When traveling outside of the United States it is often useful to be able to convert between temperatures measured using the Fahrenheit scale and temperatures measured using the Celsius scale. The standard formula for determining temperature in degrees Fahrenheit when given the temperature in degrees Celsius is, $F=\frac{9}{5} C+32$. We can write this formula using function notation by letting $F = g(C)$ and writing $g(C)=\frac{9}{3} C+32$. The function g defines a process for converting degrees Celsius to degrees Fahrenheit. a. State the meaning of g(100) and evaluate g(100). b. Solve the equation $g(C)=212$ and explain how you arrived at your answer. c. Define a function h that converts Fahrenheit temperature to its Celsius temperature. (Hint. Generalize the steps you described in part (b) that reverses of g. You can also solve $F=\frac{9}{3} C+32$ for C.) d. Determine the values of g(100) and h(212). e. Determine the values of g(h(212)) and h(g(100)) without performing any calculations. What do you notice about the relationship between g and h? f. Represent the value of g(h(n)). Represent the value of h(g(k)). What do you notice about the relationship between g and h?

¿Qué te parecería hacer servicio social en el centro de la comunidad?

The subject of Fourier series is concerned with the representation of a

$$2 \pi \text { -periodic }$$

function f as the following infinite linear combination of the set of functions

$$\{ 1 , \sin n x , \cos n x \} _ { n = 1 } ^ { \infty } : \\ f ( x ) = \frac { 1 } { 2 } a _ { 0 } + \sum _ { n = 1 } ^ { \infty } \left( a _ { n } \cos n x + b _ { n } \sin n x \right) . \quad ( 5.2 .5 )$$

In this problem, we investigate the possibility of performing such a representation. (a) Use appropriate trigonometric identities, or some form of technology, to verify that the set of functions

$$\{ 1 , \sin n x , \cos n x \} _ { n = 1 } ^ { \infty }$$

is orthogonal on the interval

$$[ - \pi , \pi ].$$

(b) By multiplying (5.2.5) by cos mx and integrating over the interval

$$[ - \pi , \pi ],$$

show that

$$a _ { 0 } = \frac { 1 } { \pi } \int _ { - \pi } ^ { \pi } f ( x ) d x \text { and } a _ { m } = \frac { 1 } { \pi } \int _ { - \pi } ^ { \pi } f ( x ) \cos m x\ d x.$$

(c) Use a similar procedure to show that

$$b _ { m } = \frac { 1 } { \pi } \int _ { - \pi } ^ { \pi } f ( x ) \sin m x\ d x.$$

It can be shown that if f is in

$$C ^ { 1 } ( - \pi , \pi ),$$

then Equation (5.2.5) holds for each

$$x \in ( - \pi , \pi ).$$

The series appearing on the right-hand side of (5.2.5) is called the Fourier series of f, and the constants in the summation are called the Fourier coefficients for f. (d) Show that the Fourier coefficients for the function

$$f ( x ) = x , - \pi < x \leq \pi , f ( x + 2 \pi ) = f ( x ),$$

are

$$\begin{array} { l l } { a _ { n } = 0 , } & { n = 0,1,2 , \ldots } \\ { b _ { n } = - \frac { 2 } { n } \cos n \pi , } & { n = 1,2 , \ldots } \end{array}$$

and thereby determine the Fourier series of f. (e) Using some form of technology, sketch the approximations to f(x) = x on the interval

$$( - \pi , \pi )$$

obtained by considering the first three terms, first five terms, and first ten terms in the Fourier series for f. What do you conclude?

Draw a graph of the cost behavior for each of the following costs incurred by the Mountain Summit Hospital. The hospital measures monthly activity in patient days. Label both axes and the cost line in each graph.

$1.$ The cost of food varies in proportion to the number of patient days of activity. In January, the hospital provided $3,000$ patient days of care, and food costs amounted to $\$ 24,000$.

$2.$ The cost of salaries and fringe benefits for the administrative staff totals $\$ 12,000$ per month.

$3.$ The hospital's laboratory costs include two components: $(a) \$ 40,000$ per month for compensation of personnel and depreciation on equipment and $(b) \$ 10$ per patient day for chemicals and other materials used in performing the tests.

$4.$ The cost of utilities depends on how many wards the hospital needs to use during a particular month. During months with activity under $2,000$ patient days of care, two wards are used, resulting in utility costs of $\$ 10,000$. During months with greater than $2,000$ patient days of care, three wards are used, and utility costs total $\$ 15,000$.

$5.$ Many of the hospital's nurses are part-time employees. As a result, the hours of nursing care provided can be easily adjusted to the amount required at any particular time. The cost of wages and fringe benefits for nurses is approximately $\$ 2,500$ for each block of $200$ patient days of care provided during a month. For example, nursing costs total $\$ 2,500$ for $1$ to $200$ patient days, $\$ 5,000$ for 201 to 400 patient days, $\$ 7,500$ for $401$ to $600$ patient days, and so forth.

Writing Direct Quotations. Complete these sentences. __________." he announced.

Unemployment programs under the Social Security Administration are controlled directly by the federal government. True or False?

Commercial contractors and home improvement merchants like Lowe's and Home Depot can purchase roofing and siding supplies from Great Plains Distributors Inc. The business owner is curious to know how different factors affect how much fiber-cement siding is sold. In the United States, the corporation operates 26 marketing districts. It gathered data on the following factors in each district: sales volume (in thousands of dollars), advertising spending (in thousands of dollars), the number of active accounts, the number of rival brands, and an assessment of market potential.

$$\begin{array}{|ccccc|} \begin{array}{c} \text { Sales } \\ \text { (000s) } \end{array} & \begin{array}{c} \text { Advertising } \\ \text { Dollars } \\ \text { (000s) } \end{array} & \begin{array}{c} \text { Number of } \\ \text { Accounts } \end{array} & \begin{array}{c} \text { Number of } \\ \text { Competitors } \end{array} & \begin{array}{c} \text { Market } \\ \text { Potential } \end{array} \\ \hline 79.3 & 5.5 & 31 & 10 & 8 \\ 200.1 & 2.5 & 55 & 8 & 6 \\ 163.2 & 8.0 & 67 & 12 & 9 \\ 200.1 & 3.0 & 50 & 7 & 16 \\ 146.0 & 3.0 & 38 & 8 & 15 \\ 177.7 & 2.9 & 71 & 12 & 17 \\ \vdots & \vdots & \vdots & & \\ 93.5 & 4.2 & 26 & 8 & 3 \\ 259.0 & 4.5 & 75 & 4 & 9 \\ 331.2 & 5.6 & 71 & & \\ \hline \end{array}$$

Find the most accurate sales predictors by performing a multivariate regression analysis. Using this, develop a histogram of the residuals and a normal probability plot. Are there any problems?

Why is the English Bill of Rights important to both the English and American people?

In many buildings, particularly outside of the United States, the ground floor of a building is labeled as G or Lobby. The first floor of the building is one floor above the ground floor. The building pictured has a lobby, 10 floors of offices, and 4 floors of garage below the lobby. a. Melanie has an office on the 9th floor and parks on the 3rd floor below the ground floor. Taylor and Cecelia are determining how many floors Melanie must go up from her car to reach her office. Taylor represents the 9th floor as 9 and the 3rd floor below ground as -3. Therefore, since 9 - 3 = 6, Melanie traveled 6 floors to get from her car to her office. Cecelia says that the ground floor to the 9 th floor is 9 floors, and from the ground floor to the 3 rd garage level is 3 floors. Melanie traveled |9| + |-3| = 9 + 3 = 12 floors. Who is correct? Explain your reasoning. b. Caleb parks his car on the 2 nd floor below ground and works on the 7th floor. How many floors must he go up from his car to reach his office? c. Lucinda is working on the 8th floor. At lunch, she goes to her car on the 4 th floor below ground, and then back up to the lobby. How many total floors does Lucinda travel? d. If Damon goes from his office on the 10th floor to a meeting on the 5th floor, how many floors does he travel and in which direction?

What is one hypothesis that could explain how a dominance hierarchy benefits the animals in a social group?

An office supply firm finds that the number of laptop computers sold in year x is given approximately by the function $f(x)=150+2 x+x^2$, where x=0 corresponds to 2010.

(a) What does f(0) represent?

(b) Find the number of laptops sold in 2016.

Identify the correct term or person that best matches the following description.

Supporter of political and social reform

For term or name, write a sentence explaining its significance in the rise of democratic ideas. social contract

Simplify by first writing in radical form.

$$16^{3 / 4}$$