Questions

2 points Find a power series representation for the function f x 1 1 4x 2 Hint Find a power series representation for the antiderivative off and differentiate this series to get a power series representation for f The n 0 term should correspond to the first nonzero coefficient in your representation otherwise your answer will be marked incorrect 1M8

Set up the definite integral required to find the area of the region between the grap of y 11 2 and y 22x 128 over the interval 1 1 10 234 23 xdx mixed numbers are not allowed 1

1 A roll of wrapping paper covers 25 cm Imani plans to wrap the cylinder below in paper a What is the surface area of the cylinder below b How many rolls of paper will Imani need to purchase 4 cm 2 cm

Notice that the graph looks like it is made of three pieces Because of this the function is called a piecewise defined function Piecewise defined functions are functions that have different descriptions for different values of the independent variable Here the piecewise defined function d where d t is the distance from home and t is time is defined over three time intervals if 0 t 20 d t 0 05t if 20 t 50 d t 1 Hint if 50 t 60 d t 0 1t 6 What is the expression for the distance d t for each time interval d t if 0 t 20 if 20 t 50 if 50 t 60

Suppose Then 6 5 f x dx 10 and f x dx 6 f x dx 7 5

The function is defined on the closed interval 3 8 and is g graph above Let g be the function defined by g x 71 t de a Find g 3 and g 0 b Does g x have a relat the interval s are g x is concave down Give a reas 1 x 3020 d Find sin 2x Wr Evaluate the following expressions t to the p x x 1 Explain your reason 3x sin 3t 1 dt In 4t dt

1 For the data set below which shows the number of days a sample of students missed class during the fall semester find the five number summary find the inter quartile range and draw a box and whisker plot Then find the mean and standard deviation of the sample 5 0 10 7 3 20 24 16 4 2 1 9 6 Five Number Summary Inter quartile Range x S Box And Whisker Plot 2 3 In problem 1 above what number represents the 75th percentile The mean price of new homes from a sample of houses in a neighborhood is 190 000 If the standard deviation is 8 000 and the data has a bell shaped distribution answer the questions that follow A What percentage of the data falls between 166 000 and 214 000 Explain your decision B Between what two values would 68 of the data fall

In earlier lessons you analyzed graphs that related time to distance traveled position and investigated scenarios where people or objects were moving according to some prescribed pattern In some situations however the description of the position of a person or object may change abruptly at certain points in time Consider this example Suppose it takes you 20 minutes to walk to your favorite bakery which is one mile from your home Upon arriving you spend 30 minutes at the bakery having a snack and talking to a friend and then realize that you need to get home quickly in order to watch your favorite TV show You decide to jog home it takes you 10 minutes to jog from the bakery to your house If you walked to the bakery at a constant speed and ran home at a constant speed then the following table describes your distance from home as a function of time Time minutes 0 Distance from Home miles 20 50 60 0 1 0 1 0 0 Complete the following table that describes the behavior of the function for the specified time intervals Hint Time Interval minutes Description of Time Interval Distance from Home Increasing or Decreasing 0 20 walking to the bakery 20 50 50 60 Rate of Change of Distance from Home miles minute 0 05 at the bakery Increasing o neither decreasing Increasing 0 B o decreasing neither increasing jogging home from the bakery A line segment with uphill slant indicates the distance is increasing and a line segment with downhill slant indicates the distance is decreasing What is of this line segment

3 2 0 1 2 1 4 1 Graph of f 5 2 5 The graph of the function shown above consists of a semicircle and three line segments Let g be the function given by g x t dt a Find g 0 and g 0 b Find all values of x in the open Interval 5 4 at which g attains a relative maximum Justify your answer c Find the absolute minimum value of g on the closed interval 5 4 Justify your answer d Find all values of x in the open Interval 5 4 at which the graph of g has a point of Inflection

Table 1 TRIAL 1 Results Phenotype of worms worms eaten start worms left in grass worms after reproduction Forest 10 1 9 18 Turquoise 10 9 1 2 Kelly 10 0 ID Olive E 110 2 8 20 16

4 Euclid Algorithm Use Euclid s algorithm to compute the following Show all your steps 1 gcd 781 994 2 gcd 67457 43521

5 Extended Euclid algorithm Does the multiplicative inverse of 59 in Z61 exist If so find it using the extended Euclid algorithm I it does not exist state why Show all your steps

3 Prime factorization a Find the prime factorization of each of these integers Show all your steps 1 108 2 1188 3 10 b Consider two positive integers a 22 x 32 x 52 x 76 and b 24 x 54 x 72 Using the GCD formula not the Euclid algorithm find the greatest common divisor of a and b No need to justifu

6 f g 110 75

Determine whether the series 2 converges conditionally converges absolutely on diverges Justify your answer 6 pts

Strong Induction Prove the following statement by strong induction There exist non negative integers x y such that n 3x 7y for all n 1

Find the zeros of f x x x 10x 8 2 then write fa in factored form The 3 zeros of f x 2x 5x 23x 10

6 Co primality Let a b be two integers We call a and b relatively primes or co primes IFF gcd a b 1 Show that the numbers 6k 5 and 7k 6 are relatively prime for every k 1 2 2

Graph f x 4 1

a A student is working on his science fair project concerning the average height of a certain native plant which he believes is normally distributed He finds 25 mature plants and measures their heights and determines the average height to be 10 2 cm s 1 2 cm Construct a 96 confidence interval on the mean height of the plants

2 Divisibility What are the quotient and remainder when 1 44 is divided by 6 2 994 is divided by 13 3 221 is divided by 16

For the following find F x F x So COS t2 dt F x 2t 5 dt night La 3x F x Int dt aloupe D S F x t dt 03 x Sandwolfof all to d 1 4 A

0 Graphy Log x 4 3 2

Solve Log x Log x 2 3 2 2

3 4 pts The Deely Laboratory analyzes specimens of a pharmaceutical product to determine the concentration of the active ingredient Such chemical analyses are not perfectly precise Repeated measurements on the same specimen will give slightly different results The results follow a Normal distribution quite closely All other conditions are met as well The Deely Laboratory has been asked to evaluate the claim that the concentration of the active ingredient in a specimen is 0 86 grams per liter The hypotheses are vs A 99 confidence interval for the true mean concentration level was calculated from the data 0 8303 0 8505 Use the confidence interval to draw a detailed conclusion about the hypothesis Be sure to indicate the appropriate significance level

What is the new equation when y x is stretched horizontally by a factor of 2 then translated right by sunits then downward by 5 units Also find vertex axis of symmetry then sketch

a A cell phone manufacturer wants to advertise their new extended life batteries They believe the lifetimes are normally distributed with 1 15 hr A test engineer takes a sample of 10 batteries and calculates a sample standard deviation of 1 3 hr Can the manufacturer confidently advertise the standard deviation in the lifetime of their new batteries Use critical values of x2 95 and X2 05 a 90 window i Find the value of the test statistic for this sample and the two critical values X0 95 x X 05 ii Can the manufacturer confidently advertise the standard deviation in the lifetime of their new batteries Circle YES or NO and briefly explain your answer YES

A sports team is interested in estimating the time required to complete a certain obstacle course 30 athletes go through the course the average completion time is 52 s s 4 s Construct a 90 lower confidence bound on the average completion time of this obstacle course

A company is interested in estimating the maximum weight that a certain machine can handle safely The weight capacity of this type of machine is normally distributed and the standard deviation is known to be 50 pounds The company takes a random sample of 15 machines and finds that the sample mean weight capacity is 1000 pounds Construct a 95 upper confidence bound on the maximum weight this machine can handle

students math scores They believe the improvement in math scores is normally distributed with 4 points A sample of 25 students is randomly selected and their math scores are measured before and after the new teaching method is implemented The sample standard deviation of the differences in math scores is 5 2 points Does the school district have evidence to support their assumed standard deviation of 4 points Use critical values of X975 and X0 025 a 95 window i Find the value of the test statistic for this sample and the two critical values ii X6 975 x X0 025 Does the school district have evidence to support their assumed standard deviation of 4 points Circle YES or NO and briefly explain your answer YES

A sports team is interested in estimating the time required to complete a certain obstacle course The standard deviation in completion time is known to be 4 s 30 athletes go through the course the average completion time is 52 s Construct a 90 lower confidence bound on the average completion time of this obstacle course

Estimating the Mean R a A student is working on his science fair project concerning the average height of a certain native plant He believes the heights are normally distributed with 1 2 cm He finds 25 mature plants and measures their heights and determines the average height to be 10 2 cm Construct a 96 confidence interval on the mean height of the plants

A flashlight has a parabolic reflector that reflects light waves from a bulb located at the focus The figure shows a vertical cross section of such a parabolic reflector with the bulb at the focus Suppose the bulb is 0 1 inch from the vertex of the reflector and the distance across the widest part of the reflector is 4 inches Suppose also that the parabola formed by the cross section is located on a grid so that the vertex is at 0 0 0 1 in x 4 in The figure may not be drawn to scale The equation for the parabola is www

Evaluate C x y ds where C is the straight line segment x t y 2 t z 0 from 0 2 0 to 2 0 0 x C x y ds Type an exact answer

3 Find the mass of a wire that lies along the curve r t 2 7 j 2tk Ost 3 if the density is 8 t x y z ds C Type an exact answer using radicals as needed

8 A trough is 4 feet long and 1 foot high The vertical cross section of the trough parallel to an end is shaped like the graph of y from x 1 to z 1 The trough is full of water Find the amount of work in foot pounds required to empty the trough by pumping the water over the top Note The weight of water is 62 pounds per cubic foot 1017 11

Variations on Mendelian Inheritance Two gene cross blue eyes Select the crosses that would lead to only blue eyed progeny Choose all that apply Check All That Apply oohh x oohh

content per 12 ounce bottle of cola is 40 milligrams You want to test this claim During your tests you find that a random sample of thirty 12 ounce bottles of cola has a mean caffeine content of 39 1 milligrams Assume the population is normally distributed and the population standard deviation is 6 9 milligrams At a 0 06 can you reject the company s claim Complete parts a through e a Identify Ho and Ha Choose the correct answer below OA Ho 39 1 Ha 39 1 OC Ho H 39 1 Ha H 39 1 E Ho H 40 H 40 B Ho H 40 Ha 40 D Ho 40 a 40 OF OF Ho H 39 1 Ha H 39 1 b Find the critical value s Select the correct choice below and fill in the answer box within your choice Round to two decimal places as needed OA The critical value is OB The critical values are 1 881

Find the P value for the indicated hypothesis test with the given standardized test statis z Decide whether to reject Ho for the given level of significance a Two tailed test with test statistic z 2 46 and a 0 08 P value Round to four decimal places as needed

Test the claim about the population mean at the given level of significance using the given sample statistics Claim 50 0 02 3 44 Sample statistics x 48 9 n 54 Identify the null and alternative hypotheses Choose the correct answer below OA Ho H 50 Ha H 50 C Ho 50 Ha 50 E Ho H 50 Ha 50 Calculate the standardized test statistic The standardized test statistic is Round to two decimal places as needed B Ho 50 Ha 50 OD Ho 50 Ha H 50 OF Ho H 50 Ha H 50

Find an equation for the conic section with the given properties The ellipse with vertices V1 1 9 and V 2 1 11 and foci F1 1 7 and F2 1 9

onsider the following exponential probability density function f x e for x 0 Which of the following is the formula for P 1 P x zo e 2 P x zo 1 e 3 P x 20 1 e Formula 2 b Find P x 1 to 4 decimals 3935 c Find P 3 to 4 decimals 1353 d Find P 5 to 4 decimals 9175 e Find P 15 to 4 decimals 5245

You may need to use the appropriate appendix table or technology to answer this question A simple random sample with n 58 provided a sample mean of 24 5 and a sample standard deviation of 4 4 Round your answers to one decimal place a Develop a 90 confidence interval for the population mean to b Develop a 95 confidence interval for the population mean to c Develop a 99 confidence interval for the population mean to d What happens to the margin of error and the confidence interval as the confidence level is increased As the confidence level increases there is a smaller margin of error and a wider confidence interval As the confidence level increases there is a larger margin of error and a wider confidence interval As the confidence level increases there is a smaller margin of error and a more narrow confidence interval As the confidence level increases there is a larger margin of error and a more narrow confidence interval

Suppose that f x is the sum of the Fourier series 3 2n f x sin nx 1 1 n sin x sin 2x sin 3x x Compute the integral 1 f x 1 sin 2x dx A 127 B 17 C 167 D 7 E 14 5 5

You may need to use the appropriate appendix table or technology to answer this question Suppose in 2021 a national research center conducted a survey of 1 501 Internet users that provided a variety of statistics on Internet users For instance in 2021 92 of adults in the United States were Internet users In 1995 only 14 of adults in the United States used the Internet a Develop a 95 confidence interval for the proportion of American adults who identified as Internet users in 2021 Round your answers to four decimal places to b The sample survey showed that 78 of its adult U S respondents said they have a broadband connection at home Develop a 95 confidence interval for the proportion of adults in the United States who say they have a broadband connection at home Round your answers to four decimal places to c The sample survey showed that 16 of adults in the United States who say their smartphones are their primary means of online access at home Develop a 95 confidence interval for the proportion of adults in the United States who say their smartphones are their primary means of online access at home Round your answers to four decimal places to d Compare the margin of error for the interval estimates in parts a b and c How is the margin of error related to the sample proportion p O The margin of error increases as p gets closer to 0 50 O The margin of error does not change as p gets closer to 0 50 O The margin of error decreases as p gets closer to 0 50

6y 11 Solve for y 9y 10

You may need to use the appropriate appendix table or technology to answer this question A simple random sample of 400 individuals provides 100 Yes responses a What is the point estimate of the proportion of the population that would provide Yes responses b What is your estimate of the standard error of the proportion o Round your answer to four decimal places P c Compute the 95 confidence interval for the population proportion Round your answers to four decimal places to

You may need to use the appropriate appendix table or technology to answer this question A simple random sample of 60 items from a population with a 8 resulted in a sample mean of 38 Round your answers to two decimal places a Provide a 90 confidence interval for the population mean to b Provide a 95 confidence interval for the population mean to c Provide a 99 confidence interval for the population mean to

access to higher education changes in the curriculum the recruitment of more professors of color and the creation of ethnic studies programs These programs were the beginning of multicultural cur riculum reform in higher education From their origins in California eth nic studies programs and departments have survived and proliferated through out the United States After serious cut backs during the budgetary crises of the 1970s and 1980s they are back bigger and stronger than ever Ethnic studies programs have been revitalized reor ganized and reconceptualized Indeed they are increasingly becoming institu tionalized The field of ethnic studies has produced a prodigious amount of new scholarship much of which is good and innovative However as is true in all dis ciplines some of the work is weak The perspectives of ethnic studies are intend ed not only to increase our knowledge base but eventually to transform the dis ciplines Their influence is being widely felt and hotly debated Today there are more than 700 eth nic studies programs and departments in the United States They are represent ed by five established professional associ ations the National Council of Black Studies the National Association of Chi cano Studies the Asian American Studies Association the American Indian Studies Association and the National Association of Ethnic Studies The Association of Puerto Rican Studies was formed in 1992 A disproportionate number of ethnic studies programs are located in public colleges and universities because these institutions are more susceptible to pub riuata cohools cific programs 3 The Department of American Eth nic Studies at the University of Wash ington Seattle was created in 1985 by bringing together programs in African American Asian American and Chicano studies 4 The Center for Studies of Ethnicity and Race in America at the University of Colorado Boulder was created in 1987 by consolidating existing programs in black studies and Chicano studies and adding new programs in Asian American and American Indian studies Aside from the West Bowling Green State University in Ohio has one of the oldest ethnic studies departments which was founded in 1979 During the decades in which ethnic studies programs were established and grew strong American society under went dramatic changes that continue to this day The civil rights movement might have removed the last vestiges of legal apartheid in the United States However other ways have been invented to deny equal opportunity to the historically mar ginalized communities of color In the 25 years since the issuance of the Kerner Commission Report which spoke of two Americas one rich one poor one white one black the gulf that divides the nation has grown wider than ever Today 1 of the population of the U S has gained control of more of the na tion s wealth than the bottom 90 This situation parallels the stark and painful inequality in much of the Third World Significant demographic changes have also taken place in the United States in the last 25 years Since 1965 when U S immigration laws eliminated the national origins quotas that favored Europeans Page 3 of 6 coming colored and increasingly more terns the U S population is rapidly be diverse language music art literature and other in race ethnicity religion cultural expressions In fact with more than half of its population already high ly diversified California provides a glimpse of the nation s future It will be an oxymoronic majority minority state by 2050 The relatively high birthrate of minority Americans as well as their low er age distribution will mean that ever increasing numbers of people of color will fill our classrooms and enter our work force In order to bring about a truly pluralis tic democracy our education system at all levels not only must reflect the nation s diversity in its student body faculty and curriculum but also must seek to achieve comparable educational outcomes for all groups in society The education reforms known collectively as multiculturalism one example of which is the integra tion of ethnic studies into the college cur riculum I have as major goals the es tablishment of democratic pluralism and the achievement of educational equity THE NATURE OF ETHNIC STUDIES What is ethnic studies First the field is distinct from global or internation al studies particularly those program known generally as area studies wit which ethnic studies is often compare and confused Area studies program arose out of American imperialism in the Third World and bear names such a African studies Asian studies and Lat in American studies These programm were designed to focus on U S Thir World relations and to train specialist

Set up the definite integral required to find the area of the region between the graph y 5x 14 of y x 2 10 and dx