第二章第五节

2-7

1. 已知y =x 3-x , 计算在x =2处当∆x 分别等于1, 0.1, 0.01时的∆y 及dy . 解 ∆y |x =2, ∆x =1=[(2+1)3-(2+1)]-(23-2)=18, dy |x =2, ∆x =1=(3x 2-1)∆x |x =2, ∆x =1=11;

∆y |x =2, ∆x =0.1=[(2+0.1)3-(2+0.1)]-(23-2)=1.161, dy |x =2, ∆x =0.1=(3x 2-1)∆x |x =2, ∆x =0.1=1.1;

∆y |x =2, ∆x =0.01=[(2+0.01)3-(2+0.01)]-(23

-2)=0.110601,

dy |x =2, ∆x =0.01=(3x 2

-1)∆x |x =2, ∆x =0.01=0.11.

2. 设函数y =f (x )的图形如图所示, 试在图(a )、(b )、(c )、(d )中分别标出在点x 0的dy 、∆y 及∆y - d y 并说明其正负.

解 (a )∆y >0, dy >0, ∆y -dy >0. (b )∆y >0, dy >0, ∆y -dy <0. (c )∆y <0, dy <0, ∆y -dy <0. (d )∆y <0, dy <0, ∆y -dy >0. 3. 求下列函数的微分:

(1)x x

y 21

+=;

(2) y =x sin 2x ; (3)1

2+=x x y ;

(4) y =ln 2(1-x );

(5) y =x 2 e 2x

;

(6) y =e -x cos(3-x );

(7)21arcsin x y -=;

(8) y =tan 2

(1+2x 2

); (9)2

2

11arctan

x x y +-=; (10) s =A sin(ωt +ϕ) (A , ω, ϕ是常数) .

解 (1)因为x

x

y 112+-=', 所以dx x

x

dy )11(2+-=.

(2)因为y '=sin2x +2x cos2x , 所以dy =(sin2x +2x cos2x )dx .

(3)因为1

)1(11

1122222++=

++⋅

-+=

'x x x x x x y , 所以dx x x dy 1

)1(122++=

.

(4)dx x x dx x x dx x dx y dy )1ln(1

2])

1(1)1ln(2[])1([ln 2--=--⋅-='-='=.

(5)dy =y 'dx =(x 2e 2x )'dx =(2x e 2x +2 x 2e 2x )dx =2x (1+x ) e 2x

.

(6) dy =y 'dx =[e -x cos(3-x )]dx =[-e -x cos(3-x )+e -x sin(3-x )]dx =e -x

[sin(3-x )-cos(3-x )]dx . (7)dx x x x dx x x dx x dx y dy 2

2

221||)12()

1(11)1(arcsin

--

=--

⋅--=

'-='=.

(8) dy =d tan 2(1+2x 2)=2tan(1+2x 2)d tan(1+2x 2)=2tan(1+2x 2)⋅sec 2(1+2x 2)d (1+2x 2) =2tan(1+2x 2)⋅sec 2(1+2x 2)⋅4x dx =8x ⋅tan(1+2x 2)⋅sec 2(1+2x 2)dx .

(9))11()

11(1111arctan 22

22

222

x x d x x x

x d dy +-+-+=

+-=

dx x x

dx x x x x x x x 4

222222

214)1()1(2)1(2)

11(11+-=+--+-⋅+-+=. (10) dy =d [A sin(ω t +ϕ) ]=A cos(ω t +ϕ)d (ωt +ϕ)= A ω cos(ωt +ϕ)dx .

4. 将适当的函数填入下列括号内, 使等式成立: (1) d ( )=2dx ; (2) d ( )=3xdx ; (3) d ( )=cos tdt ; (4) d ( )=sin ωxdx ; (5) d ( )dx x 1

1

+=

; (6) d ( )=e -2x dx ; (7) d ( )dx x

1=; (8) d ( )=sec 23xdx . 解 (1) d ( 2x +C )=2dx .

(2) d (C x +22

3

)=3xdx .

(3) d ( sin t +C )=cos tdt . (4) d (C x +-

ωω

cos 1

)=sin ωxdx .

(5) d ( ln(1+x )+C )dx x 1

1

+=

. (6) d (C e x +--22

1

)=e -2x dx .

(7) d (C x +2)dx x

1=.

(8) d (C x +3tan 3

1

)=sec 23xdx .

5. 如图所示的电缆B O A

的长为s , 跨度为2l , 电缆的最低点O 与杆顶连线AB 的距离为f ,