数学作业~
已知dx/dt*d2y/dt2=dy/dt*d2x/dt2 (d2y/dt2是二次导数) show that the points of inflection of the curve defined parametrically by x=acost+bcos2t/2 y=asint+bsin2t/2 are given by cost=-(a^2+2b^2)/3ab, a,b are constant.
解: dx/dt=-asint-bsin2t d2x/dt2=-acost-2bcos2t dy/dt=accost+bcos2t d2y/dt2=-asint-2bsin2t 根据题目得出等式: (a^2)(sint)^2+2absintsin2t+absintsin2t+(2b^2)(sin2t)^2=(-a^2)(cost)^2-2abcostcos2t-abcostcos2t-(2b^2)(cos2t)^2 整理后得(注意二倍角公式):(asint)^2+(accost)^2+3absintsin2t+2b^2+3abcostcos2t=0 a^2+2b^2+3abcost=0 cost=-(a^2+2b^2)/3ab
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