Chapter 9.EA, Problem 1EA

To determine

To verify:

The simplification of $\partial G/\partial x_1$ and $\partial G/\partial x_2$.

Answer to Problem 1EA

Solution:

The simplification of $\partial G/\partial x_1$ and $\partial G/\partial x_2$ is verified.

Explanation of Solution

Given:

The given function is,

$\begin{array}{rll}\frac{\partial G}{\partial x_1}& =& 2\left(-0.7\right)\left(0.4-0.7x_1-0.4x_2\right)+2\left(0.5\right)\left(-0.4\right)\left(0.15-0.4x_1-0.1x_2\right)\\ & & +2\left(0.4\right)\left(0.4\right)\left(-0.3\right)\left(0.25-0.3x_1-0.3x_2\right)+2\left(0.03\right)\left(x_1+x_2\right)\\ \frac{\partial G}{\partial x_2}& =& 2\left(-0.4\right)\left(0.4-0.7x_1-0.4x_2\right)+2\left(0.5\right)\left(-0.1\right)\left(0.15-0.4x_1-0.1x_2\right)\\ & & +2\left(0.4\right)\left(-0.3\right)\left(0.25-0.3x_1-0.3x_2\right)+2\left(0.03\right)\left(x_1+x_2\right)\end{array}$

Calculation:

Consider the equation for $\frac{\partial G}{\partial x_1}$. Simplify the equation further.

$\begin{array}{rll}\frac{\partial G}{\partial x_1}& =& \left[\begin{array}{l}-1.4\left(0.4-0.7x_1-0.4x_2\right)-0.4\left(0.15-0.4x_1-0.1x_2\right)\\ -0.24\left(0.25-0.3x_1-0.3x_2\right)+0.06\left(x_1+x_2\right)\end{array}\right]\\ & =& 0.56+0.98x_1+1.4x_2-0.06+0.16x_1+0.04x_2-0.06+0.072x_1+0.072x_2+0.06x_1+0.06x_2\\ & =& -0.68+1.272x_1+0.732x_2\end{array}$

Consider the equation for $\frac{\partial G}{\partial x_2}$. Simplify the equation further.

$\begin{array}{rll}\frac{\partial G}{\partial x_2}& =& -0.8\left(0.4-0.7x_1-0.4x_2\right)-0.1\left(0.15-0.4x_1-0.1x_2\right)\\ & =& -0.24\left(0.25-0.3x_1-0.3x_2\right)+0.06\left(x_1+x_2\right)\\ & =& -0.32+0.56x_1+0.32x_2-0.15+0.04x_1+0.01x_2-0.06+0.072x_1+0.06x_1+0.06x_2\\ & =& -0.395+0.732x_1+0.462x_2\end{array}$

Therefore, the simplification of $\partial G/\partial x_1$ and $\partial G/\partial x_2$ is verified.