微積分好難啊~~~~~~~~請幫我解幾個問題！！(是原文的)

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1.The plane x+y+2z=2 intersects the paraboloid z=x^2+y^2 in an ellipse. Find the point on the ellipse that is farthest from the origin.


求
max. x^2+y^2+z^2
subject to: x+y+2z = 2,
z = x^2+y^2

2013-07-12 15:44:44 補充：
2.Find the volume of the solid bounded by the paraboloid z=1-x^2-y^2 and the
plane z=0.


求
V = ∫∫∫_{(x,y,z): x^2+y^2≦1, 0≦z≦1-x^2-y^2} dV
= ∫∫_{(x,y): x^2+y^2≦1} (1-x^2-y^2) dxdy
可化成極座標.

2013-07-12 15:46:10 補充：
3.Find the length of the curve y=x^(3/2) for 0 ≤ x ≤ 4.


dy/dx = (3/2)x^{1/2}

求
S = ∫_[0,4] √[1+(3/2)^2(x^{1/2})^2] dx

2013-07-12 15:47:25 補充：
4.Let f(x,y)=x^3-6xy+y^3.
(a)Find the critical points of f(x,y).
(b)Determine whether the critical points are points of maximum, minimum
values or saddle points.



(a) 求一階偏導數, 令為 0, 解聯立方程式.

(b) 用二階導數測驗. 請參考教本.

2013-07-12 15:52:12 補充：
5.For a differential equation x^2y''-3xy'+4y=0,
(a)use z=lnx to transform such an equation into an equation with constant
coefficients;
(b)find the general solution of (a) in terms of x.



(a) 等於在告訴你解法了!

自變數改成 z, 利用連鎖律求 dy/dz 及 d^2y/dz^2.


(b) 解出 y = g(z), 再把 z = ln(x) 代入.

2013-07-12 16:06:56 補充：
6.Let Q be the solid region cut from the sphere x^2+y^2+z^2=4 by the cylinder
r=2 sinθ.



r = 2 sin(θ) 相當於 x^2+(y-1)^2 = 1.

V = ∫∫_{(x,y): x^2+(y-1)^2≦1} 2√(4-x^2-y^2) dxdy
= ∫∫_{(r,θ): r≦2 sin(θ), 0≦θ≦π} 2√(4-r^2) r dr dθ
= ∫_[0,π] ∫_[0, 2sin(θ)] (2r)√(4-r^2) dr dθ

2013-07-12 16:08:08 補充：
7.Find the local extrema of f(x)=x^2 lnx for x>0, discuss concavity and find the
point of inflection.


二階導數存在, 看二階導數即知.

2013-07-12 16:11:42 補充：
8.Find th area of the surface generated by revolving the curve 6xy=x^4+3 from
x=1 to x=3 about the x-axis.


S = ∫_[1,3] 2πy ds

y = (1/6)(x^3+3/x), y' = (1/2)(x^2-1/x^2)
ds = √[1+(1/4)(x^2-1/x^2)^2] dx = (1/2)(x^2+1/x^2) dx

2013-07-12 16:15:49 補充：
9.Find the average value of f(x,y)=xy over the quarter circle x^2+y^2 ≤ 1 in the
first quadrant.


ave(f) = ∫∫_{x^2+y^2≦1,x≧0,y≧0} f(x,y) dA / ∫∫_{x^2+y^2≦1,x≧0,y≧0} dA
= ∫_[0,π;/2]∫_[0,1] r^3 cos(θ)sin(θ) dr dθ / (π/4)

2013-07-12 16:16:49 補充：
10.Find the points on the curve 17x^2+12xy+8y^2=100 that are closest to and
farthest away from the origin.


求
max. & min. x^2+y^2+z^2
subject to 17x^2+12xy+8y^2 = 100