设x=f(t)cost-f’(t)sint,y=f(t)sint+f’(t)cost,f"(t)存在,证明: (dx)2+(dy)2=[f(t)+f"(t)]2(dt)2.

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问题 设x=f(t)cost-f’(t)sint,y=f(t)sint+f’(t)cost,f"(t)存在,证明:
(dx)2+(dy)2=[f(t)+f"(t)]2(dt)2.

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答案dx=[f’(t)cost-f(t)sint-f"(t)sint-f’(t)cost]dt =-sint[f(t)+f"(t)]dt dy=[f’(t)sint+f(t)cost+f"(t)cost-f’(t)sint]dt=cost[f(t)+f"(t)]dt 故(dx)2+(dy)2=[f(t)+f’’(t)]2(dt)2.

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