TU Wien:Mathematik 2 UE (diverse)/Übungen WS07/Beispiel 105

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Es sei <amsmath>g_u(u,v)=\frac{\partial}{\partial_u}g(u,v)=u^2-v</amsmath> und <amsmath>g_v(u,v)=\frac{\partial}{\partial_v}g(u,v)=-u+v^3</amsmath>.
Man bestimme <amsmath>h(t)=\frac{d}{dt}g(2t,t^2+1)</amsmath>.


Lösung von saufnix

Formel: <amsmath>F'=g_{u}u'+g_{v}v'</amsmath> für <amsmath>F(t)=g(u(t),v(t))</amsmath>

<amsmath>u(t)=2t\Rightarrow u(t)'=2</amsmath>

<amsmath>v(t)=t^{2}+1\Rightarrow v(t)'=2t</amsmath>

<amsmath>h(t) = g_{u}(2t,t^{2}+1)\cdot u(t)'+g_{v}(2t,t^{2}+1)\cdot v(t)'=ln(2t\cdot sin(2t)-(t^{2}+1))\cdot2+tan(-2t+(t^{2}+1)^{3}\cdot2t</amsmath>

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