ListaCalcII

5/28/2018 ListaCalcII

1/22

dydx

= y2x

y2 =C x

dydx

= 3y1x

dydx

= x2

y2

x3 y3 =C

dydx =x2y2

dydx

= x2

y3

y4

4 = x

3

3 +C

dydx

=x2y3

dydx

= 2y

y = C e

x2

2

dydx

=ey sin x

dy

dx = 1 y2

y = Ce

2x1Ce2x+1

dydx

= 1 +y2

dydx

= 2 +ey

y = log(Ce2x 1

2)

dydx

=y2(1 y)

dydx

= sin x cos2 y

y = tan1(C cos x) +n

x dydx

=y log x

dydx

= x+yxy

2tan1(y

x) = log(x2 +y2) +

C

dydx

= xyx2+2y2

dydx

= x2+xy+y2

x2

tan1(y

x) = log |x| +C

dydx

= x3+3xy2

3x2y+y3

x dydx

=y +x cos2(xy

)

y=x tan1(log |Cx|)

dydx

= yx

e

y

x

x2 y2 = c

c

dxdy

= xy

xy

(2, 3)

(x, y)

2x

1+y2

y3 + 3y 3x2 = 24

(1, 3)

1 + 2yx

=

xx0 =y y0 dy

dx=

ax+by+c

ex+f y+g


2/22

dd = a+be+f

(x0, y0)

ax+by+c= 0ex+f y+g = 0.

dydx

= x+2y42x

y

3

dydx

2yx

=x2

y = x3 +cx2

dydx

+ 2yx

= 1x2

dydx

2y= 3

y = 3

2+Ce2x

dydx

+y =ex

dydx

+y =x

y = x 1 +Cex

dydx

+ 2exy = ex

dydx

+ 10y = 1y( 1

10) = 2

10

y = 1+e

(110t)

10

dydx

+ 3x2y = x2

y(0) = 1

dydx

+ (cos x)y = 2xe sinx

y() = 0

y= (x2 2)e sinx

x2dy

dx+ y = x2e

1x

y(1) = 3e

dydt y = 2te2t

y(0) = 1

dydt

+ 2t

y= cos tt2

y() = 0

tdydt

+ (1 +t)y= ty(ln 2) = 1

y0

y y= 1 + 3 sin t

y(0) =y0

t +

dydx

+ 12

y= 2 cos xy(0) = 1

t +

y + ay=bet

a >0

>0

b

R

t

dydt

+ y4

= 3 + 2 cos 2ty(0) = 0.

t > 0

12


3/22

R

2

R3

(xy2 +y)dx+ (x2y+x)dy= 0

2xy+x2y2 =C

(ex sin y + 2x)dx + (ex cos y +2y)dy= 0

exy(1 +xy)dx+x2exydy= 0

xexy =C

(2x+ 1 y2

x2)dx+ 2y

xdy = 0

x

(x2 + 2y)dx xdy= 0

log |x| y

x2 =C

(xex+x log y+y)dx+( x

2

y

+x log x+

x sin y)dy= 0

M(x, y)

N(x, y)

M dx+N dy= 0

(y)

(y)

2y2(x+y2)dx+xy(x+ 6y2)dy= 0.

(y)

ydx (2x+y3ey)dy= 0.

x y2ey =C y2

R2

R3

R

2R

3

(1, 2)

n = (1, 3)

x+ 3y

5 = 0

(3, 1)

2x3y = 7

(x, y) = (3, 1) +t(2, 3)

(1, 2)

v = (1, 1)

(x, y) = (1, 2) +t(1, 1)

3x + 2y = 2

(2, 3)

(12

, 1)

3x+ 2y = 2

(x, y) = (12

, 1) +t(2, 3)

(1, 1, 0)

(0, 0, 1)

(2, 0, 0)

(0, 1, 0)

(1, 1, 0)

(1, 8, 4)

(1, 1, 0)

+ k


4/22

R

2

R3

(0, 1, 2)

+ + k

xy

R1= (t, 3t

1, 4t)

R2 = (3t, 5, 1t) t R

c R

R1= (t, 6t+c, 2t 8)

R2= (3t+ 1, 2t, 0)

n

(1, 1, 1), n = (2, 1, 3)

2x+

y+ 3z= 6

(2, 1, 1), n = (2, 1, 2)

2x y 2z= 5

u

v

u= (1, 2, 1), v = (2, 1, 2)

(5, 4, 3)

u = (3, 2, 1), v = (1, 2, 1)

(4, 2, 8)

(0, 1, 1), x+ 2y z = 3 (x,y,z) = (0, 1, 1) +t(1, 2, 1)

(2, 1, 1), 2x+y + 3z= 1

(x,y,z) = (2, 1, 1) +t(2, 1, 3)

(1, 2, 1)

u = (1, 1, 1) v = (1, 2, 1)

(x,y,z) = (1, 2, 1) +t(3, 0, 3)

u

v

(1, 2, 1), u = (1, 1, 2), v =(2, 1, 1)

x y+z= 0

(0, 1, 2), u = (2, 1, 3), v =(1, 1, 1)

4x+y+ 3z= 7

3 + 4

3+ 4k

u= (1, 2)

5

u= (2, 1, 3)

14

u= (0, 1, 2)

5

u= (1

2, 13

)

136

u

v

R

3

u v ||u+ v||2 = ||u||2 + ||v||2.

xy

2

(2, 8, 1)

(1, 1, 1)

1

13(+ +k)

(1, 1, 1)

(2, 1, 2)

k

(1, 1, 2)

x= 3t+ 2, y = t 1, z=t 1

(1, 1, 0)

(1, 0, 1)

(2, 3, 1)


5/22

R

2

R3

+ + k

(1, 0, 0)

+ + k

(a,b,c)

a+b+ck

(1, 0, 0)(0, 2, 0)

(0, 0, 3)

2x+ 3y+z= 0

8x y 2z+ 10 = 0

(1, 1, 1)

(1, 1,

1)

(1 + t, 1 t, t)

(1, 1, 1)

3x + 4y + 5z= 6

xy + z=

4

x+

y = z y+z = x

(1, 1, 1)

x y z+ 10 = 0

(2, 1, 2)

2x y+z= 5

(1, 2, 3), (1, 2, 3) (0, 0, 1)

(4, 2, 0)

(0, 0, 0), (1, 1, 1)

(1, 1, 2)

R2

F

F(t) = (t, 2t)

F(0)

F(1)

F

F

F(t) = (t, t2)

F

F(t) = (cos t, sin t)

t [0, 2]

F

F(t) = (2 cos t, sin t)

t [0, 2]

F(t) = (1, t)

F(t) = (t, t+ 1)

F(t) = (2t 1, t+ 2)

F(t) = (t, t3)

F(t) = (t2, t)

F(t) = (t2, t4)

F(t) = (cos t, 2sin t)

F(t) = (sin t, sin t)

R

3

F(t) = (t,t,t)

F(t) = (cos t, sin t, 1)

F(t) = (cos t, sin t,bt)

b >0

t

0


6/22

R

2

R3

F(t) = (1, t, 1)

F(t) = (1, 1, t)

F(t) = (t,t, 1)

F(t) = (1, 0, t)

F(t) = (t,t, 1 + sin t)

F(t) = (t, cos t, sin t)

F

F(t) =

(log t,t,

1 t2, t2)

F

0< t 1

F(t) = (t,

t 2t+ 1

, log(5 t2), et).

5< t < 1

2 t < 5

R3

F(t) = (t, sin t, 2)

G(t) =(3, t , t2)

F(t) G(t)

3t + t sin t + 2t2

et F(t)

(et, et sin t, 2et)

F(t) 2 G(t)

(t 6, sin t 2t, 2 2t2)

F(t) G(t) (t2 sin t2t, 6t3, t2 3sin t)

r(t)x(t)

r(t) =ti + 2j +

t2k x(t) =ti j +k

(2 + t2)i +

(t3 t)j 3tk

u(t) v(t)

u(t) = sin ti+

cos tj +tk v(t) = sin ti+ cos tj+k

1 +t

F ,G,H

A R

R3

F G= G F

F ( G+ H) = F G+ F H

F ( G+ H) = F G+ F H

R

3

limt1 F(t) F(t) =(t1t1 , t

2, t1t

)

(12

, 1, 0)

limt0 F(t) F(t) =( tan3t

t , e

2t1t

, t3)

(3, 2, 0)

R3

dFdt

d2 Fdt2

F(t) = (3t2, et, log(t2 +1))

(6t, et, 2t1+t2

)

(6, et, 22t2

(1+t2)2)

F(t) = (

3t2, cos t2, 3t)

( 233t , 2t sin t2, 3) ( 29t3t

, (2 sin t2 + 4t2 cos t2), 0

F(t) = (sin 5t, cos4t, e2t)

(5cos5t, 4sin4t, 2e2t)

(25sin5t, 16cos4t, 4e2t)


7/22

R

2

R3

F(t) = (cos t, sin t, t)

F(3

)

(x,y,z) = (1

2,32

, 3

) +

t(32 , 12 , 1), t R

F(t) = (t2, t)

F(1)

(x, y) = (1, 1) +t(2, 1), t R

F(t) = (1t

, 1t

, t2)

F(2)

(x,y,z) = (1

2, 12

, 4) +t(1

4, 1

4, 4), t R

F(t) = (t, t2, t , t2)

F(1)

(x , y, z, w) = ( 1, 1, 1, 1) +

t(1, 2, 1, 2), t R

F : I R3

I

I

t I

d2 Fdt2

(t) = F(t)

F(t) d Fdt

(t)

I

F :

R

R

3

t 0 || F(t)|| = t

dFdt

(t) d Fdt

(t) = Fd2 Fdt2

(t)

[0, +]

F

d2 Fdt2

(t)

2

||v(t)|| = 0 t T(t) = v(t)||v(t)||

T

dTdt

(t)

r(t) = (a cos wt,b sin wt)

a,b,w

d2r

dt2(t) = w2r.

R3

10(t, et)dt= (1

2, (e 1))

11(sin3t,

11+t2

, 1)dt= (0, 2

, 2)

21(3, 2, 1)dt= (3, 2, 1)

T(t) = (t, 1, et)

G(t) = (1, 1, 1)

10( T(t) G(t)) = (2e, e 3

2, 1

2)

10( T(t) G(t)) = 1

2+e

F(t)

t

t1 t2 F(t)

[t1, t2]

I=

t2t1

F(t)dt

F

[t1, t2]

F

F(t) = (t, 1, t2); t1 = 0, t2 = 2

(2, 2, 83

)

F(t) = ( 1t+1

, t2, 1); t1 = 0, t2 = 1

(2, 1

3

, 1)

F(t)

t

m

F

[t1, t2]

t2t1

F(t)dt= m v2 m v1,


8/22

v2 v1

t2 t1

F(t) =ma

f

f(x, y) =

y

x+ 1

y.

w= f(u, v)

u2 +v2 +w2 = 1, w 0.

z=f(x, y)

z= y x2.

f(x, y) =y/x

f(x, y) = x+y

xy

f(x, y) = x+y

x2+y21

f(x, y) = 2xy

x2+y2

f(x,y,z) = 2x+yz

x2+y2+z2

1

f(x,y,z) = z

x24y21

f(x, y) = x

2+y2

x2y2

f(x, y) = 2xsin(y)

1+cos(x)

f(x, y) = e

xey1+sin(x)

f(x, y) = sin(xy)

x2+y21

f(x, y) = 3x+ 2y

f(1, 1) = 1

f(a, x) = 3a+ 2x

f(x+h,y)f(x,y)h

= 3

f(x,y+k)f(x,y)k

= 2

f(x, y) = xy

x+2y

{(x, y) R2 :x = 2y}

f(2u+v, v u)

uv

z=f(x, y)

x+y 1 +z2 = 0, z 0

f(x, y) = xy

1x2y2

z=

y x2 + 2x y

z= log(2x2 +y2 1)

z2 + 4 =x2 +y2, z 0

z=

|x| |y|

f : R2 R

f(1, 0) = 2

f(0, 1) = 3

f(x, y)

f(x, y) = 2x+3y

f(x, y) = x

3+2xy2

x3y3

f(x, y) =

x4

+y4

f(x, y) = 5x3y + x4 + 3

f(x, y) = 2

x2+y2

f : R2 R

f(a, b) =a

(a, b)

a2 +b2 = 1


9/22

f(4

3, 4) = 32

3

f(0, 3) = 0

f(x, y), (x, y) = (0, 0)

f : R2 R

f(a, b) = 0

(a, b)

a2 + b2 = 1

f(x, y) = 0

(x, y) = (0, 0)

g : [0, 2[ R

f :

R2 R

=

0

[0, 2[

f(cos , sin ) = g()

f(x, y) = 1 x y

1

1

f(x, y) = 2xy

x2+y2

1, 0

1

f(x, y) = x+y

xy 1 0

f(x, y) = x

2+y2

x2y2 1, 0

1

R

f(x, y) = 31/(x2+y2)

1/e

1

4

R

y = 0

f

xz

f

f

z= 1x2+y2

f

z= y

x1

f(x, y) = 2xy2x2+y4 , (x, y) = (0, 0)

f(x, y) = 1 x2 y2

f(x, y) =x+ 3y

z= 4x2

+y2

f(x, y) = 1 +x2 +y2

z= x+y+ 1

f(x, y) =

1 x2 y2

f(x, y) = x 2y

Im(f) =

R


10/22

f(x, y) = y

x2 Im(f) = R

z= xy

x+y

Im(f) = R

f(x, y) = x

y1 Im(f) = R

z=xy

Im(f) = R

f(x, y) =x2 y2

Im(f) =

R

z = 4x2 + y2

Im(f) =

[0, +[

z= 3x24xy+y2

Im(f) =R

f(x, y) = x2x2+y2

(t) = (x(t), y(t), z(t))

xR cos t y =R sin t

z=f(x(t), y(t))

R >0

f

T(x, y) = 2x+y(oC)

xy

0oC, 3oC, 1oC

z=x2 + 2

z= |y|

z2 +x2 = 4

x2 +y = 2

x=

8z2 +x

z=

x2 +y2

z= max{|x|, |y|}

z= sin(x)

y = 1 x2 z2

z = x2 y2

xz= 1

z=x2 +y2

=

f(x, y) = 5x4y2 +xy3 + 4

fx

= 20x3y2 +y3

fy

= 10x4y+ 3xy2

z= cos xy

zx

= y sin xyzy

= x sin xy

z= x

3+y2

x2+y2

zx

= x4+3x2y22xy2

(x2+y2)2

zy

= 2x2y(1x)

(x2+y2)2

f(x, y) =ex

2y2

fx

= 2xex2y2fy

= 2yexy2

z= x2 log(1 +x2 +y2)

z= xyexy

zx =yexy(1 +xy)

zy

=xexy(1 +xy)

f(x, y) = (4xy 3y3)3 + 5x2y

fx

= 12y(4xy3y3)2 +10xyfy

= 3(4xy3y3)2(4x9y2)+5x2

z= arctan x

y

zx

= yx2+y2

zy

= xx2+y2


11/22

f(x, y) =xy

fx

=yxy1

fy =xy log x

z= (x2 +y2)log(x2 +y2)

zx

= 2x[1 + log(x2 +y2)]zy

= 2y[1 + log(x2 +y2)]

f(x, y) =

3x3 +y2 + 3

fx

= x2

3(x3+y2+3)2

fy

= 2y33(x3+y2+3)2

z= xy2

x2+y2

x zx

+y zy

=z

: R R

(1) = 4

z(x, y) = (xy

)

zx

(1, 1)

zy

(1, 1)

4

4

z(x, y)

xz

x(x, y) +y

z

y(x.y) = 0

(x, y) R2

y= 0

: R R

z=(x y)/y

z+y zx

+y zy

= 0,

x R

y= 0

z =

x sin xy

xz

x+y

z

y =z.

p= p(V, T)

pV =nRT

n

R

pV

pT

pV

= nRTV2

pT

= nRV

z = ey(x y)

z

x+

z

y =z.

: R R

f(x, y) = (x2 +y2)(xy

)

xf

x+y

f

y = 2f.

z = ex

2+y2 , x = cos y =

sin

z

=ex2

+y

2

(2x cos + 2y sin ).

z

=

z

xcos +

z

ysin .

z = z(x, y)

xyz+ z3 =

x zx zy x, y

z

zx

= 1yzxy+3z2

zy

= xzxy+3z2

z=f(x+at)

f

z

t =a

z

x.


12/22

z = f(x2 y2)

f(u)

yz

x+x

z

y = 0.

w= xy +

z4

z = z(x, y)

zx

(x = 1, y = 1) = 4

z= 1

x = 1

y = 1

wx

(x = 1, y =1)

f(x, y) = e

x2 (2y x)

2f

x+

f

y = f.

f(x, y) =

x2+y20

et2dt

fx

(x, y)

fy

(x, y)

fx

(x, y) = 2xe(x2+y2)2

fy

(x, y) = 2ye(x2+y2)2

f(x, y) =

y2x2

et2dt

fx

(x, y)

fy

(x, y)

fx

(x, y) =2xex4

fy

(x, y) =

2yey4

f(x,y,z) =xyz

f(x,y,z) =

x2 +y2 +z2

f(x,y,z) =xexyz

w= x2 arcsin y

2

w= xyz

x+y+z

f(x,y,z) = cos(xy3) +e3xyz

f(x,y,z) = sin(x2 +y2 +z2)

f(x,y,z) =xyz

s = f(x , y, z, w)

s =

xw log(x2 +y2 +z2 +w2)

f(x, y) = 3x2 +2sin(x/y2)+ y3(1

ex)

fx(2, 3) fx(0, 1), fy(1, 1) fy(1, 1)

s

estu2

r

13

r2h

cos()1+2+2

a

(bcd)

limy03+(x+y+y)2z(3+(x+y)2z)

y .

f(x,y,z) = x

x2+y2+z2

xf

x+y

f

y+z

f

z = f.

s = f(x , y, z, w)

s =ex

y zw

xs

x+y

s

y+z

s

z+w

s

w= 0.

f : R R

f(3) = 4

g(x,y,z) =

x+y2+z40

f(t)dt.

gx

(1, 1, 1)

gy

(1, 1, 1)

gz

(1, 1, 1)


13/22

z= 3x2 + 2y2

z= sin(x2

3xy)

z= (2x

2+7x2y)3xy

z=x2y2e2xy

f(x,y,z) =x2y+xy2 +yz2

fxy, fyz , fzx, fxyz

u = u(x, y)

u= 2xy

(x2+y2)2

u= cos(xy2)

u= exy

2+y3x4

u= 1

cos2 x+ey

z= f(x, y)

2

zx2 + 2

zy2 = 0

z(x, y) = x3

3xy2

z = f(x, y) = log(x2 +y2)

f(x, y) =x2

y2

f(x, y) =x2 +y2

f(x, y) =xy

f(x, y) =y3 3xy2

f(x, y) =ex sin y

f

g

z=f(x t) + g(x

t)

z

2zt2

= 2z

x2

w= f(x, y)

x= u+v

y =

u v

2w

uv =

2w

x2

2w

y2.

z= x4y3 x8 + y4

3zyxx

3zxxy

3zxyy

3zyyx

f(x,y,z) =

1x2+y2+z2

fxx+fyy +fzz = 0.

f(x, y) =exy

2

f(x, y) =x4 +y3

f(x, y) =x2y

f(x, y) = log(1 +x2 +y2)

f(x, y) =x cos(x2 +y2)

z= x3y2

z= sin xy

u= es

2r2

T= log(1 +p2 +v2)

z=

x+

3y

z

(1, 8)

dz= 1

2dx+ 1

12dy


14/22

z

x= 1, 01 y =

7.9

2.9966

z

z

x = 1, y = 8

x =

0.9, y= 8.01

z 0.049166

A

x = 2m y = 3m

x= 2, 01

y= 2.97m

A 0.03

0.03m

2m

1m

V 0.15

h = 20cm

r = 12cm

V

h

2mm

r

1mm

(1, 01)2,03

(0, 01)2 + (3, 02)2 + (3, 9)2

(1, 01)2(1 1, 98)

(0.99)3 + (2, 01)3 6(0, 99)(2, 01)

tan

+0,013,97

(4, 01)2 + (3, 98)2 + (2, 02)2

(0, 98)sin

0,991,03

1,010,97

(0, 98)(0, 99)(1, 03)

(1, 01)0,97

f(x, y) =x2 + y2

fy(1, 1)

f

x= 1

(1, 1, f(1, 1))

f(x, y) =exy

f(x, y) = x

y

x = x(t)

y = y(t)

x = 3 + 8t, y =

3 2t

T = x2 cos y

y2 sin x

dTdt

T

t

x =(3 +t)2

y = 2 t2

T = ex(y2 + x2)

dTdt

T

t

dfdt

f(x, y) = (x2 +y2)log(

x2 +y2)

(x, y) = (et, et)


15/22

f(x, y) = xex

2+y2

(x, y) =(t, t)

f(x,y,z) = x + y2 + z3 (x,y,z) = (cos t, sin t, t)

f(x,y,z) = (y2 x2)exz

(x,y,z) = (t, et, t2)

f(x,y,z) = xy

+ yz

+ zx

(x,y,z) = (et, et2, et

3)

f(x,y,z) = sin(xy)

(x, y) =(t2 +t, t3)

z=

x2 +y2 + 2xy2 x y

u

dzdu

u = sin(a+ cos b)

a

b

t

dudt

(x,y,z)

T(x,y,z) = x2 +y2 + z2

(t) = cos(t)+ sin(t)+tk T(t)

t

T(t)

t R

t=

2+ 0, 01

f(x, y) = yx

ddx

(xx)

ddx

(xx)

y

x

dydx

x2 + 2y2 = 3

x2 y2 = 7

xy

= 10

y sin x3 +x2 y2 = 1

x3 sin y+y4 = 4

ex+y

2+y3 = 0

y

x

dydx

3x2 +y2 ex = 0

(0, 1)

x2 +y4 = 1 (1, 1)

cos(x+y) =x+ 1

2

(0, 3

)

cos(xy) = 12

(1, 3

)

dxdy

x

y

F(x, y) = 0

dxdy

y

x

F(x,y,x+y) = 0 F(x,y,z)

dydx

(x,y)(t,s)

(u,v)(x,y)

x= t + s, y = t s, u= x2 + y2

v = x2

y2

(u,v)(t,s)

(u,v)(t,s)

x = t2 s2, y = ts,u = sin(x+y), v= cos(x y)

x= ts, y = ts, u= x, v= y

x= t2+s2, y=t2s2, z= 2ts,u=xv,v=xz


16/22

u = f(x,y,z)

x = r cos sin

y =

r sin cos

z = r cos

ur

u

u

ux

uy

uz

zx

zy

z= u2 +y2, u = 2x+ 7, v = 3x+y+ 7

z = u2 + 3uv v2, u = sin x, v = cos x+ cos y

z = sin u cos v, u = 3x2

2y, v =x 3y

z= u

v2, u= x+y, v=xy

f(x, y)

f(x, y) =

x2y

(2xy,x2)

log

x2 +y2

xex

2+y2

ex

2y2

ex

2y2(2x, 2y)

(x2 +y2)log

x2 +y2

xy

( 1y

,

xy2

)

xexy

3+3

f(x,y,z)

f(x,y,z) =

x2 +y2 +z2

1x2+y2+z2

(x,y,z)

xy2 +yz2 +zx2

x2 +y2 +z2

(2x, 2y, 2z)

xy+yz+xz

(x2 +y2 + 1)z

2

(2xz2(x2 + y2 +

1)z21, 2yz2(x2 + y2 +

1)z21, 2z(x2 + y2 + 1)z

2log(x2 +

y2 + 1))

f(x, y) =x2y2

f(x0, y0) (x0, y0) =

(1, 1)

(1, 1)

(1, 1)

(1, 1)

f(x, y)

f(x, y) =

xy

f(x, y) = (y, x)

2xy

f(x, y) = 2xy log 2(1, 1)

x tan x

y

f(x, y) = (tan x

y +

xy

sec2 xy

, x2y2

sec2 xy

)

f(x, y) = y x2

(t) =

(sin t, sin2 t)

y x2 =

0

(t) f((t)) = 0


17/22

f(x,y,z) = xz+yz+ xy

(t) =

(et, cos t, sin t)

2et cos t+ cos2 t sin2 t

f(x,y,z) = exyz

(t) =(6t, 3t2, t3)

f(x,y,z) =

x2 +y2 +z2

(t) =

(sin t, cos t, t)

t1+t2

f(x, y) = x2 + y2 3xy3

(1, 2)

v= (12

,32

11 163

f(x, y) = 17xy

(1, 1)

v =(

2,

2)

172

f(x,y,z) =x22xy+3z2

(1, 1, 2)

3(1, 1,

1)

143

f(x,y,z) = sin(xyz)

(1, 1, 4

)

( 12

, 0, 12

)

8 1

2

(1, 1)

f(x, y) =x2 + 2y2

15

(1, 2)

g(x, y) =x2 2y2

h(x, y) =ex sin y

(sin1, cos1)

p(x, y) =ex sin y ex cos y

T=ex + ezy+ e3z

(1, 1, 1)

f

g

f= 0 f

(f+ g) = f+g

(cf) = cf c

(f g) =fg+g f

( fg

) = gffg

g2

g=0

f(x, y) = x

2y2x2+y2

(1, 1)

(x, y)

x >0

y >0

f

(1, 1, 1)

T(x,y,z) =

ex22y23z2


18/22

e8

14e2

f(x, y) = 2x2y

(1, 1, f(1, 1))

z= 4x + 2y 4

(x,y,z) =

(1, 1, 2) +t(4, 2,

1)

f(x, y) = x3 + y3 6xy

(1, 2, f(1, 2))

f(x, y) =x2 + y2

(0, 1, f(0, 1))

z = 2y 1

(x,y,z) =

(0, 1, 1) +t(0, 2, 1)

f(x, y) = cos x cos y

(0, /2, f(0, /2))

f(x, y) = 3x2y xy

(1, 1, f(1, 1))

z= 8x+2y+8

(x,y,z) =

(1, 1, 2) +t(8, 2, 1)

f(x, y) = cos x sin y

(0, /2, f(0, /2))

f(x, y) =xex

2y2

(2, 2, f(2, 2))

z = 9x8y

(x,y,z) =

(2, 2, 2) +t(9, 8, 1)

f(x, y) = 1/(xy)

(1, 1, f(1, 1))

(1, 1, 2)

(1, 1, 1)

f(x, y) =xy

x+ 6y 2z= 3

z= 2x+y

f(x, y) =x2 +y2

z= 2x+y 5

4

z= 2x+y

f(x, y)

(1, 1, 3)

fx (1, 1) fy (1, 1)

23

13

(1, 1, 1)

(x,y,z) = (1, 1, 1) +

t(2, 1, 3)

f(x, y) = x

3

x2+y2

f

z = z(x, y)

x2

a2+

y2

b2 + z

2

c2 = 1

x0xa2

+ y0yb2

+z0zc2

= 1

(x0, y0, z0), z0= 0

xyz= 8, (1, 1, 8)

x2y2 +y z+ 1 = 0, (0, 0, 1)

cos(xy) =ez 2, (1, , 0)

exyz =e, (1, 1, 1)

m

(x,y,z)

M

F =


19/22

PmMr5

r

r = x + y + zk

r = x2 +y2 +z2

P

V F = V

F

V

x2 + 2y2 + 3z2 = 10, (1,

3, 1)

xyz2 = 1, (1, 1, 1)

x2

+ 2y2

+ 3xz= 10, (1, 2, 1/3)

y2 x2 = 3, (1, 2, 8)

xyz= 1, (1, 1, 1)

xyz

= 1, (1, 1, 1)

x2 + 2y2 = 3, (1, 1)

xy= 17, (x0, 17/x0)

cos(x+y) = 1/2, x= /2, y= 0

exy = 2, (1, log 2)

e(x

2+y2+z2) =e3, (1, 1, 1)

2x2 + 3y2 +z2 = 9, (1, 1, 2)

xyz = 1, (1, 1, 1)

xyz2 = 4, (1, 1, 1)

x2 +y2 +z2 = 1

(1, 1,

3)

t = 0

xy

S1 : x

2 +y2 +z2 = 6

S2 : 2x

2 + 3y2 +z2 = 9

S1 S2

(1, 1, 2)

(1, 1, 2)

S1

S2

x2 y2 +z2 = 1

2x2 y2 +

5z2 = 6

(1, 1, 1)

f(x, y) =

2x2 +y2 2xy+x y

x2 y2 + 3xy x+y

x3 y2 +xy+ 5

x3 +y3 xy

x4 +y4 + 4x+ 4y

x5 +y5 5x 5y

f(x, y)

x2 + 3xy+ 4y2 6x+ 2y


20/22

x2 +y3 +xy 3x 4y+ 5

x3 + 2xy+y2

5x

x2 +y2 + 2xy+ 4x 2y

x2 4xy+ 4y2 x+ 3y+ 1

x2 +xy+y2

y2

3 + 2x2 xy+y2

x2 xy+y2 + 1

x+ 2y

z= 4

z= x5y+

xy5 +xy

z =

log(x2 +y2 + 1)

z= e1+x2+y2

(0, 0)

z =

x3 +y3

z = x

33x1+y

2

(u, t)

R(u, t) = u2(1 u)t2et

0 u 1 t 0

E

T

T(, T) =

2k5T5

h4

c4

x5

ex

1

x = hckT

h

k

c

T

E = E(, T)

=hc

kTx0

5 x0 5ex0 = 0

f(x, y) = (x2 + 3y2)e1x

2y2

(0, 0, 0), (1, 0, 0), (0, 1, 0)

(0, 0, 1)

f(x,y,z)

C2

(x0, y0, z0) Df

(x0, y0, z0)

f

H(x,y,z)

H1(x,y,z)

H=

2fx2

2fxy

2fxz

2fxy

2fy2

2fyz

2fxz

2fyz

2fz2

H1 =

2fx2

2fxy

2fxy

2fy2

2fx2

(x0, y0, z0) > 0H1(x0, y0, z0)>0 H(x0, y0, z0)>0

(x0, y0, z0)

2f

x2 (x0, y0, z0) < 0

H1(x0, y0, z0)>0 H(x0, y0, z0)

0, y >0

f(x, y) =xy x2 + 4y2 = 8

f(x, y) =x2+2xy+y2

x+2y1 =

0

f(x, y) =x2 + 2xy + y2

x2 + y2 =

1

f(x, y) = 3x+ 2y

2x2 + 3y2 3

f(x, y) = xy

2x+ 3y 10, x

0, y 0

f(x, y) =x+y x2 +y2 = 1

f(x, y) =x y

x2 y2 = 2

f(x, y) =xy

x+y = 1

f(x, y) = cos2 x+ cos2 y

x+y =

/4

f(x, y) =

x2 + 16y2

xy = 1 x > 0, y > 0

x2 +16y2 = 8

(2, 1

2)

x+ 2y = 1

(12

, 14

)

y = x2

(14, 1)

(2, 4)

f(x,y,z) = x+ 2y + z

x2 + 2y2 +z2 = 4

4

(1, 1, 1)

4

(1, 1, 1)


22/22

x+ 2y

3z= 4

(27 , 47 , 67)

T

x2+y2+z2 = 1

T(x,y,z) =xz+yz

xyz=

1 x > 0, y > 0

(1, 1, 1)

f(x, y) = 2 0 0x + xy/8

{(x, y) R2 :x2 + 2y2 30000}

( c3

)3

xyz x 0, y 0

z 0

x+y+z=c (c >0)

100cm2

523

y

x

z

y

x

z

= 1.

F(x,y,z) = 0

x = f(y, z), y =g(x, z), z=h(x, y)

z= f(x, y)

fxx +

fyy = 0

(x0, y0) fxx(x0, y0)= 0

f

(x0, y0)

f

x2 +y2 1

x2 +y2 = 1

f

ListaCalcII

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