If $\mathbf{F}(x, y)=P(x, y) \mathbf{i}+Q(x, y) \mathbf{j}$ is a vector field in a domain D with div F = 0 and curl F = 0, then the complex function $g(z)=P(x, y)+i Q(x, y)$ is analytic in D. _____

A humor piece published in the British Medical Journal ("Parachute use to prevent death and major trauma related to gravitational challenge: systematic review of randomized control trials," Gordon, Smith, and Pell, BMJ, 2003:327) notes that we can't tell for sure whether parachutes are safe and effective because there has never been a properly randomized, double-blind, placebo-controlled study of parachute effectiveness in skydiving. (Yes, this is the sort of thing statisticians find funny . ... ) Suppose you were designing such a study: How would you randomize this "experiment"? How would you make the experiment double-blind?

Suppose that f is a continuous additive function on

$$\mathbb{R}.$$

If c := f (1), show that we have f(x) = cx for all

$$x \in \mathbb{R}.$$

An office employs six customer service repres entatives. Each day, two of them are randomly selected and their customer interactions are monitored for the purposes of improving customer relations. In how many ways can the repres entatives be chosen?

Suppose that a student has 500 vocabulary words to learn. If the student learns 15 words after 5 minutes, the function $L(t) = 500(1 - e^{-0.0061t})$ approximates the number of words L that the student will learn after t minutes. How many words will the student learn after 60 minutes?

Compute the first-order partial derivatives.

$$w = \frac { x } { y + z }$$

How many ways can you arrange two of the first three letters of the alphabet?

three siblings are three different ages. The oldest is twice the age of the middle sibling, and the middle sibling is six years older than one-half the age of the youngest. Write a composite function that gives the oldest sibling’s age in terms of the youngest. Explain how you arrived at your answer.

Simplify each sum or difference.

$$( 9 x - 3 y ) - ( 3 x - 9 y )$$

An athletic store sells about 200 pairs of basketball shoes per month when it charges $120 per pair. For each$2 increase in price, the store sells two fewer pairs of shoes. How much should the store charge to maximize monthly revenue? What is the maximum monthly revenue?

Use the extended Euclidean algorithm to find the greatest common divisor of the given numbers and express it as a linear combination of the two numbers. 4158 and 1568

Solve each equation by factoring. $3 x^{2}-x-2=0$

A carton of milk is poured into a bowl. Describe the changes that occur in the milk's shape and volume.

Use the most appropriate method to solve each equation on the interval

$$[0,2\pi).$$

Use exact values where possible or give approximate solutions correct to four decimal places.

$$3 \tan ^ { 2 } x - \tan x - 2 = 0$$