課程名稱︰微積分甲下
課程性質︰暑修
課程教師︰朱樺
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰106/09/08
考試時限（分鐘）：170
試題 :
2 -1 3
∂ tan y ln(y+1)+cos x
(1) (5%) Evaluate ------ (----------------------).
∂x∂y e^(sin y)
-1 x
(2) (5%) Find unit vectors u in which the rate change of f(x,y) = tan (---)
y
at the point P(3,4) in the direction u has value 4/25.
(3) (10%) Let S1 be the surface z = x^2+y^2 and S2 the surface 2x^2+y^2+z^2=7.
(a) Find the angle between S1 and S2 at the point P(-1,1,2).
(b) let C be the curve of intersection of S1 and S2. Find the equation of
tangent line to C at P.
(4) (10%) Let u = f(x,y) where x = e^s cos t and y = e^s sin t. Assume that u
has continuous second order partial derivatives. If
2 2 2 2 2
∂u ∂u ∂u ∂u ∂u
A ------ + B ------ = ------ + ------ + C ------ ,
2 2 2 2
∂s ∂t ∂x ∂y ∂x∂y
find A, B and C.
(5) (10%) Find the extreme values of f(x,y,z) = xy+z^2 on the ball
x^2+y^2+(z-1/2)^2≦1.
1 y y^2
(6) (10%) Evaluate ∫∫ ----------- dx dy.
0 y^3 √(1-x^2)
(7) (10%) Write five other iterated integrals that are equal to the integral
1 1 1-y
∫∫ ∫ f(x,y,z) dz dy dx.
0 1-x 0
□ □ □
(a) ∫ ∫ ∫ f(x,y,z) dz dx dy
□ □ □
□ □ □
(b) ∫ ∫ ∫ f(x,y,z) dy dx dz
□ □ □
□ □ □
(c) ∫ ∫ ∫ f(x,y,z) dy dz dx
□ □ □
□ □ □
(d) ∫ ∫ ∫ f(x,y,z) dx dz dy
□ □ □
□ □ □
(e) ∫ ∫ ∫ f(x,y,z) dx dy dz
□ □ □
(8) (10%) Let B be the solid bounded by 4x^2+4xy+2y^2+z^2 = 4. Find
∫∫∫ (x^2+y^2+z^2) dV.
B
x sin z y^z
(9) (10%) Let F =〈x+xy+xyz, ∫ ------- dz, x 〉. Find ▽‧F and ▽×F.
y z
(10) (10%) Let C be the curve (x^2+y^2)^3 = x^2. Find ∫ ∣y∣ds.
C
x^2 y+xy^2 -x^3-x^2 y
(11) (10%) Find ∫ F‧dr, where F =〈-------------, ------------〉and C is the
C (x^2+y^2)^2 (x^2+y^2)^2
closed (not simple) curve formed by the graphs of y = (x^2-1)(x-2) and
y = -(x^2-1)(x-2) oriented counter-clockwise.
(12) (10%) Find the area of the part of the sphere x^2+y^2+z^2 = b^2 that lies
inside x^2+y^2 = a^2 where 0<a<b.
2 x^2 2y^2 2 3z^2
(13) (10%) Evaluate ∫ F‧dr, where F =〈-y +e ,x+e ,z +e 〉and C is
C
the curve of intersection of x^2+y^2 = 1 and y+z = 2 oriented
counter-clockwise as viewed from above.
(14) (10%) Evaluate ∫∫ (x cos z + y^2 + z sin x) dS where S is the sphere
S
x^2+y^2+z^2 = 4.
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