※ 引述《stman (小強)》之銘言：
: sin(πx)
: 1.If f(x)=exp(g(x)), where g(x)=∫ √(1+t^2)dt. find f'(1)
: 0
f'(x) = (exp(g(x)))*(g'(x))
g'(x) = (√(1 + (sinπx)^2))*(πcos(πx))
g'(1) = (√(1 + (sinπ)^2))*(πcosπ)
= (√(1 + 0^2))*(π)*(-1) = -π
sinπ
g(1) = ∫ √(1 + t^2) dt
0
0
= ∫ √(1 + t^2) dt = 0
0
f'(1) = (exp(g(1))*(g'(1))
= (exp(0))*(-π) = -π
: 2.The base of a solid is the ellipse x^2+4y^2=4, and every parallel cross
: sections perpendicular to the x-axis are equilateral triangles. Find the
: volume of the solid?
: 3.Let S be the surface of the solid E thar lies above the cone z=√(x^2+y^2)
: and below the sphere x^2+y^2+z^2=z and F(x,y,z)=(z,y,sin(x+y)). Evaluate
: ∫∫ F‧dS =?
: S
: 4.Use the formula x^x=e^(xlnx) and the Maclaurin series for e^x to derive
: the formula 1 x ∞ n-1 n
: ∫ x dx= Σ((-1) )/(n )
: 0 0
: 5.Let f(x,y)=8xy-2x-4y+5. Find the absolute minimum value of the function
: f(x,y) in the set D, where D is the region bounded by the parabola y=x^2
: and the line y=4.
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