ЛЕКЦІЇ ² ВПРАВИ

( k)
7.9.6. ( )
m
y = a + x , m∈ N; y (0) = ?
7.9.7. y = e x , y (k) (0) = ? 7.9.8. y = sin x, y (k) (0) = ?
7.9.9. y = cos x, y (k) (0) = ?
Ðîçâ’ÿçàííÿ.
7.9.1. Ïîñë³äîâíî äèôåðåíö³þþ÷è, îòðèìàºìî
(ó)′ = ó′ =5õ 4 +21õ 2 , (ó′)′ = y′′ =20õ 3 +42õ,
(ó′′)′ = ó′′′ =60õ 2 + 42; ó (4) =(ó′′′)′ = 120x,
ó (5) =(ó (4) )′ = 120 = 5!, ó (6) =(ó (5) )′ =0.
y′ lnx ′ 1
2
= = = x , y′′ =− x
x
−
3
, y′′′ = 2x − (4) 4
, y =− 6x − ,
(5) −5
24
y = 24x
=
5 .
x
y′ = xlnx ′ = ln x+ 1, y′′
= lnx+ 1 ′ = lnx ′ = x − . Òåïåð
−1
7.9.2. ( )
7.9.3. ( ) ( ) ( )
1
ñêîðèñòàºìîñÿ ðåçóëüòàòàìè ðîçâ’ÿçàííÿ ïðèêëàäó 7.9.2.
Ïðè öüîìó ìàòèìåìî
7.9.4. s ( arctg 2x)
y
24
= 24x
=
5 .
x
(6) −5
( x)
′
′ = ′ = =
1 2 +
( 2x)
2
+ ( x) 2 1 4x
2
2
( + x
′
) x
2 2
( 1+ 4x
) ( 1+
4x
)
21 4 16 16
s′′ =− =− , s′′
2 2
( − 1)
=
25
.
7.9.5. Ñïî÷àòêó ðîçãëÿíåìî âèïàäîê, êîëè k ≤ m. Äèôåðåíö³þþ÷è
k ðàç, îòðèìàºìî:
y′ = m x m-1 , y′′ = m(m − 1) x m-2 ,
y′′′ = m(m − 1) (m − 2) x m-3 , … ,
,
y (k) = m(m − 1) … (m – k +1)x m-k . (7.9.1)
ßêùî ó ôîðìóë³ (7.9.1) ïîêëàñòè k = m, òî îòðèìàºìî, ùî
y (m) = m(m − 1) … 1= m! . (7.9.2)
Òåïåð ðîçãëÿíåìî âèïàäîê, êîëè k > m. Îñê³ëüêè m! º
ñòàëà, òî î÷åâèäíî, ùî y (m+1) = 0. À öå îçíà÷àº, ùî äëÿ áóäüÿêîãî
k > m â³ðíà ôîðìóëà
y (k) = 0. (7.9.3)
Îá’ºäíàºìî ôîðìóëè (7.9.1) — (7.9.3). Ó ðåçóëüòàò³ îòðèìàºìî:
⎧ − − + <
( k)
y = x = m k=
m
⎪
⎩
0, k>
m.
m−k
mm ( 1)...( m k 1) x , k m;
( ) ( k
m
) ⎪
⎨ !, ;
(7.9.4)
7.9.6. Àíàëîã³÷íî, ÿê ³ â ïîïåðåäíüîìó ïðèêëàä³, ìîæíà
ïîêàçàòè, ùî
⎧ − − + + <
( k)
y = a+ x = m k=
m
⎪
⎩
0, k>
m.
m−k
mm ( 1)...( m k 1)( a x) , k m;
m
(( ) ) ( k ) ⎪
⎨ !, ;
²ç ôîðìóëè (7.9.5) âèïëèâàº, ùî
⎧ − − + <
( k)
y (0) = a+ x (0) = m!, k=
m;
⎪
⎩
0, k > m.
7.9.7. Ìàºìî
Îòæå,
m−k
mm ( 1)...( m k 1)( a) , k m;
m
(( ) ) ( k ) ⎪
⎨
′ ′
( ) ( )
( k)
(7.9.5)
(7.9.6)
x x x x x
y′ = e = e , y′′
= e = e ,..., y = e . (7.9.7)
( k)
( ) ( k
x
) 0
y (0) = e (0) = e = 1 . (7.9.8)
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