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Calculus

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Title: Calculus Subject: Chapter 1 Author: Ming-Long Liu Created Date: 9/14/1996 10:56:36 AM Document presentation format: A4 (210x297 ) – PowerPoint PPT presentation

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Title: Calculus


1
??? ?????
  • ????
  • ??
  • ?????
  • ?????
  • ????
  • ????????
  • ????
  • ???????

2
?????
  • ????????????????????,????????????????,????????????
    ?????????????,?? (function)?????????
  • ??????????????????????????????????????????????????
    ?????????????????????,????????

1-1 ??
3
?????
  • ???????
  • ?????? x ?,?????? p x2 ???
  • ??????? x ? y ???????
  • ?????????

1-1 ??
4
?????
  • ??1-1? A ? B ??????????? A ???????,? B
    ???????????,????????,???? A ?? B ???????? A
    ????????(domain),B ????????(codomain)?

??? (independent variable)
??? (dependent variable)
??????? f(x) ?????,???? f ???(range)??
1-1 ??
5
?????
  • ?????????
  • ?????????
  • ???????????
  • ?? y2 x2 5 ?????????

1-1 ??
6
???????
  • ??????????????

1-1 ??
7
?????(Polynomial Function)
  • ? a0,a1,...,an ???,n ?????,???????????????????????
    ?
  • ?????????? R?a0,a1,...,an ???????????(coefficient)
    ?
  • ? an?0 ,?? an ? P(x) ?????(leading
    coefficient),?? P ? n ???? (polynomial of degree
    n)?

1-1 ??
8
?????
  • ????(constant function) f(x) a,a?0
    ,???????????? f(x) 0 ???????,????????
  • ???????????? (linear function),???? P(x) ax
    b,a?0?
  • ???????????? (quadratic function),???? P(x) ax2
    bx c,a?0?

1-1 ??
9
????? ???
  • ? p(x) ? q(x) ???????,???
  • ?????? (rational function) ???????? ?
    ???????????????
  • ? k ???0???,r ?????,??? f(x) kxr ????? (power
    function)????????? r ???????

1-1 ??
10
????? ? ????
  • ??
  • ???????(absolute value function),????? R?
  • ?????? x,??? x ??????? x ???????? 2.3
    2,-p - 4, 5 5 ???? f(x) x
    ??????(Gauss function)???????(greatest integer
    function),????? R?

1-1 ??
11
??????
  • ??????? (piecewise-defined function)
  • ? t -2,t 0 ? t ½ ???

1-1 ??
12
???????
  • ??1-2 ? f ? g ?????,??? f g,? f - g ,? f .g
    ?? f / g ??????

1-1 ??
13
????????????
  • ? ,g(x) 2x 1?? f g,f
    - g, f .g ?? f / g ???? x 2 ??????????????

1-1 ??
14
????(Composition Function)
  • ??1-3? f ? g ?????,? f ? g ?????,? f?g ??,???
  • ????? g ?????????g(x) ? f ?????????? x ??????
  • ?????
  • ? f(x) (x 1)2,g(x) 3x1?????????
  • (1) f?g (2) g?f (3) f?f (4) g?g

1-1 ??
15
??????????
  • ??? ,
    ??????? g?f ??????

1-1 ??
16
?????????
  • ? ,
    ?????? f?g?????

1-1 ??
17
??????
  • ???,??????????????????
  • ??? h(x) (x2 1)10 3 ?????????????

1-1 ??
18
?????
  • ??1-4 ????? f(x),????? A,???????,????? (x, f(x))
    ?????,?? ,???? f(x) ???,??
  • ??????????
  • ??1 ??? f ?????????? x,????? f?????? f(x) ?
  • ??2 ??????????? (x, f(x)) ?
  • ??3 ???2???????????????

1-2 ?????
19
?????????
  • ???? f(x) x3 ????

1-2 ?????
20
????????
  • ???? ????

1-2 ?????
21
??????????
  • ????
    ????

1-2 ?????
22
??????
  • ??????????????????,????????? (vertical-line test)?

1-2 ?????
23
???????
  • ????????????????
  • ???? (cost function)
  • ???? (revenue function)
  • ???? (profit function)?
  • ???????? (supply) ??? (demand) ????
  • ? x ?????????????????,???
  • C(x) ?? x ?????????
  • R(x) ?? x ?????????
  • P(x) ????? x ?????????

1-3 ???????
24
???
  • ?????? x ???????(???)?
  • ????????6??
  • ??????
  • ?????10???????????
  • ?????2???????????

1-3 ???????
25
???
  • ??????
  • ?????10???????????
  • ?????2???????????

1-3 ???????
26
????
  • ? P(x) 0 ?,????????? R(x) C(x) ? x????????
    (break-even quantity)?y R(x) ? y C(x)
    ????????????? (break-even point)????????,?????????
    ,??????????

1-3 ???????
27
??????
  • ??????????,???? x ??????? C(x) 5x2 40x
    600????? x ??????? R(x) 400x - 25x2?????????

1-3 ???????
28
????
  • ???? (demand function)?????????? p
    ?????????????(???????),x ??????,? x D(p)
    ?????????,???????????(???,????,??????)???????????
    ?(????,?????,???????)?
  • ??????????

1-3 ???????
29
?????????????
  • ????????????? x D(p) -30p8100?? p ?????,D(p)
    ????? p ??????????
  • ????200??,???????
  • ?????2??,?????????
  • ??? D(p) ?????

1-3 ???????
30
????
  • ???? (supply function)?????????? p
    ??????????????????(??????????),x ??????? x
    S(p) ?????????,?????????????????????,??????,?????
    ?,???????
  • ??????????

1-3 ???????
31
?????????
  • ?????????????(???)????? x S(p) 50p-150
  • (1) ????4??,????????
  • (2) ??????????,??????????????

1-3 ???????
32
????
  • ??????,???(????)??,?????,??????,?????,????????????
    ? (pe, xe) ????? (equilibrium point),???????? pe
    ?????? (equilibrium price)?
  • ???????, ?????????????????????,???? xe,??????
    (equilibrium quantity)?
  • ??????????,????????,??,???????????????????????,???
    ?????,???????????

1-3 ???????
33
????
1-3 ???????
34
?????????????
  • ?????????????????x D(p) 590 - 5p,x S(p)
    p2 3p - 70 ?
  • (1)????? (2)???? (3)????

1-3 ???????
35
?????
  • ???????????????????????????????--??????
  • ??????????? x ??????? a ?,?? f(x) ???????? f(a)
    ?????,?? f(a) ? x a ??????
  • ??????? ?????? x ??1?,??
    f(x) ??????

1-4 ?????
36
?????
  • ??1-5 ? x ???????(????) a ,??? f(x) ????????
    L,????? x ?? a ?,f(x) ???? L (??f(x)?? a ????
    L),???
  • ??????? x ?? a ?,??? f(x) ??????,??,????? x
    ?? a ?,f(x) ???????

1-4 ?????
37
???
  • ?
  • ?

1-4 ?????
38
????????
  • ?????????,???????

1-4 ?????
39
?????? L ???
  • ???????????????????????,?????????????????

1-4 ?????
40
???
  • ? ,?
    ?
  • ? ,? ?

1-4 ?????
41
????
  • ??1-1 ?? ? ???
  • ?????????????? c, ?
  • ?? h(x) x ? a ????? a,? ?
  • ?????????
  • ?????????
  • ?????????
  • ?????????
  • ???????????
  • ?????????

1-4 ?????
42
???
  • ???????
  • ??????
  • ??1-1??????
  • ? f(x) x - 4, ,? f(x)g(x)
    1,??
  • ,?
    ?????,????,?

1-4 ?????
43
???
  • ???????
  • ??????
  • ???????????
  • ? ,a 3,? ?

1-4 ?????
44
???????
  • ??1-6 ? x ? a ???(? x lt a )?????(????) a ?,????
    f(x) ???????? L,????? x ?? a ?,f(x) ????? L (??
    f(x) ?? a ????? L)????
  • ????(left-hand limit)
  • ??1-7 ? x ? a ???(? x gt a )?????(????) a ?,????
    f(x) ???????? L,????? x ?? a ?,f(x) ????? L (??
    f(x) ?? a ????? L)????
  • ????(right-hand limit)

1-4 ?????
45
?????
  • ????????
  • ????,????????

1-4 ?????
46
????????????
  • ? ,?
    ? ?

1-4 ?????
47
????????
  • ? f(x) ?? a ???? L ?,f(x) ?? a ?????????????,????
    L???,? f(x) ?? a ??????????????? L ?, f(x) ?? a
    ??????????? L??????????
  • ??? x ?? 0 ?,
  • ?? ??????

1-4 ?????
48
????
  • ????(continuous function)??????????????,??????????
    ?????????????,??????,???????????????
  • ?????????????,?????????????,??????????????,???????
    ????
  • ???? f(x) ?? x c ?? (f(x) is continuous at x
    c),??? f ??????? (c, f(c)) ?,??????????
  • ??? f ????? (c, f(c)) ???????,??????? x ??? c ?,
    f(x) ????? f(c) ??????,??? f ?? c ????,??????
    ???? f(c) ?

1-5 ????
49
????
  • ??1-8 ??? f(x) ???????,???? f(x) ?? x c
    ??(continuous at x c)
  • (1) f(c) ????
  • (2) ???
  • (3) ?
  • ???????,????????????,?? f(x) ?? x c
    ???(discontinuous at x c)?

1-5 ????
50
?????????
  • ?????????? c ????,??????

1-5 ????
51
????
  • ???? f(x) ????(????)????????,??? f(x) ????????,?
    f(x) ?(?????)?????????????????????,??????????,????
    ????? (one-sided continuity)?
  • ??? a, b ????????? a,???? a ????,?
    ,?? f(x) ?? x a ,???(continuous
    from the right at x a)????,?
    ,?? f(x) ?? x b, ???(continuous from
    the left at x b )?

1-5 ????
52
???????
  • ???? f(x) k ???(???)??,?? f(x) k ???(-8, 8)
    ???????
  • ????(identity function),f(x) x ???????,?? f(x)
    x ??????
  • ?? ,?? ???0,
    8)???????

1-5 ????
53
?????
  • ???????????????????,????????????
  • ??1-2??? f(x) ? g(x) ?? c ???,????????
  • (1) f(x) g(x) , f(x) - g(x) ? f(x)g(x) ?? c
    ????
  • (2) ? g(x)?0,? ?? c ???
  • (3) ?? c ??,?? n ??? 1 ?????? n
    ???,????? f(x) ? 0?

1-5 ????
54
??????????????
  • ??1-3
  • (1) ???????????????????,????????(-8, 8)???????
  • (2)?????????? 0 ????????
  • ?????????????????
  • ????????
  • g(x) x3 1 ? h(x) x2 4 ??(-8,
    8)???????????,????? (g?h)(x) (x2 4)3 1 ?
    (h?g)(x) (x3 1)2 4 ???(-8, 8)???????

1-5 ????
55
?????
  • ???????? x 2 ?????
  • ?????????????? y x ?????????

1-5 ????
56
????????
  • ?????????? x ??????(????,? x ?????),?? f(x)
    ?????????????,??????????? (limit at infinity),???
  • ?
  • ?????? x ?8,???? x ??????????,x ?-8???? x
    ?????????

1-6 ????????
57
????????
  • ???? ,???????? x ?????,??? 0?
  • ?????,????? x ??????, ???? 0,??
    ???, ???????? 0???????? x ???????,
    ????????,??????? x ??????, ?????? 0?

1-6 ????????
58
??????????
  • ??1-4 ? t ???????? x t ???,?
  • ?? c ?????
  • ?????????
  • ???????
  • ????????

1-6 ????????
59
???
  • ???????? x ??????,??? ???? 0 ?????????? x
    ??????,????????????? y 0?????,??? x ????
    ?????? (asymptote)?
  • ??1-9? ? ,?????? y L ? y
    f(x) ???????????

1-6 ????????
60
??????
  • ???????????????

1-6 ????????
61
????
  • ?? ?? x ?? a ,?,??? f(x)
    ?????????(????,?????)???????
    ?????????????????,?????????
  • ??? f(x) ? x ??? a ? f(x) ??????,?? ?

1-7 ????
62
????
  • ????? x ? a ????? a ?,??????????
  • ????,??????????,????????

1-7 ????
63
??????
  • ??1-5 (1)? n ????????,?
  • (2)? n ????????,?
  • ???????? ???????

1-7 ????
64
?????
  • ????????
  • ??? ???,?
  • ??????? x ?( 2 ?)????2 ?,??????????,?????????????
    x 2?
  • ??1-10?? (?-8),?
    (?-8),??????? x a ??? f(x) ?????????

1-7 ????
65
??????????
  • ????????????????????????????? 0
    ??????????????,???????,????????? 0 ? x
    ?,??????????????????
  • ??1-6 ? ,?? P ? Q ???????? Q(a)
    0 ? P(a) ? 0,??? x a ??? f ?????????

1-7 ????
66
????
  • ??? ???????

1-7 ????
67
??????????????
  • ??1-7 ? ,?? P ? Q ????????
    P(x) ????? Q(x) ???,?
  • ???
  • ???? 1-6 ??? 1-7,????????????????????,?????????

?
?
?
?
?
1-7 ????
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