Creating uses. List five garments in your wardrobe. Describe two new ways to wear each one.

Show that, for x>0

$$x/1+x^2<tan-^1x<x$$

Se encontró el cadáver de una mujer en el bosque. Algunas fibras de pelo encontradas en el cuerpo fueron enviadas al laboratorio criminalístico para su análisis. Las puntas del pelo pegado al cuerpo eran grises, pero las puntas del pelo mostraban que había sido teñido. La distancia desde la raíz del pelo hasta el comienzo de la zona teñida medía $8\text{ mm}$. Los investigadores determinaron que el cabello de la víctima se había teñido por última vez el $1$ de agosto de $2004$. Suponiendo que el cabello crece a razón de $0.44\text{ mm}$ al día, ¿en qué fecha murió la mujer aproximadamente? Explique su respuesta.

For an interarrival distribution F having mean $\mu$, we defined the equilibrium distribution of $F$, denoted $F_e$, by

$$F_e(x)=\frac{1}{\mu} \int_0^x[1-F(y)] d y$$

a) Show that if $F$ is an exponential distribution, then $F=F_e$.

b) If for some constant $c$,

$$F(x)= \begin{cases}0, & x<c \\ 1, & x \geq c\end{cases}$$

show that $F_e$ is the uniform distribution on $(0, c)$. That is, if interarrival times are identically equal to $c$, then the equilibrium distribution is the uniform distribution on the interval $(0, c)$.

c) The city of Berkeley, California, allows for two hours parking at all nonmetered locations within one mile of the University of California. Parking officials regularly tour around, passing the same point every two hours. When an official encounters a car he or she marks it with chalk. If the same car is there on the official's return two hours later, then a parking ticket is written. If you park your car in Berkeley and return after three hours, what is the probability you will have received a ticket?

Give a graph of the function f , and identify the limits as

$$x \rightarrow \pm \infty$$

, as well as locations of all relative extrema, inflection points, and asymptotes (as appropriate).

$$f ( x ) = \left\{ \begin{array} { l l } { \frac { 1 } { 2 } x ^ { 2 } , } & { x \leq 0 } \\ { - x ^ { 2 } , } & { x > 0 } \end{array} \right.$$

For the following problem,

a. Plot the graph using a window set to show the entire graph, when possible. Sketch the result.

b. Give the y-intercept and any x-intercepts and the locations of any vertical asymptotes.

c. Give the range.

Power function $f(x)=0.3 x^{1.5}$ with the domain $0 \leq x \leq 9$

Refer to the example below.

a. In Part B, why did you multiply both sides of the equation by 4a?

b. In Part C, discuss why you added $b^2$ to each side to complete the square.

c. Given the expression under the radical sign, $b^2$ - 4ac, is positive, how many solutions will the quadratic formula give for a quadratic equation? Discuss.

d. If the expression under the radical sign, $b^2$ - 4ac, is 0 , how many solutions will the quadratic formula give for a quadratic equation? What if the expression is negative?

e. Another method of deriving the quadratic formula is to first divide each term by a, and then complete the square. Complete the derivation below. (In this derivation, you will use the quotient property of square roots, which says that $\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$. For a square root of a fraction, this property allows you to simplify the numerator and denominator separately. For instance, $\sqrt{\frac{5}{9}}=\frac{\sqrt{5}}{\sqrt{9}}=\frac{\sqrt{5}}{3}$.)

$$\begin{aligned} a x^2+b x+c & =0 \\ a x^2+b x & =-c \quad\quad\quad \text {Subtract c from both sides.}\\ x^2+-x & =-\frac{c}{a} \quad\quad\quad \text {Divide each term by $a$.} \end{aligned}$$

$$\begin{aligned} x^2+\frac{b}{a} x+\left(\frac{b}{2 a}\right)^2 & =\left(\frac{b}{2 a}\right)^2-\frac{c}{a} \\ (\quad \quad)^2 & =\frac{b^2-4 a c}{4 a^2} \quad\quad\quad \text {Factor the left side, and write the right side as a single fraction.} \end{aligned}$$

$$\begin{aligned} & x+\frac{b}{2 a}=\pm \frac{\sqrt{b^2-4 a c}}{\sqrt{4 a^2}} \quad \quad \quad\text {Apply the quotient property of radicals.}\\ & x+\frac{b}{2 a}=\pm \frac{\sqrt{b^2-4 a c}}{} \quad \quad \quad\text {Simplify the radical in the denominator.} \\ & \end{aligned}$$

$$x=\frac{-b \pm \sqrt{b^2-4 a c}}{2 a} \quad \quad \quad\text {Solve for x.}$$

Example:

Solve the general form of the quadratic equation, $a x^2+b x+c=0$, by completing the square to find the values of x in terms of a, b, and c.

A Subtract c from both sides of the equation.

$a x^2$ + bx =

B Multiply both sides of the equation by 4a to make the coefficient of $x^2$ a perfect square.

$$4 a^2 x^2+\quad x=-4 a c$$

C Add $b^2$ to both sides of the equation to complete the square. Then write the trinomial as the square of a binomial.

$$\begin{aligned} & 4 a^2 x^2+4 a b x+b^2=-4 a c+ \\ & (\quad \quad)^2=b^2-4 a c \\ & \end{aligned}$$

D Apply the definition of a square root and solve for $x$.

$$\begin{aligned} & =\pm \sqrt{ \quad} \\ 2 a x & =-\quad \pm \sqrt{ \quad} \end{aligned}$$

x = __________

The formula $x=\frac{-b \pm \sqrt{b^2-4 a c}}{2 a}$ is called the quadratic formula. For any quadratic equation written in standard form, $a x^2$ + bx + c = 0, the quadratic formula gives the solutions of the equation.

A fair coin is tossed seven times. Find the probability of obtaining at most five heads

Solve $\left(x^{2}+4\right) \frac{d y}{d x}=2 x-8 x y$ subject to y(0)=-1.

What is an example of alliteration in The Hill We Climb?

Teachers who expect more from students often get more. This is an example of a(n) ____________ prophecy.

Base your answers to question on the following diagram of Earth. Some of the latitude and longitude lines have been labeled. Points $A$ through $\mathrm{E}$ represent locations on Earth's surface.

The latitude and longitude of which location are closest to those of New York State?

(1) $\mathrm{A}$

(2) B

(3) C

(4) D

From the list below, supply the words needed to complete the paragraph. Some words will not be used.

$$\begin{array}{llll}\text { recoil } & \text { bombast } & \text { paleontology } & \text { panache } \\ \text { flaunt } & \text { elicit } & \text { shibboleth } & \text { aggrandize } \\ \text { obviate } & \text { endemic } & & \end{array}$$

When Dr. Carter is not fulfilling his duties as the professor of .... at Ganton College, he is in the jungles of South America supervising excavations of ancient creatures. During the last dig, Dr. Carter .... when a poisonous snake tried to bite him. Now he wears tall boots and gloves to .... the risk of dangerous animals .... to the region. He will definitely .... the need for such safety measures. Some of the incoming scientists will accuse him of self ..... despite the fact that the sociable doctor always avoids using too much .... when speaking about safety; he'll ..... his findings when he returns to the campus at the end of the summer. He can put his .... to better use in the classroom.

Confirm your understanding of the text by creating a timeline of the story's events.