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Question Number 74300 by ~blr237~ last updated on 21/Nov/19
Letconsiderγ:I→R2aparametriccurve 1)Provethatifa<bandγ(a)≠γ(b)thenthereexistt0∈]a,b[ suchasγ′(t0)iscolineartoγ(b)−γ(a) 2)Showthatifγisregularandthefunctionf:I→Rt→f(t)=∣∣γ(t)−O(0,0)∣∣ismaximalint0∈I Then∣Kγ(t0)∣⩾1f(t0)
Answered by mind is power last updated on 21/Nov/19
γ(t)=(x(t),y(t)) γ(a)≠γ(b) ⇒x(a)≠x(b)ory(a)≠y(b) x(a)≠x(b) g(t)=y(t)(x(a)−x(b)−x(t)(y(a)−y(b)) g(a)=−y(a)x(b)+x(a)y(b) g(b)=y(b)x(a)−x(b)y(a)=g(a) g(a)=g(b)meanvalues⇒∃t0∈]a,b[∣g′(t0)=0 g′(t0)=y′(t0)(x(a)−x(b))−x′(t0).(y(a)−y(b))=0 sinceγ(a)≠γ(b)wehave3cases ifx(a)=x(b)⇒x′(t0)=0 x(a)≠x(b)&y(a)−y(b)=0⇒y′(t0)=0 You can't use 'macro parameter character #' in math modeYou can't use 'macro parameter character #' in math mode ⇔y′(t0)x′(t0)=y(a)−y(b)x(a)−x(b) y′(t0)x′(t0)iscoeficuentoftangentinto ⇒int0,γ′(t0)iscolineartoγ(b)−γ(a) 2)calculecorbur
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