Dz Dotrovina

Domáce zadanie

1. ročník Bc.

Funkcia viac premenných Dotyková rovina a normála ku grafu funkcie dvoch premenných Napíšte rovnice dotykovej roviny a normály ku grafu funkcie z = f (x, y) v danom dotykovom bode. 1. z = 3x2 + 2y 2

v bode

[−1, 2, 11] [6x − 8y + z + 11 = 0, x = −1 + 6t, y = 2 − 8t, z = 11 + t, t ∈ R]

2. z = x2 + xy 2 + z 3

v bode

[1, 0, 1] [2x − z − 1 = 0, x = 1 + 2t, y = 0, z = 1 − t, t ∈ R]

3. z =

p

19 − x2 + 2y 2

v bode

[1, 3, 6] [x − 6y + 6z − 19 = 0, x = 1 + t, y = 3 − 6t, z = 6 + 6t, t ∈ R]

4. z = ex cos y

v bode

[0, 0, 1] [x − z + 1 = 0, x = t, y = 0, z = 1 − t, t ∈ R]

5. z = exy − 3

[1, 0, −2]

v bode

[y − z − 2 = 0, x = 1, y = t, z = −2 − t, t ∈ R]

6. z = e2x+y + x2 − 2y − 3

v bode

[0, 0, −2] [2x − y − z − 2 = 0, x = 2t, y = −t, z = −2 − t, t ∈ R]

7. z = √

x x2 +y 2

v bode

[4, −3, 45 ] [9x + 12y − 125z + 100 = 0, x = 4 + 9t, y = −3 + 12t, z =

8. z = arctg

y x

4 5

[2, 2, π4 ]

v bode

[x − y + 4z − π = 0, x = 2 + t, y = 2 − t, z =

9. z = ln

p

x2 + y 2

− 125t, t ∈ R]

v bode

π 4

+ 4t, t ∈ R]

[1, 0, 0] [x − z − 1 = 0, x = 1 + t, y = 0, z = −t, t ∈ R]

10. z =

x+y x−2y

v bode

[3, 1, 4] [3x − 9y + z − 4 = 0, x = 3 + 3t, y = 1 − 9t, z = 4 + t, t ∈ R]

11. xy + xz + yz + 1 = 0

v bode

[1, 2, −1] [x + 3z + 2 = 0, x = 1 + t, y = 2, z = −1 + 3t, t ∈ R]

1

12. x2 + 3y 2 + 2z 2 = 9 v bode

[2, −1, 1] [2x − 3y + 2z − 9 = 0, x = 2 + 2t, y = −1 − 3t, z = 1 + 2t, t ∈ R]

13.

x2 4

− y2 −

z2 9

= 1 v bode

√ [4, 2, 3]

14.

x2 4

+

y2 9

z2 25

=1

[2, 3, 5]

−

v bode

√ √ √ [3x − 3 2y − z − 3 = 0, x = 4 + 3t, y = 2 − 3 2t, z = 3 − t, t ∈ R]

[15x + 10y − 6z − 30 = 0, x = 2 + 15t, y = 3 + 10t, z = 5 − 6t, t ∈ R]

15. z − y − ln xz = 0

v bode

[1, 1, 1] [x + y − 2z = 0, x = 1 − t, y = 1 − t, z = 1 + 2t, t ∈ R]

2