ĐẠO HÀM CẤP CAO

[Pages:4] - T?I LIU HC TP MIN PH?

O H?M CP CAO

A.T?M TT GI?O KHOA.

1. Cho h?m s y f x c? o h?m f 'x . H?m s f 'x c?n gi l? o h?m cp 1 ca h?m s f x . Nu h?m s f 'x c? o h?m th? o h?m ? c gi l? o h?m cp 2 ca h?m s f x , k? hiu l? y'' hay f ''x . o h?m ca o h?m cp 2 c gi l? o h?m cp 3 ca h?m s f x , k? hiu l? y''' hay f''' x . Tng t, ta gi o h?m ca o h?m cp n 1 l? o h?m cp n ca h?m s f x , k? hiu l? yn hay fn x , tc l? ta c?:

yn yn1 ' n N, n 1 .

2.o h?m cp 2 ca h?m s f(t) l? gia tc tc thi ca chuyn ng s=f(t) ti thi im t. B.PHNG PH?P GII TO?N. DNG 1: T?nh o h?m cp cao ca h?m s. 1.PHNG PH?P

?p dng trc tip nh ngha: yn yn1 ' t?nh o h?m n cp m? b?i y?u cu.

V? d: T?nh o h?m n cp ? ch ra ca c?c h?m s sau:

a). y x sin 2x,y'''

b). y cos2 x,y'''

c). y x4 4x3 3x2 1, y(n)

d). y x4 sin 2x, y(4) e). y sin2 2x, y(5)

f).

y

3x 1 ,

y(4)

x 2

LI GII

a). C? y' x' sin 2x x.(sin 2x)' sin 2x 2x cos 2x

y '' (sin 2x)' (2x)'cos 2x 2x(cos 2x)' 4 cos 2x 4x sin 2x

y''' 4(cos 2 x)' (4 x)'sin 2 x 4 x(sin 2 x)' 8 sin 2 x 4 sin 2 x 8 cos 2 x

12 sin 2 x 8 cos 2 x .

b). Ta c? y cos2 x 1 1 cos 2x y ' sin 2x 2 y '' 2 cos 2x y''' 4 sin 2x

c). y x4 4x3 3x2 1

y ' 4x3 12x2 6x y'' 12x2 24x 6 y''' 24x 24

Group:

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y(4) 24 y(5) 0 ... y(n) 0 . d). y x4 sin 2x

y ' 4x3 2 cos 2x y'' 12x2 4 sin 2x

y ''' 24x 8 cos 2x y(4) 24 16 sin 2x

e). y sin2 2x 1 1 cos 4x 2 y ' 2 sin 4x y '' 8 cos 4x y ''' 32 sin 4x y(4) 128 cos 4x y(5) 512 sin 4x

f).

3x 1

y

,

y( 4)

x 2

/

y'

7

y ''

7

x

22

14

(x 2)2

x 24 x 23

/

/

y '''

14

x2

3

42

y(4)

42

x2

4

168

x 26 x 24

x 28

x 25

DNG 2: T?m o h?m cp n ca mt h?m s

PHNG PH?P Bc 1: T?nh y', y'', y''' . Da v?o c?c o h?m va t?nh, d o?n c?ng thc t?nh y(n) .

Bc 2: Chng minh c?ng thc va d o?n l? ?ng bng phng ph?p quy np.

Ch? ?: Cn ph?n t?ch k c?c kt qu ca o h?m y',y'', y''' t?m ra quy lut d o?n c?ng thc y(n) ch?nh x?c.

V? d 1: T?m o h?m cp n ca h?m s y sin x n N*

LI GII

Bc

1:

Ta

c?:

y

'

cos

x

sin

x

1.

2

;

y

''

sin

x

sin

x

2

2

D o?n:

yn

sin x n

1 ,n N*

2

Bc 2: Chng minh 1 bng quy np:

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 - T?I LIU HC TP MIN PH?

n 1 : 1 hin nhi?n ?ng.

Gi s

1

?ng vi

n k 1

ngha l? ta c?:

yk

sin x k

ta phi chng minh

1

c?ng

2

?ng vi n k 1 ngha l? ta phi chng minh

yk1

sin x

k1

2

2

/

Tht

vy

:

v

tr?i

2

yk1

yk /

sin

x

k

2

cos

x

k

2

sin

x

k

1

2

=v

phi

2

2 ?ng, ngha l? 1 ?ng vi n k 1.

Bc

3:

theo nguy?n

l?

quy

np

suy

ra

yn

sin

x

n

,

n

N*

.

2

V? d 2: T?m o h?m cp n ca h?m s y 1 n N* x 3

LI GII

Ta c?:

y' 1/

1

x 32

1/

1!

x 32

;

y'' 12 . 1.2 3 12 .

2! .

3

x 3

x 3

D o?n: yn 1n n!

1 ,n N* .

n1

x 3

Chng minh 1 bng phng ph?p quy np:

n 1 : 1 hin nhi?n ?ng.

Gi s 1

?ng

vi

n k 1

,

ngha

l?

ta

c?:

yk

1k

k!

k1

x 3

ta phi chng minh 1

c?ng

?ng vi n k 1 , ngha l? ta phi chng minh:

yk1

k1

1

k1 !

2

k2

x 3

Tht vy: v tr?i

/

2

yk1

yk /

1

k

k!

x3

k1

1

k1

.

k!

x3

k1 2

.

x3

k1 /

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k1

1 .

k!k 1

k1

1 .

k 1!

vt 2

k2

x3

k2

x 3

Vy 2 ?ng ngha l? 1 ?ng vi n k 1.

Theo

nguy?n

l?

quy

np

ta

suy

ra

yn

1n

.

x

n!

3n

1

,n N* .

DNG 3: Chng minh ng thc:

B?i 11:

a). Cho h?m s y x sin x . Chng minh x.y'' 2y' sin x xy 0

b). Cho h?m s : y 2x x2 chng minh: y3.y'' 1 0

c). Cho h?m s: y x tan x chng minh: x2.y'' 2 x2 y2 1 y 0

d). Cho h?m s: y x 3 chng minh: 2y'2 y 1.y''

x 4

LI GII

a). Cho h?m s y x sin x . Chng minh x.y'' 2y' sin x xy 0 Ta c? y ' x sin x/ y' x'.sin x x.sin x/ y' sin x x cos x

y'' sin x x cos x/ sin x/ x cos x/ cos x x'.cos x x.cos x/ 2 cos x x sin x

1 x 2 cos x x sin x 2sin x x cos x sin x x2 sin x 0

2x cos x x2 sin x 2x cos x x2 sin x 0 0 0 (pcm).

b). Cho h?m s : y 2x x2 chng minh: y3.y'' 1 0

/

Ta c?: y ' 2x x2 y '

1

.

2x x2

/

1x

.

2 2x x2

2x x2

1 x/ . 2x x2

y''

/

2x x2 . 1 x

2x x2

1 x .1 x

2x x2

2

2

2x x2

2x x2

2x x2 1 x2

1

.

2

3

2x x2 . 2x x2

2x x2

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