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Chapter 0ne Limits and Rates of Changeup down return end up down return end 1 1.4 The Precise Definition of a Limit 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 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 that if all x that 0|x-a|then|f(x)-L|.Another notation for is f(x)L as x a.Geometric 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 In the definition,the mai3Example 1 Prove that Solution Let be a given positive number,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)-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|.up down return end Example 6 If prove that(6)6Example 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 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(2)exists.up down return end 1.5 Continuity If f(x)not co9Example is discontinuous at x=2,since f(2)is not defined.Example is continuous at x=2.Example 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 ifup down return end Example 10(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,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(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(2)Definition A function f(11(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(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().Example On what intervals is each function continuous?up down return end(5)THEOREM (a)any polynom12(8)THEOREM If g(x)is continuous at a and f(x)is continuous 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 continuous at b and ,then up down return end(8)THEOREM If g(x)is conti13Example Show that there is a root of the equation 4x3-6x2+3x-2=0 between 1 and 2.up down return end Example Show that there is a141.6 Tangent,and Other Rates of ChangeA.Tangent(1)Definition The Tangent line to the curve y=f(x)at point P(a,f(a)is the line through P with slope provided that this limit exists.Example Find the equation of the tangent line to the parabola y=x2 at the point P(1,1).up down return end 1.6 Tangent,and Other Rates 15B.Other rates of changeThe difference quotient is called the average rate change of y with respect x over the interval x1,x2.(4)instantaneous rate of change=at point P(x1,f(x1)with respect to x.Suppose 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 called the increment of x)is x=x2-x1 and the corresponding change in y is x=f(x2)-f(x1).up down return end B.Other rates of changeThe di16 (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 which intersectsthe circle once and only once.How to give the definition of tangent line to a curve?For example,up down return end (1)what is a tangent to a17 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 Fig.(a)In Fig.(b)there a18Let 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 QLet us see the tangent to a ci19 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(1,1).Q is a point on the curve.Q y=x2 Pup down return end 20Then 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 And So we can guess that slope of the tangent to the parabola at(1,1)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 Then we can say that the slope21Suppose 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 5 seconds.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 Suppose that a ball is dropped22So we can approximate the desired 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 can 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:In general,the following notation:We see that when x is close to26(1)Definition:We write Guess the value of .Notice that the function is not defined at x=1,and x1 f(x)0.5 0.666667 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 like)by taking x to be sufficiently close to a but not equal to a.Sometimes we use notation f(x)L as x a.(1)Definition:We write Gues27 Example 1 FindExample 2 FindNotice that as x a which means that x approaches a,x may a and x may a.Example 3 Discuss ,where The function H(x)approaches 0 as x approaches 0 and x0.So we can not say H(x)approaches a number asx a.up down return end Example 1 FindExample 2 Fi28One-side Limits:Even though there is no single number that H(x)approaches as t approaches 0.that is,does not exist.But as t approaches 0 from left,t0,H(x)approaches 1.Then we can indicate this situation symbolically by writing:up down return end One-side Limits:Even though t29We writeAnd 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 sufficiently 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 writeHere 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 (if n is even,we assume that 38Example 6.CalculateExample 1.FindExample 2.FindExample 3.CalculateExample 4.CalculateExample 5.Calculatewhereup down return end Example 6.CalculateExample 1.39If f(x)is a polynomial or rational function and a is in the domain of f(x),then(1)THEOREM if and only ifExample:Show thatExample:If ,determine whether exists.Example:Prove thatdoes not exists.Example:Prove that does not exists,where value of x is defined as the largest integer that is less than or equal to x.up down return end If f(x)is a polynomial or rat40(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(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 thenExample:Show thatup down return end(2)THEOREM If f(x)g(x)41
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