Suppose $f$ is differentiable at points on a closed path $\gamma$ and at all points in the region G enclosed by $\gamma,$ except possibly at a finite number of poles of $f$ in G. Let Z be the number of zeros of $f$ in G, and P the number of poles of $f$ in G, with each zero and pole counted as many times as its multiplicity. Show that $\frac{1}{2 \pi i} \oint_{\gamma} \frac{f^{\prime}(z)}{f(z)} d z=Z-P.$ This formula is known as the argument principle.

Garcia Co. had three major business transactions during 2014.

(c) Garcia Co. wanted to make its 2014 income look better, so it added 2 more weeks to the year (a 54-week year). Previous years were 52 weeks.

Instructions

In each situation, identify the assumption or principle that has been violated, if any, and discuss what the company should have done.

use Archimedes’ Principle, which states that when a sphere of radius r with density

$$d_s$$

is placed in a liquid of density

$$d_L=62.5lb/ft^3$$

,it will sink to a depth h where .

$$\dfrac{π}{3}(3rh^2-h^3)d_L=\dfrac{4}{3}πr^3d_s$$

Find an approximate value for h if:

$$r=5 ft$$

and

$$d_s=20lb/ft^3$$

Solve each minimum problem using the duality principle. Minimize

$$C = x_1 +4x_2 + 2x_3 +4x_4$$

subject to the constraints 2

$$\left\{ \begin{array}{ccccccc} x_1 & & +x_3 & \geq 1 \\ & x_2 & & +x_4 \geq1 \\ - x_1 &-x_2 &-x_3 &-x_4 \geq -3 \\ x_1 \geq 0, & x_2 \geq 0, & x_3 \geq 0 & x_4 \geq 0 \end{array} \right.$$

What is the expected result when you roll a single die? Specify 40 rolls, and then click on "roll the dice." After the last roll, click on "show mean." Does your expected sum agree with the green mean? What happens to the plotted points as the number of rolls increases? If the picture is not clear, click on "roll the dice" for a new total of 80 rolls. Describe what you see in a sentence. What is the name for this principle?

Solve each minimum problem using the duality principle. Minimize

$$C = x_1 +2x_2 + 3x_3 +4x_4$$

subject to the constraints 2

$$\left\{ \begin{array}{ccccccc} x_1 & & +x_3 & \geq 1 \\ & x_2 & & +x_4 \geq1 \\ - x_1 &-x_2 &-x_3 &-x_4 \geq -3 \\ x_1 \geq 0, & x_2 \geq 0, & x_3 \geq 0 & x_4 \geq 0 \end{array} \right.$$

Rayleigh’s criterion is used to determine when two objects are barely resolved by a lens of diameter d. The angular separation must be greater than $\theta_{R}$ where $\theta_{R} =1.22 \lambda /d$ . In order to resolve two objects 4000 nm apart at a distance of 20 cm with a lens of diameter 5 cm, what energy (a) photons or (b) electrons should be used? Is this consistent with the uncertainty principle?

Refer to the figure and use the principle of superposition to Find the component of the current $i$ through $R_{3}$ that is due to $V_{S 2}$.

$$\begin{aligned} & V_{S 1}=V_{S 2}=450 \mathrm{~V} \\ & R_{1}=7 \Omega \quad R_{2}=5 \Omega \\ & R_{3}=10 \Omega \quad R_{4}=R_{5}=1 \Omega \end{aligned}$$

Object A has mass m$_A$ = 10 kg and initial momentum p$_{A,i}$ = (21, -7, 0) kg·m/s, just before it strikes object B, which has mass m$_B$= 14 kg. Just before the collision object B has initial momentum p$_{B,i}$= (3, 6, 0) kg·m/s. Therefore, what does the Momentum Principle predict that the total final momentum of the system will be, just after the collision?

Use the uncertainty principle to estimate the binding energy of the $\mathrm{H}_2$ molecule by calculating the difference in kinetic energy of the electrons between when they are in separate atoms and when they are in the molecule. Take $\Delta x$ for the electrons in the separated atoms to be the radius of the first Bohr orbit, $0.053 \mathrm{~nm}$, and for the molecule take $\Delta x$ to be the separation of the nuclei, $0.074 \mathrm{~nm}$. [Hint: Let $\left.\Delta p \approx \Delta p_x.\right]$

The Chess Club at Asiniboyne High School consists of 6 seniors and 11 juniors. Presently, the club president and the club secretary are both seniors. (They must be different people.) If the students were selected randomly for these offices, what is the probability both would be seniors?

a. Show how to find the answer to this question using the General Multiplication Rule for probability.

b. Show how to find the answer using the Multiplication Principle of Counting and the definition of probability.

Applying Le Chatelier's principle, in which direction will the equilibrium shift, (if at all)?

$$\mathrm{CH}_4(\mathrm{~g})+2 \mathrm{~O}_2(\mathrm{~g}) \rightleftharpoons \mathrm{CO}_2(\mathrm{~g})+2 \mathrm{~H}_2 \mathrm{O}(\mathrm{g})+802.3 \mathrm{~kJ}$$

(a) if the temperature is increased
(b) if a catalyst is added
(c) if $\mathrm{CH}_4$ is added
(d) if the volume of the reaction vessel is decreased

Suppose you attempt to pick up a very heavy object. Before you tried to pick it up, the object was sitting still—its momentum was not changing. You pull very hard, but do not succeed in moving the object. Is this a violation of the Momentum Principle? How can you be exerting a large force on the object without causing a change in its momentum? What does change when you apply this force?

Magnificent Blooms is a florist specializing in floral arrangements for weddings, graduations, and other events. The firm has a fixed cost associated with space and equipment of $100 per day. Each worker is paid$50 per day. The daily production function for Magnificent Blooms is shown in the accompanying table.

$$\begin{array}{cc} \begin{array}{c} \text { Quantity of labor } \\ \text { (workers) } \end{array} & \begin{array}{c} \text { Quantity of floral } \\ \text { arrangements } \end{array} \\ \hline 0 & 0 \\ \hline 1 & 5 \\ \hline 2 & 9 \\ \hline 3 & 12 \\ \hline 4 & 14 \\ \hline 5 & 15 \\ \hline \end{array}$$

Calculate the marginal product of each worker. What principle explains why the marginal product per worker declines as the number of workers employed increases?