Question

微积分 高数？

（1）微分 (2)求极值

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Answer 1

(1)
y=e^x/x
dx = [(xe^x -e^x)/x^2]dx
(b)
y=sinx.cosx
=(1/2)sin2x
dy=cos2x dx
(c)
y=ln(x^2+1)
dy= [2x/(x^2+1) ]dx
(d)
y=arcsinx-arccosx
dy
=[1/√(1-x^2) +1/√(1-x^2)]dx
=[2/√(1-x^2)] dx
(e)
y=e^(x^2)
dy=2x.e^(x^2) dx
(f)
y=arctan(2x)
dy
=[2/(1+4x^2)]dx
(2)
f(x)=x^2.e^x
f'(x)=(2x+x^2).e^x
f'(x)=0
(2x+x^2).e^x=0
x(2+x).e^x=0
x=0 or -2
f''(x)= [(2x+x^2)+2+2x].e^x = [x^2+4x+2].e^x
f''(0)= 2 >0 (min)
f''(-2) = -2e^(-2) <0 (max)
min f(x) = f(0)=0
max f(x) = f(-2)=4e^(-2)