www.5615.net > 参数方程x=tCost y=tsint求二阶导数。 主要是过程...

参数方程x=tCost y=tsint求二阶导数。 主要是过程...

x'=-tsint+cost x''=-tcost-sint-sint=-tcost-2sint y'=tcost+sint y''=-tsint+cost+cost=-tsint+2cost

解:x=e^tsint,y=e^tcost,则dx/dt=e^t(sint+cost),dy/dt=e^t(cost-sint)dx/dt=2e^tcost,dy/dt=-2e^tsint从而dy/dx=(dy/dt dx/dt - dx/dt dy/dt)/(dx/dt) =-2(x+y)/(x+y)

先求dx=(cost-tsint)dt, dy=(sint+tcost)dt然后dy/dx=(sint+tcost)/(cost-tsint)根据x=tcost ; y=tsint; y/x=tant所以dy/dx=[y+yarctan(y/x)]/[x-yarctan(y/x)]

φ(t)=acost,ψ(t)=bsint,φ'(t)=-asint,ψ'(t)=bcost,φ"(t)=-acost,ψ"(t)=-bsint,φ'3(t)=asint故d^2y/dx^2=(-abcost*cost-absint*sint)/asint=-b/sint

用石头兄长的想法. 我认为也可这样做:将x=tcost,y=tsint,化为x^2+y^2=t^2.这是隐函数. 求导数得, 2x+2yy~=0 y~=-x/y.再求导数: y~~=-(y-xy~)/y^2 而 y=tsint的导数是:y~=sint+tcost. 将x、y、y~代入y~~中,得 y~~= [ sintcost+t(cost)^2-sint]/[t(sint)^2] 说明:y~代表一阶导数,y~~代表二阶导数. 上述对否我也没有验证过.我学习时是按石头兄长的办法.

dy/dx=(dy/dt)/(dx/dt)=(cost-tsint)/(1-sint-tcost)d^2y/dx^2=(dy/dx)/(dx/dt)=[(-sint-sint-tcost)(1-sint-tcost)-(cost-tsint)(-cost-cost+tsint)/(1-sint-tcost)^3=[(2sint+tcost)(sint+tcost-1)+(cost-tsint)(2cost-tsint)]/(1-sint-tcost)^3后面的展开计算化简了.

dx/dt=cost-tsintdy/dt=sint+tcostdy/dx=(sint+tcost)/(cost-tsint)

dx/dt=e^tsint+e^tcost dy/dt=e^tcost-e^tsint dy/dx=(e^tcost-e^tsint)/(e^tsint+e^tcost) =(cost-sint)/(sint+cost)

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