- 数学作业~已知dx/dt*d2y/dt2=dy/dt*d2x/d
- 已知dx/dt*d2y/dt2=dy/dt*d2x/dt2 (d2y/dt2是二次导数）
show that the points of inflectn 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