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1. I = ∫<0, 1>xe^(-x^2)dx
= -(1/2)∫<0, 1>e^(-x^2)d(-x^2)
= -(1/2)[e^(-x^2)]<0, 1> = (1/2)(1-1/e).
2. 用罗比塔法则得
原式 = lim<x→0> -2sin(4x^2)/(3x^2)
= lim<x→0> -2(4x^2)/(3x^2) = -8/3
3. f = e^[arctan(y/x)]ln(x^2+y^2)
f'<x> = e^[arctan(y/x)]{1/[1+(y/x)^2]}(-y/x^2)]ln(x^2+y^2)
+ e^[arctan(y/x)]*2x/(x^2+y^2)
f'<x>(1, 0) = 0 + 2 = 2
= -(1/2)∫<0, 1>e^(-x^2)d(-x^2)
= -(1/2)[e^(-x^2)]<0, 1> = (1/2)(1-1/e).
2. 用罗比塔法则得
原式 = lim<x→0> -2sin(4x^2)/(3x^2)
= lim<x→0> -2(4x^2)/(3x^2) = -8/3
3. f = e^[arctan(y/x)]ln(x^2+y^2)
f'<x> = e^[arctan(y/x)]{1/[1+(y/x)^2]}(-y/x^2)]ln(x^2+y^2)
+ e^[arctan(y/x)]*2x/(x^2+y^2)
f'<x>(1, 0) = 0 + 2 = 2
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