaches as t approaches 0. that is, does not exist.)(lim0 xHxBut as t approaches 0 from left, t0, H(x) approaches 1. Then we can in

27、dicate this situation symbolically by writing:)x(Hlimxup down return end We writeLxfax)(limAnd say the left-hand limit of f(x) as x approaches a (or the limit of f(x) as x approaches a from left) is equal to L. That is, we can make the value of f(x) arbitrarily close to L by taking x to be sufficien

28、tly close to a and x less than a.And say the right-hand limit of f(x) as x approaches a (or the limit of f(x) as x approaches a from right) is equal to L. That is, we can make the value of f(x) arbitrarily close to L by taking x to be sufficiently close to a and x greater than a.We writeLxfax)(limHe

29、re x a+ ” means that x approaches a and xa.(2)Definition: Here x a- ” means that x approaches a and x0)11.where n is a positive integer,up down return end Example 6. Calculate) 1743(lim233xxxx1025911lim5242xxxxx)2(102lim3331xxxx12344lim221xxxx11lim221xxxxx)(lim1xgx010011)(2xxxxxxgExample 1. FindExam

30、ple 2. FindExample 3. CalculateExample 4. CalculateExample 5. Calculatewhereup down return end If f(x) is a polynomial or rational function and a is in the domain of f(x), then .limf(a)f(x)ax(1) THEOREM if and only ifLxfax)(limLxfax)(lim)(limxfaxExample : Show that. 0|lim0 xxExample: If 8581)(xxxxxf

31、,determine whether exists. f(x)x8limExample: Prove thatxxx|lim0does not exists. Example: Prove that xx2limdoes not exists, where value of x is defined as the largest integer that is less than or equal to x.up down return end (2) THEOREM If f(x) g(x) for all x in an open interval that contains a (except possibly at a) and the limits of f and g exist as x approaches a, then.limlimg(x)f(x)axax(3)SQUEEZE THEOREM If f(x) g(x) h(x) for all x in an open interval that contains a (except possibly at a) and then,limlimLh(x)f(x)axax.limLg(x)axExample: Show that. 0)1arctan(lim0 xxxup down return end