91ç-ç‚Bù ‚25

91ç-ç‚Bù ‚25

1. (10%)°-D( r = 1 + cos θ 5CÅ£F¨ˇ5Þ
2. (10%)q f (x, y) = 2x²− xy + y²− 2x + y

(a) ° f Êõ P = (0, 0) £j² −→u = (√

3/2, 1/2) 5j²ûb (df ds)^−→u ,P? (b) Êõ P = (0, 0) T, µø_j² −→v íj²ûb (df

ds)^−→v ,P |×?

3. (10%)q (a, b, c) ÑÞ xyz = 1 ,5øõ, w2 a, b, c > 0
°âúè™Þ x = 0, y = 0, z = 0 ¸¬ (a, b, c) 5~ÞFˇA5ñ

4. (10%)°

Z ³

0

Z ¹

√^x

3

e^y3 dydx 5M

5. (15%)°â7 x²+ y²+ z² ≤ 1 £Æ6 (x − 1

2)²+ y² ≤ 1

4 ó>¶M, 7Ê z ≥ 0 T5 ñ

6. (15%)° f (x, y) = x² 2 −√

3xy − y²

2 Êx²+ y² ≤ 1 ,5"ú”×M£wêÞ5õ

7. (15%)qÞ 2y + 4z = 5 DÞ z² + 4x² + 4y² ó>, )( Γ
à Lagrange multiplier ¶, °Ê Γ ,, |Ô¡Ÿõíõ

8. (15%)I z = f (x, y), 7 x = u²− v², y = 2uv
z ∂z

∂x £ ∂z

∂y, à u, v £ ∂z

∂u, ∂z

∂v [ ý|V

9. (15%) q f (x, y) Å— ∂f

∂x = ∂f

∂y

(a) °ø_øŸ, ùŸ£úŸíÖá f (x, y), Å—,H‘K

(b) I u = x + y, v = x − y
° ∂f

∂v
(c) â5!‹, 5?ú¤ f -Bó!?

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