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\title{M 5710 - Fall 2013 \\
Quiz 1}
\author{Your name ......................}
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1. Solve, using Lagrange multipliers method
$$
\min_{x} {x_1^2+ x_2^2+ x_3^3 - x_1 + x_2 + 3 x_3} \mbox{ subject to }  x_1+ 2 x_2=1.
$$

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2. The network is shown in the figure:

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\noindent
Derive the formula for the current through the resistor $R_5$. 

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3. The triangular lattice was the first model used by Cauchy to derive equation of elasticity.
Using the symmetry, the problem can be reduced to the two-rods network. Compute vertical $u_V$ and horizontal $u_H$ displacements caused by vertical force $F=(0, 1)$. What are the specific displacements $\epsilon$
$$
\epsilon_V=\frac{u_H\V}{\mbox{vertical distance}}; \quad \epsilon_H=\frac{u_H}{\mbox{horizontal distance}}; 
$$ 
and the ratio between them? 
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