guia resuelta primer parcial analisis

7/21/2019 guia resuelta primer parcial analisis

1/607


2/607


3/607


4/607

=23

1

2+ 1 +

1

2+

5

12 2

1

4

1

13

2

6+

1

4+

2

3

=

2

3

6 + 12 + 6 + 5

24

12 12 4 + 3 + 812 =

2

3

7

12

1

4

5

12

=

2

3+

7

121

4+

5

12

=8 + 7 3 + 5

12

=17

12


5/607

=25 2

1

2+ 1 1

5+

3

101

4

=2

5 2

5 + 10 2 + 310

14

=2

5 2

63

3

51

4

=2

56

5+

24

1

2

= 45

+1

2

=8 + 5

10

= 310


6/607

ab

n=

b

a

n

(a)nm = m

an

1

4

4 36

22=

1

4

1

6

22

=

1

41

2

62

2=

1

4

1

362

=

9 1

36

2

=

836

2

9

2

=

9

22

= 814


7/607

8 1

2

2+

6 12

2 12=

7

2

2+

7

2

2 12

=

49

4 +

49

4

12

=

2 49

4

= 2 494

= 7

2

2


8/607

= 34+7

312

= 311

312

= 31112

= 31

=

3

1

1=

1

3

=

5 4 106+2

2

8 105=

5

210

4

105

=

5

2 105(4) =

5

2 109

=

1

5

2

10 108= 1

4

108

= 1

2 10 82=

1

2 104=

1

20000=

=

4

813

+

49

16+

3

64

27

2+

5

1

32

426

23 + 3

72

+ 12

= 33 +7

4+

4

3

2+

1

2

4+ 24 + 34

= 27 +7

4+

16

9 +

1

16+ 81

= 108 +16

9 +

28 + 1 + 1

16

= 108 +16

9 +

158

30

16

= 72 108 + 16 8 + 15 9

72

= 8039

72


9/607

= 25 + 9 + 4

81 +38

3

27+

3

33 3

= 34 + 3 23

+ 3 3

3

= 37 3 2

3 + 3 3

3

= 109

3 + 3 3

3


10/607

2

2

33

2

= 2

4 9

3

6

= 2

5

3

=

10

3

2 2

3+ 32 = 4332 =8 96 = 176

2 +23

32

= 2 1 = 3

2 +2

3

32

=

6 + 23

32

=

(4) (1)2

= 2


11/607

a

b

a+

b

n+ 1 n

=

n+ 1 n

n+ 1 +

n

n+ 1 +

n

=

n+ 1

2+

n+ 1 n

n+ 1 n (n)2n+ 1 +

n

= n+ 1 n

n+ 1 +

n

=

1n+ 1 + n

n3 + 3n2 +n

n2 + 1 =

n (n2 + 3n+ 1)n n+ 1

n

=

n2 + 3n+ 1

n+ 1n

n


12/607

2x = 1 12x = 2

x = 2

2

x = 1

5x = 7 25x = 9

x = 9

5

x = 95

6x+ 1 + 4 = 7x 3

6x 7x = 3 513x = 8x =

8

13

9x 3 = 2 (2x+ 4)

9x 3 = 4x+ 89x+ 4x = 8 + 313x = 11

x = 11

13


13/607

(1 x) 3 = (1 +x) 23 3x = 2 + 2x

3x 2x = 2 35x = 1

x = 1

5

2x+ 3

2 (x 1) = 6x 23(x 1)

(2x+ 3) (3) = (6x 2) 26x 9 = 12x 4

18x = 5

x = 5

18

2 (x 1)

3

(x 1


14/607

2 +

34

x4 10x2 + 1 =x2(x2 10) + 1

2 + 32 = 2 + 223 + 3 = 5 + 26

x=

2 +

3

x4 10x2 + 1 = x2(x2 10) + 1=

2

6 + 5

2

6 + 5 10

+ 1

=

2

6 + 5

2

6 5

+ 1

= 2

62

52 + 1

= 24 25 + 1= 0

x2

x=

2 +

3

(a+b) (a b) = a2 b2


15/607

R

a

b

c

R

a < b a+c < b+c

a < b

c >0 a c < b c

a < b

c b c

0

2x 2 + 1 2x 3 x 3

2

x (,32

)

2x 2 + 1 2x 3 x 3

2

x (, 32

]

2x+ 6x >10 11 8x > 1 x > 18 x

1

8, +


16/607

2 x

x

2 x >0

2 x 0

2> 0 + x x 4 2> 4(2 x) 2> 8 4x 4x >8 2 4x >6 x > 6

4=

3

2

x 2

2

2 x


17/607

2

2 x>4 x

3

2, 2

A(x)B(x)

A(x)B(x) >0

A(x)> 0

B(x)> 0

A(x)< 0

B(x)< 0

A(x)B(x)

0

B(x)< 0

A(x)< 0

B(x)> 0

A(x) B(x)

0

4

2

2 x>4 2

2 x 4> 0

2 4(2 x)2 x >0

4x 22 x >0

0


18/607

4x 6> 0 2 x >0

4x >6 x 32

x


19/607

x+ 2 < 0 x 3> 0

x < 2 x >3

x

x+ 2 > 0 x 3< 0

x > 2 x 1 x 3x+ 1

1> 0

(x 3) (x+ 1)x+ 1

>0

4x+ 1

>0

x+ 1 > 0 x < 1 x (, 1)

x (, 1)


20/607

52

< 47

< 611

0

r >0

log(x+ 102)

x= 0

2 + log(x)

x >0


24/607

log(a b) = log(a) + log(b)

loga

b

= log(a) log(b)

log ab =b log(a)

10log(x) =x

log

4x2

= log(1) = 0

(x 2) log(4) = 0x 2 = 0x= 2

log

25x3

= log

1

8

= log(1) log(8) = log(8)

(5x 3) log(2) = log(23

) = 3 log(2) (5x 3)log(2) = 3log(2)5x= 0x= 0

10log(x+7) = 10100 x+ 7 = 10100 x= 10100 7


25/607

10log(x23x+1)=100=1

x2 3x+ 1 =1x(x 3) = 0x= 0

x= 3


26/607

P = (x, y) R

x

P

y

P

y

x

P

o= (0, 0)

Sy(P) = (x, y)Sx(P) = (x, y)So(P) = (x, y)

P = (1, 3)

P

P


27/607


28/607


29/607


30/607

V(x) =x(30 2x)(40 2x)

R

V(x)

V(x)

xR

x1530 2x0

30 2x

x 0

V

V

= 30 2x >0

0< x 0

V

(0, 15)

V(x)

x= 0

x= 15

x = 6

x

6

x

V(x)


31/607

0

500

1000

1500

000

500

000

500

y

2 4 6 8 10 12 14x

V(x)

x = 0

x = 15

x= 6

x y=V(x)


32/607

x

y

x

y

X= base

y= altura

x

y

20 2x+ 2y = 20

y= 10 x

(f)

x

y >10

20

x 10

y 0

(f) = (0, 10)

0

2

4

6

8

10

y

2 4 6 8 10x

y= 10 x


33/607

x = lado

x = lado

h 2=x+y

2

2

A(x) =

x x2

=x2

2

A

(A) = (0, +)

A(x)

x

0

2

4

6

8

10

y

1 2 3 4x

x y


34/607

x

(x R)((x, y)

(f) (x, y)

(f) y = y

(f)


35/607

(, 0)

(0, +)

x= 0

(f) = [2, 2]

(1, 0) (1, 2)

(2, 1) (0, 1)

x= 0

1

x =1

x = 1

1

(, 0) (1, +)

(0, 1)

x= 1

x= 0


36/607

x

y


37/607

f(x) =mx+b

m

b

f(0)

y

m >0

m


38/607

f(x) =34 x+174

(x0, f(x0)) (x1, f(x1)) (f) f

m

m=f(x1) f(x0)

x1 x0 m= 2 53 1=

34=

3

4

m

b

(1, 3)

(80, 3)

(f)

m= 3 3

80 (1)= 0 b= 3

f(x) = 3

(0, 4)

(3, 0)

(f)

f(x) =mx+b f(0) =b

f(0)

(0, 4)

(f)

f(0) = 4

b= 4

m

0 = 3m+ 4 m= 43

f(x) = 4

3x+ 4

b

f(a) = 0

0 =a m+b

a m= b


39/607

a = 0

b = 0

f(x) = m x m R m= 0

f(x) = 0

a = 0

m= b

a b= 0

m = 0

b

f(x)

f(x) = 0

ab

R

f(0) =17

4

b

f(0) = 3

b

f(2) = 34 (2) + 4 =8

3+ 4 =

20

3

m = 34 m = 0 m = 43

f(x) = 0 m= 0


40/607

f(x)

-3 1

5

f(x)


41/607

f(x)

4

2

2

4

6

y

4 3 2 1 1 2 3 4x

f(x)


42/607

f(x) =m x+b= x+b 3 = 2 +b b= 1

f(x) =x+ 1

b= f(x)

m

x

b= 5

0

1 = 5

f(x) = 5

b= f(x) m x= 4 (2) 3 f(x) = 2x+ 2

f(x) =x+b

b

f(x)

4

2

0

2

4

y

4 2 2 4x

y=x+ 1

x= 1

m >0


43/607

f(x)

10

8

6

4

2

0

2

4

6

8

10

y

4 2 2 4x

f(x) = 5

m= 0

f(x)

4

2

0

2

4

y

4 2 2 4x

f(x) = 2x+ 2

x= 1

m


44/607

f(x)

f(x) =x+b

b > 0

b0

0

x= 0

y


45/607

g(0) = 32

g(100) = 212

g

g(x) =mx+b

b= 32

m=212 32

100 0 =

95

180

100

g(x) =9

5x+ 32

h(x)

h(32) = 0 h(212) = 100

m = 100 0212 32=

9

5

b=h(x) m x

x= 32

b= 0 59 32 = 160

9

h(x) =59

x 1609

g(x)

h(x)

g(x)

h(x)

g (h (x)) = g

5

9x 160

9

=9

5 5

9

x

160

9 + 32= x 160

5 + 32

= x 32 + 32= x

f(x) 1f(x)

f1(x) f(x)


46/607

h (g (x)) = h

9

5x+ 32

=

5

9

9

5x+ 32

160

9

= x+ 32 5 160= x+ 160 160= x

g (h (x)) =h (g (x)) =x


47/607

f(x) =ax2 +bx+c

a,b,c R

a = 0

a >0

f(x) =x(x 4) 4 =x2 4x 4


48/607

f(x) =ax2

+bx+c

f(x) =a (x xv)2 +yv

f(x) =a (x r1) (x r2)

D= b2 4ac

d >0 f R

d= 0 f

d 0

a


49/607

xv = 0 yv =f(xv) = 0

2 = 0

a= 1> 0

f(x)

4

2

0

2

4

y

4 2 2 4x

xv = 0 yv = 0 r1= r2= 0 (f) = [0, +) xv

xv

= 0 y

v = f(x

v) = 0

a =

1 < 0

f(x)

4

2

0

2

4

y

4 2 2 4x

xv = 0 yv = 0 a= 1< 0 (f) = (, 0] xv


50/607

xv = 0 yv = f(0) =3 a = 1 > 0

x2 3 = 0 x2 = 3 x=

3

xv yv

f(x)

4

2

0

2

4

y

4 2 2 4x

xv = 0 yv = 3 r1= 3 r2= 3 a= 1< 0 (f) = [3, +)

yv y

D >0

D= b2 4ac


51/607

a

xv yv a= 1 xv = 5

yv = 0

f(x)

5

4

3

2

1

1

2

y

2 4 6 8x

xv = 5 yv = 0 r1= r2= 0 a= 1 (f) = (, 0]

xv x


52/607

a= 2< 0

xv = 0 yv = 0 r1=r2= 0

f(x)

(, xv) = (, 0) (0, +)

f(x) 0x R r1=r2=xv = 0

xv f(x) x= 0

r1 = 0 r2 = 3 a =2

xv

xv =r1+r2

2

xv =3

2

yv =f(xv) = 2 32

3

2 3

=

9

2


53/607

f(x)

, 32

32

, +

f(x)

f = (, 0) (3, +) f

f

f+ = (0, 3)

f

xv =

32

f

f(x) = 2

x 12

2

r1= 0 r2=1

2

a= 2

xv = b2a

= 12 (2)=

1

4

yv = 18

+1

4=

1

8

f(x)

, 1

4

14

, +

f = (, 0) 12 , + f+= 0, 12 f(x)

xv =

14

f(x) = (x+ 1)2 = (x (1))2

r1 = r2 =

1

a= 1> 0

xv =

1

yv = 0

f(x)

(1, +)

(, 1)

f

a >0

f+ =R{1}

f

0

f(x)

x= xv = 1


54/607

r1= 3 r2= 5 a= 20

xv =

3+52

= 1

f

(, 1))

(1, +)

f+ = (3, 5) f= (, 3) (5, +) f

x= xv = 1


55/607

a= 5< 0

f

xv

xv = b2a

= 102

(

5)

= 1

yv f(x) xv

f(xv) =f(1) = 5 12 + 10 1 = 5

5


56/607

f(x) =x3

8

6

42

0

2

4

6

8

y

3 2 1 1 2 3x

f(x) = (x 2)3

10

8

6

4

2

0

2

4

6

8

10

y

1 1 2 3 4x

f(x) =x3


57/607

f(x) =x3 1

10

8

6

4

2

0

2

4

6

8

10

y

4 3 2 1 1 2 3 4x

f(x) =x3

f(x) =x4

5

10

15

20

y

3 2 1 1 2 3x


58/607

f(x) =ax+b

cx+d

adbc = 0

adbc = 0

c = 0

c = 0

f(x) =ax+b

cx+d=A+

B

x e

A= a

c

e= dc

B = bcadc2

B >0

(f)

++

b < 0

+

+

(f) = R {e}

x = e

B

x1 x2

P =

(x1, f(x1)) Q= (x2, f(x2))


59/607

B >0

B >0 ++ b < 0

+

+

(f) = R {e} x= e

f

x= e

y= A


60/607

y = 0

x = 0

x1 =1 x2 = 1

P = (1, 4) Q = (1, 4)

f)

++

(f) = R{0}

x10 x1 < x2

f(x1)< f(x2)

(x1, x2) (f)

f(x) = 4

x

10

8

6

4

2

0

2

4

6

8

10

y

10 8 6 4 2 2 4 6 8 10x

f

(, 0) (0, +)


61/607

y = 0

x = 0

x1 =1 x2 = 1

P = (1, 4) Q = (1, 4)

f)

+

+

(f) = R{0}

x10 x1 < x2

f(x1)< f(x2)

(x1, x2)

(f)

f(x) = 4

x

10

8

6

4

20

2

4

6

8

10

y

10 8 6 4 2 2 4 6 8 10x

f

(, 0) (0, +)


62/607

y = 0

x1= 2 x2= 4 P = (2, 4)

Q= (4, 4)

f)

++

(f) = R{3}

x10 x1 < x2

f(x1)> f(x2)

(x1, x2) (f)

f(x) = 4

x3

10

8

6

4

2

02

4

6

8

10

y

10 5 5 10 15x

f

(, 3) (3, +)


63/607

y = 2

x = 3

x1 = 1 x2 = 4

P = (1, 0)

Q = (4, 6)

f)

++

(f) = R{3}

x10 x1 < x2

f(x1)> f(x2)

(x1, x2) (f)

f(x) = 4

x3+ 2

10

8

6

4

2

02

4

6

8

10

y

10 5 5 10 15x

f

(, 3) (3, +)


64/607

y = 4

x = 2

x1 = 1 x2 = 3

P = (1, 1)

Q = (3, 17)

f)

++

(f) = R{2}

x10 x1 < x2

f(x1)> f(x2)

(x1, x2) (f)

f(x) = 4x+5

x2

20

10

0

10

20

y

10 5 5 10x

f

(, 3) (3, +)


65/607

y = 3

x =1

x1 =2 x2 = 0

P = (2, 4) Q = (0, 2)

f)

+

+

(f) = R

{1}

x10 x1 < x2

f(x1)> f(x2)

(x1, x2) (f)

f(x) = 3x+2

x+1

10

8

6

4

2

02

4

6

8

10

y

4 3 2 1 1 2 3x

f

(, 1) (1, +)


66/607

f(x) =

x

1

0

1

2

3

y

1 1 2 3 4 5x

(f) = R 0

f(x) = x

3

2

1

0

1

y

1 1 2 3 4 5x

(f) = R 0


67/607

f(x) =

x+ 3

1

0

1

2

3

y

4 2 2 4x

(f) = R 3

f(x) = |x 2|

1

1

2

3

4

5

6

y

4 2 2 4 6 8x

(f) = R


68/607

(f) =R x2 0x R x2 + 4 0 x Rx2 + 4

0

x 8

0 x 8

f= [8, +)

x

f x2 9 0 x2 9 |x| 3

f= (, 3] [3, +)

x

f

x (x 1) 0

x 0

x 1 0

x 1

x 0

x 1 0

x 0

f= (, 0] [1, +)


69/607

ff(1) =f(f(1)) =f(2 (1)2 +5(1) =

f(3) = 3

f h(1) = f(h(1)) = f(4) =

32 20 = 12

g f(1) f(1) = 3 / (g) = R {3}

h g(2) =h(g(2)) =h 15

= 2 1

5 6 = 230

5 = 28

5

f g(x) = f

1

x+ 3

= 2

1

x+ 3

2+ 5 1

x+ 3

= 2

(x+ 3)2+

5

x+ 3

=

2 + 5

(x+ 3)

(x+ 3)2

= 5x+ 17

(x+ 3)2

g h(x) = 1(2x2 + 5x) + 3

= 1

2x2 + 5x+ 3

((f g) h) (x) =

(f g) (h(x))= 2 + 5 ((2x 6) + 3)

((2x 6) + 3)2

= 10x 13

(2x 3)2

f h(x) = 2(2x 6)2 + 5(2x +)= 2(4x2 24x+ 36) + 10x 30= 8x2 38x+ 42


70/607

f f(x) = 2 2x2 + 5x2 + 5x= 2(4x4 + 20x3 + 25x2) + 5x

8x4 + 40x3 + 50x2 + 5x

f g(x) =g f(x)

(f (g h)) (x) = f(g h (x))= f(g(h (x)))

= (f

g) (h (x))

= ((f g) h) (x)

f g h t(x)

(((f g) h) p) (x)


71/607

f : R R m = 0

x

y

f

y = 3x 5 y+ 53

=x

f1(x) =

x+ 5

3

f

f1(x) =1

3x+

5

3

x 0

f(x) = 2x2 1

f :RR f

f : [0, +

)

R

1

f

m m = 1m

13

y0 f f1(y0)


72/607

y= 2x

2

1

2

1

0

1

2

3

4

y

1 2 3 4 5x

f= [0, +)

f= [1, +)

y= 2x2 1 x2 = y+ 12

|x| =

y+ 1

2

x 0 |x| =x x= y+ 12 f1(x) =

x+ 1

2

f1 : [1, +) R 0 f1 (x) =

x+ 1

2

f= [5, +) x+ 5 0 +

x [5, +)

f = (, 3]

x

y

f(x)


73/607

y = 3 x+ 5 x+ 5 = 3 y x+ 5 = (3 y)2 x= (3 y)2 5 x= y2 6y+ 4 f1(x) =x2 6x+ 4

f1 : (, 3] R f1(x) =x2 6x+ 4

f : R R f1(x) = 3x

f

f= [3, +)

xv = 3 a = 1 > 0 xv

f(xv) = 5

(f)

f= [5, +) =

(f1)

x

y

f

y= x2 6x+ 4 (x 3)2 5

(x 3)2 =y + 5 |x 3| =

y+ 5

x 3 |x 3| =x 3 x 3 =

y+ 5

x= 3 + y+ 5 f1(x) = 3 + x+ 5

f1 : [5, +) = R

f1(x) = 3 +

x+ 5

f(x)

f=

(, 3] x 3


74/607

f= [

5, +

)

y= x2 6x+ 4 (x 3)2 5

(x 3)2 =y + 5 |x 3| =

y+ 5

x 3 |x 3| = (x 3) 0 x 3 =

y+ 5

x= 3

y+ 5

f1(x) = 3

x+ 5

f1 : [5, +) = R

f1(x) = 3 x+ 5


75/607

x >0

(x >0) xx

=

x

xx

x>0=

x2x

=

x2

x =

x

f

1

x

=

1x

+ 1

1x

+ 1

=

1x

+ 11+xx

=

1+x

x

x+1x

=x+ 1

x x

x+ 1

= x

x

x+ 1

x+ 1

= x

x f(x)

=

x f(x)

f1x = x f(x)


76/607

f(x) = 2x

2

1

0

1

2

3

4

5

6

y

3 2 1 1 2 3 4 5x

f

f= R

f= R >0

f(x) = 12x

2

1

0

1

2

3

4

5

6

y

3 2 1 1 2 3 4 5x

f

f= R

f= R >0


77/607

f(x) = 3x

2

1

0

1

2

3

4

5

6

y

3 2 1 1 2 3 4 5x

f

f= R

f= R >0

f(x) =

13

x

2

1

0

1

2

3

4

5

6

y

3 2 1 1 2 3 4 5x

f

f= R

f= R >0


78/607

r >0, r = 1 R

rx : R R>0

logr(x) :R>0 R

(x, y)

(rx)

(y, x)

(logr(x))

f(x) = log2(x)

4

2

2

4

6

y

2 2 4 6 8x

f

f= R>0 f= R

f(x) = log 1

2(x)

4

2

2

4

6

y

2 2 4 6 8x

f

f= R>0 f= R


79/607

f(x) = log

3(x)

4

2

2

4

6

y

2 2 4 6 8x

f

f= R>0 f= R

f(x) = log 1

3(x)

4

2

2

4

6

y

2 2 4 6 8x

f

f= R>0 f= R

log 12

(x) = log2(x)

log 13

(x) = log3(x)

x

1


80/607

(ln(x)) = R>0 x

2x >0 x >0

(f) = R>0

x/f(x) = 1

2x= e x= e2

f

{x: 3x2 + 2x >0}

a >0

3x2 + 2x >0 x (, r1) (r2, +)

3x2 + 2x= 3x

x

2

3

r1 = 2

3 r2= 0

(f) =

, 2

3

(0, +)

e ln 2, 7182


81/607

f(x) = 1 3x2 + 2x= e 3x2 + 2x e= 0 x=2

4 + 12e

6

x=2 2

1 + 3e

6

x=1

1 + 3e

3

f(x) = 1 x=1 1 + 3e3


82/607

f : R>0 R y = ln(2x) ey = 2x x = 12 ey

f1 : R R>0 f1(x) = 12

ex

(f) = R

f

f(1) = ln(5) =f(1)

f

(f) := [0, +)

x2 + 4

ln(x)

f(x)

[0, +)

(f) = [ln(4), +)

y = ln x2 + 4 ey =x2 + 4 x2 =ey 4 |x| = ey 4

x 0 |x| =x x= ey 4

f1 : [ln(4), +) [0, +) f1 (x) =

ex 4

f1 (x)

(f)

f= f1

(f) = (, 1)(1, +)

(f) = (1, +)

(f) = R


83/607

y= ln

x2 1 ey + 1 =x2 |x| = ey + 1

x >0 |x| =x x= ey + 1

f1 : R (1, +) f1(x) =

ex + 1

(f) = R

0 (f) = [6, +

)

y = 2x + 5 y 5 = 2

x

x= ln(y 5)ln(2)

x=

ln(y 5)ln(2)

2 x= ln

2(y 5)ln2(2)

f1 : [6, +) R

f1(x) =ln2(x 5)

ln2(2)

(f1)

f1(x)

(f1) = (5, +)

f1(x)

f

f

(f) = R

(f) = R0

y= ex+3 ln(y) =x+ 3 x= ln(y) 3


84/607

f1 : R0 R f1(x) = ln(x) 3

(f) = R>0 f (f) =

R>1

y = ex2 x2 = ln(y)

(x >0) x= ln(y)

f1 : R>1 R>0 f1(x) =

ln(x)


85/607

sin(x ) = sin(x)

2

cos(2x+) = cos(2x)

sin

x+

2

= cos(x)

1

f(x) = sin(x )


86/607

f(x) = cos(2x)

f(x) = cos(2x+)


87/607

f(x) = sin(x+

2)

2

2

2

3 2

2-

-


88/607

sin(x)

2

sin(x) = 1

2

[0, 2)

S[0,2)

S=

x0+ 2k : x0 S[0,2) k Z

[0, 2)

sin(x) =1

2 x=

6 x=

6 =

5

6

S[0,2)=

6,5

6

S= 6+ 2k : k Z 5

6+ 2k : k

Z

x R|cos(x)| 1

[0, 2)

x= 0 x=

S= {2k : k Z} {+ 2k : k Z}

R


89/607

x R

sin(x+ y) = sin(x)cos(y) + cos(x) sin(y)

sin(x y) = sin(x)cos(y) cos(x) sin(y)

cos(x+y) = cos(x)cos(y) sin(x)sin(y)

cos(x y) = cos(x) cos(y) + sin(x)sin(y)

sin(x+x) = sin(x)cos(x) + cos(x) sin(x)

= 2 sin(x)cos(x)

sin(2x) = 2 sin(x) cos(x) x R

x R

cos(x+y) =cos(x)cos(y) sin(x)sin(y)

cos

x+

4

= cos(x)cos

4

sin(x)sin

4

= cos(x)

2

2 sin(x)

2

2

=

2

2 (cos(x) sin(x))

cos

x+

4

=

2

2 (cos(x) sin(x)) x R


90/607

cos(x)

(cos(x)) = [0, ]

sin(x)

2

, 2

arc cos(x) : [1, 1] [0, ]

arcsin(x) : [1, 1]

2,

2

cos(x) arc cos(x)

sin(x) arcsin(x)

x= sin

4

=

2

2


91/607

x= cos () = 1

(x [1, 1]) arc cos(x) 0

cos2(x) = 1 sin2(x)

sin (arcsin (x)) = x

sin2 (x) = (sin(x))2

cos(arccos(x)) =

cos2 (arcsin (x))

=

1 sin2 (arcsin (x))

=

1 (sin (arcsin (x)))2

=

1 x2

(x

(arcsin(x)) cos (arcsin (x)) =

1 x2


92/607


93/607

f(x)

x= 3

x 1

x= 1

x= 4

f(3) = 3 + 2 = 1 f(1) = 1 f(4) = 3 4 4 = 8

y= f(x)

y

(0, y)

f(x)

y

(

,

1)

(1, +

)

y


94/607

f(x)

f(x)

x= 3

x= 1

x= 4

x 4

f(3) = 13 + 2 = 1 f(1) = 1

1 + 2=

1

3 f(4) =

1

4 + 2=

1

6

y= f(x)

y

(0, y)

f(x)

y 11,

12 (0, +)

y


95/607

x

0 x 100,000

100,000

x

100,000

200,000

x 200,000

f(x)

f(x) =

0

0 x 100000x100000

1000 1

2 100000< x 200000

50 + x

2000001000

x >200000

50

f(x)

200000

200000

$200000

$50

$530

50 +x 200000

1000 = 530

x 200000 = 1000 480 x = 200000 + 480000 x = 680000

$680000


96/607

f(x)

f

(1, 2)

(2, 0)

m= 2 01 2= 2

3 f(x) = 2

3x+b

0 =f(2) = 43

+ b b= 43

f(x) = 23

x+4

3

g(x)

2

2

g(x) =a (x 2) (x+ 2)

g(1) = 2

2 =a(3)(1) = 3a a= 23

g(x)) = 23 (x 2) (x+ 2)

x/f(x)> g(x)

a g(x)

|||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||


97/607

f(x)> g(x) x (, 1) (2, +)


98/607

a2 b2 = (a b) (a+b)

cosh2(x) sinh2(x) = (cosh(x) sinh(x)) (cosh(x) + sinh(x))= ex +ex

2 e

x

ex

2ex +ex

2 + e

x

ex

2

=

2ex

2

2ex

2

= ex ex= ex+(x) =e0 = 1

cosh2(x) sinh2(x) = 1

x R

cosh(x) = sinh(x) cosh2(x) = sinh2(x) cosh2(x) sinh2(x) = 0

x

R cosh2(x)

sinh2(x) = 1


99/607


100/607

x= 0

f(1) = 3, 55Kg

4, 55 = 2 3, 55 f(2) = 4, 55Kg


101/607

ab

cd

= ab d

c

f(n+ 1)f(n)

= 23(n+1)

4(n+ 1) + 14n+ 1

23n

= 23n+3

4n+ 54n+ 1

23n

= 8(4n+ 1)

4n+ 5

= 8

4n+ 1

4n+ 5

f(n+ 1)f(n)

= 84n+ 14n+ 5

f(2)

f(1)= 8 5

9=

40

9

f(3)

f(2)= 8 9

13=

72

13

f(4)

f(3)= 8

13

17=

104

17

f(5)

f(6)= 8 17

21=

136

21

f(6)

f(5)= 8 21

25=

168

25


102/607


103/607

(1, +)

(, 2]

|x|


104/607

x2 0

2x


105/607

1< 4

x

x 4

x

x (0, +)

x (3, 3)

3< x 2< 3 1< x


106/607

2 x 4 3 x 6 2 x 6

x < 3

x > 1

1 < x < 3 = (1, 3)

(, 3) (1, +) = (1, 3)

1< x


107/607

1< x


108/607

12;23;34;45 ;56


109/607

3

p N

q N /

3 =p

q

p

q

pq

3 = p2

q2

3 |p 3 | q

|

|

p

q

3

p

3

q

p2 = 3q2 3|p2 3|p

3

p

p

3|p

3

3 | q

3|p p= 3t

t N 3q2 =p2 = (3t)2 = 9t2 q2 = 3t2 3| q2

3| q

3 / Q

p

q

p

q

p

q

32

24

12


110/607

3, 141592<

3, 14< 3, 141Q

<

0, 001 = 3, 140592 < 0, 001 = 3, 140592 >3, 14

3, 14< 0, 001<

0, 001 RQ

x / Q

y Q x y / Q

z Q

x y= x Q

x= yQ

+ zQ

Q

/ Q 0, 001 Q 0, 001 / Q


111/607

n 1 n N 1n 1 n N 1 A

1 A

1 = max(A) = sup(A)

sup(A)

max(A)

1

A

n N, n > 0 1

n > 0

0

A

0 = nf(A)

n

1n

0 = nf(A)

> 0 a A/ 0 a < 0 +

a

A

a = 1

n n N

1

> 0

n0 N n0 > 1 1n0 <

a= 1n0

A

0 a


112/607

12

1

2 n

n+ 1 n+ 1 2n n 1

12

B

12 B

1

B

1 nn+ 1

n+ 1 n

1 = sup(B)

>0 b B / 1 < b 1

1 < nn+ 1

= n+ 1 1

n+ 1 =

n+ 1

n+ 1 1

n+ 1= 1 1

n+ 1

< 1

n+ 1

> 1n+ 1

n+ 1 > 1

n0 >

1 1

R

b = n0

n0+1

1 < b

1 = sup(B)


113/607

1 /

B

1 B n N / 1 = nn+ 1

n+ 1 =n 0 = 1

1

B

B

nf(C) = 0

0 / Csup(C) = 7

7 / C

1 = nf(N) =mn(N)

0

E

0 E

0 = mn(E)

E

n 1n2

n 1> M n > M 1

M >0

n= [M] + 1

n 1

n2 > M

E

mn(F) = 1

max(F) = 4


114/607

G=

6 1

10n : n N

10n 1 110n

1 110n

1 6 110n

5

G

5 G

5 = mn(G)

6 1

10n 6 6

G

> 0 n N /n > 1

1n

n N 10n > n 110n

< 1

n<

n N / 110n

< ( >0)

110n

> ( >0)

6

1

10n >6

(

>0)

6 = sup(G)( >0)

H= (1, 3)

I= (, 3)(3, +)


115/607


116/607

2n 1n+ 2

2 2n 1 2n+ 4 1 4

1, 99 = 2

= 0, 01

1, 99< p=2n 1

n+ 2 2 < 2n 1

n+ 2 2n+ 4 (n+ 2) 5

n+ 2 > 5

n > 5 2

t < 2

t= 2

>0

n > 5

2

p= 2n1

n+2

n=

5

1, 99 = 2 0, 01

n= 5

0, 01= 500


117/607

25001500+1

= 999502

1, 990039 > 1, 99

t 0/t= 2

t 0/t= 2

n=

5

p=2n 1

n+ 2

> t


118/607

n= 1001 1

n= 0, 000999< 0, 001

x >0 1

x>0

R

n N/n > 1x 1

n< x


119/607

A B

a A, a B

c R

B c bb B c aa A

a B

{

} {

}

nf(B) nf(A) sup(A) sup(B)

A= (1, 2) B= (0, 2)

A= (1, 2) B= (0, 3)


120/607

A

a =

1> 0

(r1, r2) r1 r2

x2 3x+ 2 = (x 1) (x 2) r1 = 1 r2= 2

A= (1, 2)

nf(A) = 1

sup(A) = 2

A

f(x) = 3x2 3x+ 2

f(x) x= 32

f32

= 0, 25

f(0) = 2

B

B

x (0, 2) y (0,25, 2)

B= (0,25, 2)

nf(B) = 0, 25

sup(B) = 2

B

C


121/607

C= [

0,25, +

]

nf(C) = mn(C) =

0, 25


122/607

a1 =1

2 a2=

2

3 a3=

3

4 a4=

2

5 a5=

5

6

b1= 1 b2= 2

32 b3=

22

53 b4 =

23

73 b5=

24

93

c1 = 1 c2 = 14

c3=1

6 c4= 1

24 c5 =

1

120

(n N) n!

n! =n

(n

1)

(n

2)

(n

3)

2

1

5! = 5 4 3 2 1 = 120


123/607

d1=cos()

1 = 1

d2=cos(2)

2 =

1

2

d3 = cos(3)

3 = 1

3

d4=cos(4)

4 =

1

4

d5 = cos(5)5

= 15


124/607


125/607

an cn

cn= 1n2

+ 1=

2

n+ 2

cn :

23

; 12

; 25

; 13

; 27

; 14

; etc

an

an = bn cn = 1 + (1)n

2 2n+ 2

=1 + (1)n

n+ 2

an=1+(1)n

n+2 a100=

1

2

51

102

151

a200 = lmn

an= 0

an=(1)n+1 a100= 1 a200 = 1

an=n+1n

a100 =

101100

a200=

201200

lmn

an= 0

1; 1

2; 1

3; etc

bn=1 + (1)n+1

n+ 1 : 1, 0,

1

2, 0,

1

3, 0,

1

4, 0,etc

n+ 1


126/607

cn

1, 2, 3, 4, 5,etc

cn=(1)n + 1

2 n

2 : 0, 1, 0, 2, 0, 3, 0, 4, 0, 5,etc

an = bn+cn=1 + (1)n+1

n+ 1 +

(1)n + 12

n2

an=1 + (1)n+1

n+ 1 +

(1)n + 12

n2

a100= 50 a200 = 100

a1 = 1

a2 = 2a1 = 2

a3 = 2a2 = 4

a4 = 2a3 = 8

an = 2n1 a100= 299a200 = 2199 lm

nan = +


127/607

n N

an > 10 an>1000

an> 10 an > 1000

M > 0R

n0 M an> Mn n0

an

an=n 5

22 57

4

an> M

n 52

2> M+

57

4

n 52

2> M+

57

4

n3

n 5

2 > 0

an> M n 52

>

M+

57

4

n > 52

+

M+

57

4


128/607

n

3

an> M n > 52

+

M+

57

4

n0

M= 10

n0 =

52

+

10 + 574

+ 1 = 8

n 8

an>10

M= 1000 n0 = 52+ 1000 + 574

+ 1 = 35

n 35

an>1000

an+1 an = 2n+1 2n =

2n(2 1) = 2n >0 an+1> an(n N)

n0 an0 > M

M

an > M 2n > M+ 100 n > ln(M+ 100)ln(2)

M= 10

n0 =

ln(110)

ln(2)

+ 1 = 7

n 7

an>10

M= 1000

n0 = ln(1100)ln(2)

+ 1 = 11

n 11

an>1000

1, 9 an 2, 1 0, 1 an 2 0, 1

|an 2| 110 1n+1 110 110

110


129/607

|an 2| n 1 1

= 1

10

n0 = 1

( 110 ) 1 = 9

(n 9) |an 2| 110

1, 9 an 2, 1

= 1

1000

n0= 1

( 11000 )1 = 999

(n 999) |an 2| 11000

1, 9999 an 2, 001

|sin(n)| 1

1n 1

10

1n 1

1000

n0 = 10 n0= 1000


130/607


131/607

14

< an 1 < 14 34

< an< 54

an


132/607

lmn

an = lmn

n2

4n+ 2 3n 1

n2

n

2

5 + 4n2

5

=

lmn

an = lmn

n+

n2 5n

n

1 + 3n

1

= +

lmn

an= lmn

n3

1 + 2

n3n2 1 1n2 = lmn0

n3

n4

1

1 + 2

n31 1n21= 0

lmn

an = lmn

n2

n2

22 1

n23 + 2

n2

3

=

23

lmn

an = lmn

n2

44 + 3

n2

n

2

3 + 4000n2

3

=4

3


133/607

lmn

an = lmn

n (1)n

1 1n

+ 1

2

= 12

an=

bn n+ 2n+1

2

2(n2 +1)

2n2 + 1 cn

lmn

n+ 2n+1

2

= lmn

2 n+ 2n+ 1

= lmn

2 nn

1 + 2n

1 + 1

n

= 2

lmn

2(n2

+1)

2n2 + 1

= 2 2n2

2n2

1 + 1

2n2

1

= 2

bn= n+2

(n+12 )

an

cn = 2(n2+1)

2n2+1

an

lmn

an= 2


134/607

lmn bn= 2

??? lmn b2n1 = 2 an bn

lmn cn= 2 ???

lmn c2n = 2

an cn

an

an

lmn

bn= 2 lmn

b2n1= 2

> 0

n0

N/ n

n0 |bn 2| < n n0 2n 1 n |b2n1 2| <

lmn

b2n1 = 2


135/607


136/607

(

n

n2)

|an

2

|<

lmn

an = 2

12

an =

1 + r + r2 + r3 + r4 + + rn

r= 12

an

an = 1 +r+r2 +r3 +r4 + +rn

r an = +r2

+r

3

+r

4

+ +rn

+r

n+1

an r an = 1 rn+1

an(1 r) = 1 rn+1

an

an=1 rn+1

1 r =1

2

an

an= 2

12n

lmn

an = lmn

2 12n

= 2


137/607


138/607

lmn

an = lmn

n2 +n 2 n

n2 +n 2 +nn2 +n 2 +n

= lmn

n2 +n 2 n2n2 +n 2 +n

= lmn

n1

1 2n

n

1 + 1n 2

n2+ 1

2

=1

2

lmn

an = lmn

n2 + 1

n2 3 + 3

n2 + 1 + n2 3 + 3n2 + 1 +

n2 3 + 3

= lmn

(n2 + 1) (n2 n+ 3)n2 + 1 +

n2 3 + 3

= lmn n

11 2nn

1 + 1n2

+

1 1n

+ 3n2

2

=12

lmn

an = lmn

n

2

n2

2 1n2

3 + 2n2

23

+nn

323 1

n

2 + 3n

=

2

3+

3

2


139/607

lmn

an = lmn

n

n+ 2 n

n+ 2 +

n

n+ 2 +

n

= lmn

n (n+ 2 n)

n+ 2 +

n

= lmn

2

n

n

1 + 2n

+ 1

2

= 1

lmn

an = lmn

n

n+ 2 n

n+ 2 +

n

n+ 2 +

n

= lmn

n(n+ 2 n)

n

1 + 2n

+ 1

= lmn

+n2n

21 + 2

n+ 1

2

= +

lmn

an = lmn

n

n+ 1 + n

n+ 1 n2

= lmn

n

2 n+1n + 1n

21 + 1

n+ 1

n2

= lmn

1

n(1+ 1n)n2

0

+ 1

1 + 1n

+ 1n2

1

= 1


140/607


141/607

an bn

cn

cn = an bn

a := lmn an b := lmn bn

c:= lmn cn

an bn


142/607

an=bn= n

lmn

an bn = lmn

n n= 0

an= 2n bn=n

lmn

an bn = lmn

2n n= lmn

n= +

an bn +

(+) (+)

an=bn= n

lmn

anbn

= lmn

1 = 1

an=n

2

bn = n

lmnan

bn = lmnn= +

an bn +

an=bn=

1n

lmn

anbn

= lmn

1 = 1

an=

1n

bn= 1n2

lmn

anbn

= lmn

n2

n = +

an bn

00


143/607

an=e

n

bn= 1n

lmn

abnn = lmn

en

1n =e1 =

1

e

an=e

n

bn = 1n2

lmn

abnn = lmn

en

1n2 =e

1n =e0 = 1

an bn

00

an=

1n

bn = n

lmn

an bn = lmn

1

n n= 1

an=

1n

bn= n

2

lmn an bn = lmn1

n n2

= +

an bn

0

an= 1n

bn = en

lmn

bann = lmn

(en)1n =e1 =e

an= 1n2 bn=en

lmn

bann = lmn

(en)1n2 = lm

ne

1n =e0 = 1

an bn

0


144/607

+

lmn

an= +

an > 0

an> 0n

lmn(an+bn) = lmn

+

an1 +01

an bn 1

= +

L 0

an L


145/607

lmn

anbn

= lmn

0

1

bn

an0

= 0

an>0

N

abnn = ebnln(an)

lmn

e

+bn ln(

0+an )

= 0


146/607


147/607

lmn

an = lmn

01

n

sin(n)

= 0

lmn

an = lmn

(1)n(n+ 2 n)(

n+ 2 +

n)

n+ 2 +

n

n+ 2 +

n

= lmn

(1)n (n+ 2 n)n+ 2 +

n

= lmn

(1)n

2n+ 2 +

n

0

= 0

an (1)n 1n

an

lmn an=

an a2nn1 a2n1n1

an ank


148/607

lmn

an= 0

rn n 0 |r| 1, r R

lmn

an= 0

rn n 0 |r|


149/607


150/607

lmn

an = lmn

n

5n

2

5

n+ 1

= lmn

5 n

2

5

n+ 1

1= 5

lmn

an = lmn

n

n4 + 11

2

= lmn

n

n4

1 + 1

n4

12

= lmn nn1

4

n1 + 1n41

12

= 1

(1 + (1)n)n

2n

an = 0 2nn


151/607

2n

n

an a2n1 0 a2n1n 0 a2nn +

an L R an

an

an

an

an

an +


152/607

lmn

an= + (M >0) n0 N n n0 an> M

M = 1

n0 N n n0 an > 1 an0 = 0 an0+1= 0 an

lmn

an= +

an

bn =

2n

n +

an

lmn

bn = L lmn

bn+1=L

bn b1 b2 b3 b4 b5 b6 bn+1 b2 b3 b4 b5 b6 b7

R +

bn bn+1 bn(n

N)

bn+1 bn 2n+1

n+ 1 2

n

n

2 n+ 1n

= 1 +1

n


153/607


154/607

lmn

an = lmn

3 9n + cos(n)2 9n + sin(n)

= lmn

9n

9n

3 +

0cos(n)

9n

2 + sin(n)9n0

= 3

2


155/607

bn

1 + 1

e

bn

1

lmn

1 +

3n+ 1

3n 5 1n

= lmn

1 +

3n+ 1 3n+ 53n 5

n

= lmn

1 + 13n5

6

3n5

6

e

2

63n 5 n

= e2


156/607

1

1

1

4

3+

n +

lmn

1 + 3

3n+ 12n+1

= lmn

1 + 13n+13 3n+1

3

e

2

3n+ 1

3 n

= e2

lmn

1 + 1

n2

n2

e

0

1n

=e0 = 1


157/607

lmn

1 + 1n17

n

17

e

0

17

=e17

+

lmn

an= +

lmn

1 +

2n+ 6

3n2 5n2+2

2n+1

= lmn

1 + 13n25

2n+6

+

3n252n+6

e

2n+63n25n2+22n+1

13

=e13

=0ann 0 lmn (1 +an)

1an =e

lmn

sin(n)

n2 n= lm

n

1

n

0

sin(n)

= 0

lmn

1 +sin(n)

n2

1( sin(n)

n2 )

e

sin(n)

n2 n0

=e0 = 1


158/607

cos2(x) = 1 sin2(x)

lmn

cos2

1

n

1sin2( 1n) = lm

n

1

1 cos2

1

n

1 sin2( 1n)

1

=

1 sin2

1

n

1 sin2( 1n)

e

1

= e1

bn =

Cos(n)5n3+1

lmn bn= 0

bn= 0 n N

lmn

bn(2n2 + 3) = lm

ncos(n)

0

2n2 + 3

5n3 + 1= 0

lmn

an= lmn

(1 +bn)

1bn

e

bn(2n2+3)0

lmn

an=e0 = 1


159/607

lmn

an+1an = lmn n+ 12n+2 2

n+1

n =

1 n+ 1

n 2

n+1

2 2n+1 =1

2 1 lm

nan= +

lmn

an+1an = lmn (n+ 1) 2n+1(n+ 1)! n!n2n

= lmn

n+ 1

n+ 1n!

n!2

n

2n

2n

= lmn

0

2n

= 0< 1

lmn

an= 0


160/607

lmn

|an+1||an| =l lmn

n

|an| =m y l= m

lmn

|an+1||an| = lmn

(n+ 1)!

n! = lm

nn+ 1 = + >1

lmn

n

n! = +

(r >0 R), lmn

rn

n! = 0

lmnan+1an = lmn r

n+1

(n+ 1)!n!rn

= lmn

r

n+ 1= 0< 1


161/607


162/607

lmn

|an+1||an| = lmn

22(n+1)+1

(2(n+ 1))! (2n)!

22n+1

= lmn

22n+3

(2n+ 2)! (2n)!

22n+1

= lmn

4 22n+1(2n+ 2)(2n+ 1) (2n)!

(2n)!

22n+1

= lmn

4

(2n+ 2)(2n+ 1)= 0< 1

lmn

an= 0

lmn

n|an| = lmn

12

+ 2n

=12


163/607

lmn

|an+1||an| = lmn

n+ 1!

(n+ 1)n+1 n

n

n!

= lmn

nn

(n+ 1)n = lm

n

n

n+ 1

n= lm

n

n+ 1 1

n+ 1

n= lm

n

1 1

n+ 1

n

= lmn

1 1n+ 1

n+1

e1

1

n

n+1

= e1


164/607


165/607


166/607

bn

lmn

n

|bn| = lmn

1 +

1

n

n=e >1

lmn

1

an= +

an

1

anan>0n

> bn an< 1

bn n0

lmn

an= 0

+

lmn

an= +


167/607


168/607


169/607

a4n =

cos(4n) +

sin(2n) = 1 n N lmn

a4n= 1

a4n+1 = cos ((4n+ 1) ) + sin

(4n+ 1)

2

= cos(4n+) + sin

2n+

2

= cos(pi) +sin

2 = 1 + 1 = 0 ( n N) lmn

a4n+1 = 0

lmn an

a2n =

=1n(1)

6n+1

+ 4 = 3 ( n N) lm

na2n= 3

a2n1 =

=1n

(1)3(2n1)impar +1 4 = 3 ( n N) lmn a2n1 = 3

lmn an


170/607

cos(2n) = 1 ( n N) cos((2n 1)) = 1 ( n N)

a2n= 6n+ 1

10n 2 n N lmn a2n= 0

a2n1 = 3(2n 1) + 15(2n 1) 2=

(6n 2)(10n 7)( n N)

lmn

a2n1= 1

lmn an

an

a5n

a5n+1

a5n = 5n

5n n 1

a5n+1= 2 +

1

5n+ 1n2

lmn an

cos(2n) = 1 ( n N)cos ((2n 1) ) = cos (2n ) = 1 ( n N)a2n =

1 +

1

2n

2nne

a2n1 =

1 12n 1

2n1n e1

lmn an


171/607


172/607

L= lmn an =L ank

bn

lmn

bn=L

lmn

bn = L L= 0

L >0

lmn

bn=L


173/607

an+1an

= 2 > 1 an+1 > an an

(nN)

(n N)

(n N)

an = 2

n1 (n N)

a1= 1 = 211 = 20 = 1

an= 2

n1

an+1 = 2

n

an+1= 2an= 2(2n1) = 2(n1)+1 = 2n

(n N) (n N)an = 2n1


174/607

an

an

0< a1=1

3


175/607

n= 6

an

0< an < 1n N


176/607


177/607

a1 =

13

0< a1


178/607

L= lmn

an+1 lmn

1

2(an)

L (1 an)1L =12

L(1 L)

L= 12

L(1 L) 2L= L L2

L+L2 = 0 L(L+ 1) = 0

L 0

L= 0

lmn an


179/607


180/607

an a1 =

1; a2 = 1

2; a3 =

13

; a4 = 1

4

an = 1nn

N

a1 = 1 =

11

an =

1n

an+1= 1

1 +

1

an

=H.I1

1 +

1

1n

= 1

1 +n

= 1

n+ 1an+1=

1

n+ 1

an= 1

nn N lm

nan= 0

an a1 = 1; a2 =

3; a3 =

3

30 an

>0

2n+1

5n

>

0

an L 0

L= lmn

an+1= lmn

Lan

2

52n+ 15n

=

2

5L

L= 0


187/607


188/607

c1 = 1000 10 %

c2 = c1+c1 10

10 0 =c1+c1 1

10=c1

1 +

1

10

=c1 11

10

K

10 %

1110

10 %

K

x %

1 + x

100

K

1 x100

cn

c1= 1000 c2= 1000

1110

10 %

c3= 1000 1110

910 10 %c4= 1000 11102 910 10 %

c5= 1000

1110

2 910

2

10 %

c6= 1000

1110

3 910

2

10 %

c7= 1000

1110

3 910

3

10 %

an

an

cn=

1000 99

100

n12

1000 99100

n2 1 11

10


189/607

cn n 2n n 2n 1

c2n1= 1000

99

100

n1

c2n = 1000

99

100

n1

11

10

99100

= 0, 99 < 1

cn

c2n

c2n1

cn lm

nc2n = L

lmn

c2n1=L lmn

cn=L

lmn

cn= 0

cn L 0

cn

9 %

1 9100

= 91100

L

L


190/607

c1= 1000 c2= 1000

1110

10 %

c3= 1000

1110

91100

9 %

c4= 1000

1110

2 91100

10 %

c5= 1000

1110

2 91100

2

9 %

c6= 1000

1110

3 91100

2

10 %

c7= 1000 1110

3

91100

3

9 %

cn

cn =

1000 1001

1000

n12

1000 10011000

n21 11

10

c2n1= 1000

1001

1000

n1

c2n= 1000

1001

1000

n1

11

10

10011000

= 1, 001> 1

cn c2n c2n1

+

lmn cn= +


191/607


192/607

an 0

an lmn an = L 0

L >0

1 = lmn

Lan+1

anL

= lmn

1n+ 1

n

r= r 0

L= 0


193/607


194/607

an = 3n2 + 1

n 0

an

bn

n

n

bn=

n+12 +1n+1

2

= 1 + 2n+1

11+ 2

n+1

bn

b2n1 = 1 +1

nn 1

b2n= 11 + 2

2n+1

n 1

bn lm

nb2n = L

lmn

b2n1 = L lmn

bn = L

lmn

bn = 1

L

L


195/607


196/607

an>0n N

an lm

nan = L

+

M = lmn bn = lmn3an

2an+ 1 = lmnan

3

an

2 + 1an

L

= 3

2+

1

L

L

R

32

L= +

L R M < 3

2

M

2

L= + M

32

M


197/607

lmn

3n

n+ 1 = lm

nn 3

n

1 + 1n0

= 3

an bn cn lmn an = lmn cn = L

lmn bn=L

an=n5

6n

lmn

an= 0

lmn

n

|an| = lmn

n

n5

n

6n= lm

n

nn1

56

=1

6


198/607


199/607

f(an

f(an) an f

f

an f

1

4

lmn

an = 1+

1

f(x)

4x

an

f(x)

f(an) 4

+

1


200/607


201/607


202/607

lmn

n

7notn

1 2

n

n2= lm

n7

1 1n2

n2

e1

2

= 7e2 = 7e2

0, 94< 1

lmn

5 3an= 0

bn = 5

lmn

3an= 5

cn =13

lmn

an =5

3


203/607

xn 3

1

lmn

xn=L R 1< L 3

L

1

xn >1n N

x1>1 xn

L= 1

L

1< L 3

1

2xn

lmn

1 2xnL

= 1 2L

1< L 3 13 1

L


204/607

an a

a= 0

lmn

an = lmn

n2

6

a+ 3bn2

+ 2n12

n6

n

4

5 3n3

+ 4n4

= lmn

+n2

a=0

a+0

3bn2 +0

2n 112 5 3

n30

+ 4n40

5

=

a = 0

lm

nan=

a

4

a

b

a = 0 a= 0

an= 3bn4 + 2n5n4 3n+ 4

lmn

an = lmn

n4

3b+ 2n12

n4

n

4 5 3n3

+ 4n4

= lmn3b+

0 2n7

5

3

n30 + 4

n40

=3b

5

4

3b5

= 4 3b= 20 b= 203

a= 0

b= 20

3


205/607


206/607


207/607

n N xn+1 xn

xn+1 xn 14

+x2n xn

x2n xn+1

4 0

xn 122

0

xn

xn

L

a

L

an+1 an L

L= lmn

xn+1 = lmn

1

4+

L2x2n =

1

4+L2

L

L2

L+1

4= 0

12

L=1

2

0< a 1

2 n N 0< an 12


208/607

x1 = a

0< x1

12

guia resuelta primer parcial analisis

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