北京大学期末复习-04物院高数期末试题.pdf

2004-2005 1. ( 26 ) (1)(8 ) ℄ u = f x , y ,f (, ) y x (2)(8 ) u = ln(x2 + 2y + 5z2) P (√2, −1, 1) k = (1, 1, −1) (3)(10 ) f (x + 2y + 3z, x2 + y2 + z2) = 0 z = z (x, y) f 2. ( 16 8 ) (1) y = √1 − 3x + x2 − √1 + 2x − x3 x = 0 Æ x2 (2) (0, 1) Æ z = x3 − x2 + xy + x − y . 3. ( 18 9 ) (1) (1, 2, −1) (−2, 3, 7) x − 1 y z + 1 = = −1 2 1 (2) ℄ |a| = 1, |b| = 2, |c| = 3, |a + b + c| = 17 + 6√3 a ⊥ b, a, c = π/3, b, c =? 4. (10 ) x → +∞  ()() 1 1 1 − cos x 10 ; arctanx; e ; (lnx) ; csc ; x x √ 1 x −x ln x ln(1 + 4 ); (ln x) ; e ; x ; x; x 5. (10 ) f (x, y, z, w) = x y z w 1 1 1 1 1 + + + = , a 0, x 0, y 0, z 0, w 0 x y z w a ai 0(1 ≤ i ≤ 4) √ 4 ≤ 4 a a a a . 1 1 1 1 1 2 3 4 + + + a1 a2 a3 a4 6. ( 10 ): (1) (4 ) ℄ P (x, y, z) S F (x, y, z) = 0, G(x, y, z) = 0 F G P S (X, Y, Z ), (2)(6 )