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Chapter 0ne Limits and Rates of Change

Chapter 0ne Limits and Rates of Change. up down return end.

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Chapter 0ne Limits and Rates of Change

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  1. Chapter 0ne Limits and Rates of Change up down returnend

  2. (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|< . 1.4 The Precise Definition of a Limit How to give mathematical description of 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. up down returnend

  3. Another notation for is f(x) L as x a. y=f(x) y y=L+ y=L y=L -  o x a- a a+ In the definition, the main part is that for arbitrarily >0, there exists a >0 such that if all x that 0<|x - a|< then |f(x) - L|<  . Geometric interpretation of limits can be given in terms of the graph of the function up down returnend

  4. Example 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|<. Example 2 Prove that Example 3 Prove that up down returnend

  5. (4)DEFINITION OF LEFT-SIDED LIMIT If for every number  >0 there is a corresponding number >0 such that |f(x) - L|<  whenever 0< a-x <, i.e, a- <x <a. (5)DEFINITION OF LEFT-SIDED LIMIT If for every number  >0 there is a corresponding number >0 such that |f(x) - L|<  whenever 0< x - a <, i.e, a<x <a + . Example 4Prove that Similarly we can give the definitions of one-sided limits precisely. Example 5Prove that up down returnend

  6. (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 M>0 there is a corresponding number >0 such that f(x)>M whenever 0< |x - a|< . Example 6 If prove that up down returnend

  7. (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 numberN<0 there is a corresponding number >0 such that f(x)<N whenever 0< |x - a|< . Example 5Prove that Example Prove that up down returnend

  8. Example Prove that Example Prove that Example Prove that Similarly, we can give the definitions of one-side infinite limits. up down returnend

  9. (1) Definition A function f(x) is continuous at a number a if (2) exists. . (3) 1.5 Continuity If f(x) not continuous at a, we say f(x) is discontinuous at a , or f(x) has a discontinuity at a . Note that: (1) f(a) is defined A function f(x) is continuous at a number a if and only if for every number >0 there is a corresponding number >0 such that |f(x) - f(a) |<  whenever |x - a|< . up down returnend

  10. Example is discontinuous at x=2, since f(2) is not defined. Example is continuous at x=2.. (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 Example Prove that sinx is continuous at x=a. up down returnend

  11. Example At each integer n, the function f(x)=[x] is continuous from the right and discontinuous from the left. (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 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) isn’t 0.) up down returnend

  12. (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 (). (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 Example On what intervals is each function continuous? up down returnend

  13. (7) THEOREM If f(x)is continuous at b and , then y y=N x a b (8) THEOREM If g(x)is continuous at a andf(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)=N up down returnend

  14. Example Show that there is a root of the equation 4x3- 6x2 + 3x -2 =0 between 1 and 2. up down returnend

  15. (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. 1.6 Tangent, and Other Rates of Change A. Tangent Example Find the equation of the tangent line to the parabola y=x2 at the point P(1,1). up down returnend

  16. The 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. B. Other rates of change 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 returnend

  17. 1.1 The tangent and velocity problems (1) what is a tangent to a circle? The tangent to a circle is a line which intersects the circle once and only once. How to give the definition of tangent line to a curve? Can we copy the definition of the tangent to a circle by replacing circle by curve? For example, up down returnend

  18. Fig. (a) Fig. (b) L1 L2 In Fig. (b) there are straight lines which touch the given curve,but they seem to be different from the tangent to the circle. up down returnend

  19. Let us see the tangent to a circle as a moving line to a certain line: Q P Q' So we can think the tangent to a curve is the line approached by moving secant lines. up down returnend

  20. Example 1: Find the equation of the tangent line to a parabola y=x2 at point (1,1). Q is a point on the curve. xmPQ 2 3 1.5 2.5 1.1 2.1 1.01 2.01 1.001 2.001 Q P y=x2 up down returnend

  21. 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 is y-1=2(x-2) i.e y=2x-3. 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 And up down returnend

  22. (2)The velocity problem: 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 5 seconds. Solution 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? up down returnend

  23. Time interval Average velocity(m/s) 5<t<6 53.9 5<t<5.1 49.49 5<t<5.05 49.245 5<t<5.01 49.049 5<t<5.001 49.0049 So 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:

  24. The above table shows us the results of similar calculations of average velocity over successively smaller time periods. It also appears that as time period tends to 0, the average velocity is becoming closer to 49. So the instantaneous velocity at t=5 is defined to be the limiting value of these average velocities over shorter time periods that start at t=5. up down returnend

  25. xf(x) xf(x) 1.0 2.000000 3.0 8.000000 1.5 2.750000 2.5 5.750000 1.8 3.440000 2.2 4.640000 1.9 3.710000 2.1 4.310000 1.95 3.852500 2.05 4.152500 1.99 3.970100 2.01 4.030100 1.995 3.985025 2.001 4.003001 1.2 The Limit of a Function Let us investigate the behavior of the function y=f(x)=x2-x+2 for values of x near 2. up down returnend

  26. We see that when x is close to 2(x>2 or x<2), 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:

  27. Guess the value of . (1) Definition: We write up down returnend and 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 (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 xa. Example 1 Solution Notice that the function is not defined at x=1, and x<1 f(x) x>1 f(x) 0.5 0.666667 1.5 0.400000 0.9 0.526316 1.1 0.476190 0.99 0.502513 1.01 0.497512 0.999 0.500250 1.001 0.499750 0.999. 0.500025 1.0001 0.499975

  28. Notice that as xa which means that x approaches a, x may >a and x may <a. Example 3Discuss , where xa. Example 1Find Example 2Find The function H(x) approaches 0 as x approaches 0 and x<0, and it approaches 1, as x approaches 0 and x>0. So we can not say H(x) approaches a number as up down returnend

  29. Even though there is no single number that H(x) approaches as t approaches 0. that is, does not exist. One -side Limits: But as t approaches 0 from left, t<0, H(x) approaches 0. Then we can indicate this situation symbolically by writing: But as t approaches 0 from right, t>0, H(x) approaches 1. Then we can indicate this situation symbolically by writing: up down returnend

  30. Here xa- ” means that x approaches a and x<a. Here xa+ ” means that x approaches a and x>a. (2)Definition: 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 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. 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 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. up down returnend

  31. y y=f(x) x a O b What will it happen as xaor xb? See following Figure: up down returnend

  32. x 1/x2 ±1 1 ±0.5 4 ±0.2 25 ±0.1 100 ±0,05 400 ±0,01 10000 ±0.001 1000000 y y=1/x2 x O (3)Theorem: if and only if Example: Find . up down returnend

  33. (4)DEFINITION: Let f be a function on both sides of a, except possibly at a itself. Then means that values of f(x) can be made arbitrarily large ( as we please) by taking x sufficiently close to a ( but not equal to a). The another notation for this is f(x) as xa, which is read as “the limit of f(x), as x approaches a, is infinity” or “f(x) becomes infinity as x approaches a” or “f(x) increases without bound as x approaches a” . To indicate the kind of behavior exhibited in this example, we use the notation: Generally we can give following Example Find up down returnend

  34. y y=f(x)=ln|x| x (5)DEFINITION: Let f be a function on both sides of a, except possibly at a itself.Then means that values of f(x) can be made arbitrarily large ( as we please) by taking x sufficiently close to a ( but not equal to a). The another notation for this is f(x) - as xa, which is read as “the limit of f(x), as x approaches a, is negative infinity” or “f(x) becomes negative infinite as x approaches a” or “f(x) decreases without bound as x approaches a” . Obviously f(x)=ln|x| becomes large negative as x gets close to 0. up down returnend

  35. Remember the meanings of xa- and xa+ . Similar definitions can be given for one-side infinite limits. (6)DEFINITION: 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: up down returnend

  36. Example Find and up down returnend

  37. and 1. 2. 3. 4 5. 1.3 Calculating limits using limit laws LIMIT LAWS Suppose that c is a constant and the limits exist. Then up down returnend

  38. (if n is even,we assume that ) where n is a positive integer, 6. 7. 8. where n is a positive integer, 9 10. where n is a positive integer, (if n is even, we assume that a>0) 11. where n is a positive integer, up down returnend

  39. Example 1. Find Example 2. Find Example 3. Calculate Example 4. Calculate Example 5. Calculate Example 6. Calculate where up down returnend

  40. If f(x) is a polynomial or rational function and a is in the domain of f(x), then (1) THEOREM if and only if Example : Show that Example: If ,determine whether exists. Example: Prove that does 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 returnend

  41. (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 then Example: Show that up down returnend

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