Use the branch-and-bound method to find the optimal solution to the following IP:

$$\begin{aligned} \max z &=7 x_{1}+3 x_{2} \\ \text { s.t. } & 2 x_{1}+x_{2} \leq 9 \\ & 3 x_{1}+2 x_{2} \leq 13 \\ & x_{1}, x_{2} \geq 0 ; x_{1}, x_{2} \text { integer } \end{aligned}$$

Use the branch-and-bound method to solve the following IP:

$$\begin{aligned} \max z &=x_{1}+2 x_{2} \\ \text { s.t. } & x_{1}+x_{2} \leq 10 \\ & 2 x_{1}+5 x_{2} \leq 30 \\ x_{1}, x_{2} & \geq 0 ; x_{1}, x_{2} \text { integer } \end{aligned}$$

State University must purchase 1,100 computers from three vendors. Vendor 1 charges $500 per computer plus a delivery charge of$5,000. Vendor 2 charges $350 per computer plus a delivery charge of$4,000. Vendor 3 charges $250 per computer plus a delivery charge of$6,000. Vendor 1 will sell the university at most 500 computers; vendor 2, at most 900; and vendor 3, at most 400. Formulate an IP to minimize the cost of purchasing the needed computers.

Use each of the following terms in a separate sentence. overharvesting

Prove the following Quotient Rule, where f, g are differentiable:

$$\nabla \left( \frac { f } { g } \right) = \frac { g \nabla f - f \nabla g } { g ^ { 2 } }$$

Find the sum or difference. Write your answer in standard form. $\left(-2 x+4 x^{2}-8\right)+\left(2-2 x^{2}+x\right)$

Graph $f(x)=\log _{4}(x)+1$. State the domain and range.

Find all solutions of $Ax = 0$ for the matrix

$$A=\left( \begin{array}{rrr}{1} & {0} & {-2} \\ {0} & {1} & {3} \\ {0} & {0} & {0}\end{array}\right)$$

. Express your answer in parametric form.

Which two major groups of fishes evolved from the early jawed fishes and still survive today?

Repeat the previous problem, only for the violet light. (a) $15^{\circ}$ (b) $20^{\circ}$ (c) $41^{\circ}$(d) $45^{\circ}$ (e) $49^{\circ}$

Let $\overline { X }$ be the mean of a random sample of size n from a $N \left( \theta , \sigma ^ { 2 } \right)$ distribution, $- \infty < \theta < \infty , \sigma ^ { 2 } > 0$ Assume that $\sigma ^ { 2 }$ is known. Show that $\overline { X } ^ { 2 } - \frac { \sigma ^ { 2 } } { n }$ is an unbiased estimator of $\theta ^ { 2 }$ and find its efficiency.

In a MOS cascode amplifier, the cascode transistor is required to raise the output resistance by a factor of 50. If the transistor is operated at $V_{O V}=0.2 \mathrm{V},$ what must its $V_{A}$ be? If the process technology specifies $V_{A}^{\prime}$ as $5 \mathrm{V} / \mu \mathrm{m},$ what channel length must the transistor have?

Prove that an invertible matrix A is normal if and only if $A^{*} A^{-1}$ is unitary.

A Sunco oil delivery truck contains five compartments, holding up to 2,700, 2,800, 1,100, 1,800, and 3,400 gallons of fuel, respectively. The company must deliver three types of fuel (super, regular, and unleaded) to a customer. The demands, penalty per gallon short, and the maximum allowed shortage are given in Table 29. Each compartment of the truck can carry only one type of gasoline. Formulate an IP whose solution will tell Sunco how to load the truck in a way that minimizes shortage costs. TABLE 29:

$$\begin{matrix} \text{Type of Gasoline} & \text{Demand} & \text{Cost per Gallon Short (\$)} & \text{Maximum Allowed Shortage}\\ \text{Super} & \text{2,900} & \text{10} & \text{500}\\ \text{Regular} & \text{4,000} & \text{8} & \text{500}\\ \text{Unleaded} & \text{4,900} & \text{6} & \text{500}\\ \end{matrix}$$