Functions Questions and Answers

blem 3 Before kerosene can be used as jet fuel federal regulations require that the pollutants in it be removed by passing the kerosene through clay Suppose that the clay is in a pipe and that each foot of the pipe will remove 20 of the pollutants that enter it a If Po is the initial number of pollutant and P x represents the quantity of pollutants left after z feet pipe derive an expression for P x Poez and assume that Po 1 Hint Find r b Graph the function and use it to explain what will happen the amount of pollutant in kerosene if we continue to increase the length of the pipe c Given the initial pollutant is Po 1 how long does the pipe need to be to insure that 50 of the pollutants have been eliminated

The set of ordered pairs represents a function Write a rule that represents the function 0 0 1 2 2 16 3 54 4 128 y Simplify your answer Type an expression using x as the variable

Debra s company makes solid spherical metal balls for various industrial uses A customer wants iron balls that have a diameter of 9 in If Debra must make 30 of these balls how much iron will she need If necessary refer to the list of geometry formulas For your calculations do not round any intermediate steps and use the button on the ALEKS calculator Round your answer to the nearest hundredth 3 in X

A flower bed is in the shape of a rectangle It measures 7 yd long and 3 yd wide Chau wants to use mulch to cover the flower bed The mulch is sold by the square foot Use the facts to find the area of the flower bed in square feet 2 X 3 Conversion facts for length 1 foot ft 12 inches in 1 yard yd 3 feet ft 1 yard yd 36 inches in

An actor bought a pearl diamond and ruby necklace for his famous wife for 57 000 In 2011 this necklace was auctioned for 11 140 200 The auction price was what percent of the original purchase price The auction price was of the original purchase price Round to the nearest integer as needed

While collecting data the environmentalists found an injured male from the western area and two females from the eastern population These deer were transported back to the facility for treatment and rehabilitation The deer were kept in the same enclosure for over a year Despite observed attempts of the male to mate while the females were in heat no viable offspring were produced Genetic tests were run to determine if any problems were preventing fertilization and all came back with normal results More individuals were brought to the facility and again the deer from the different populations were unsuccessful in their attempts to breed Prompt Write a scientific claim that explains the impact the highway construction had on the whitetail deer populations Claim Evidence 8

2 01 29 00 Recent financial statement data for Harmony Health Foods HHF Incorporated is shown below Income before interest and taxes Interest expense Income before tax Income tax Net income Current liabilities 10 Bonds long term Total liabilities Shareholders equity Common stock Retained earnings Total shareholders equity Total liabilities and equity HHF s debt to equity ratio is Note Round your answer to 2 decimal places Multiple Choice 189 380 569 202 299 501 1 070 116 38 78 22 56

1 3 Are the populations in Exercises 1 1 and 1 2 concep tual or physical populations 1 4 How would you obtain a sample of size 20 from all of the possible distances that you might throw a baseball Is the population that you are sampling from conceptual or physical 1 5 A group of 100 dental patients is randomly divided into

A true false quiz has 6 questions a How many different ways can the quiz be filled out b How many different ways can the quiz be filled out if the student does not answer all true or all false In both parts assume each question must be answered with true or false

The point 15 16 is on the graph of y f x a A point on the graph of y g x where g x f x is b A point on the graph of y g x where g x f x 7 is c A point on the graph of y g x where g x f x 12 is d A point on the graph of y g x where g x f x is e A point on the graph of y g x where g x f x is f A point on the graph of y g x where g x 9f x is

b C 15 3 C 15 12 b S ra Sizde A a daki ifadelerin e itlerini bulunuz a C 12 4 C 9 2 C 7 3 15 3 1 1513

b Set the first derivative f x 0 to find any critical values Tip You may wish to graph f x by itself to explore if it has any zeros It can be helpful to know that the formula of f x factors Select the correct choice below and if necessary fill in the answer boxes within your choice A The function f has a critical number at x 1 3 at this critical number the second derivative f x 0 Type an exact answer OB There are no critical numbers c What conclusion can be made from your answer to part b Select the correct choice below and if necessary fill in the answer box within your choice OA At the critical number x OB At the critical number x OC At the critical number x D At the critical number x E No conclusion can be made using the second derivative test at this critical number O the function f is concave down and corresponds to a relative minimum of f x the function f is concave down and corresponds to a relative maximum of f x the function f is concave up and corresponds to a relative maximum of f x the function f is concave up and corresponds to a relative minimum of f x d Set the formula for the derivative f x 0 to find any possible inflection points 1 3 f x 0 at x Use the table feature for the graph of f x to examine the concavity of the graph of f x What can you conclude Select the correct choice below and if necessary fill in the answer boxes within your choice OA There is a point of inflection at x B There is a point of inflection at x down to where the graph of f changes from concave down to concave up The slope of the graph of f at this inflection point is where the graph off changes from concave up to concave down The slope of the graph of f at this inflection point is

Answer the questions for the function f x 3x 3x x 3 b Set the first derivative f x 0 to find any critical values Tip You may wish to graph f x by itself to explore if it has any zeros It can be helpful to know that the formula of f x factors Select the correct choice below and if necessary fill in the answer boxes within your choice A The function f has a critical number at x 1 3 at this critical number the second derivative f x 0 Type an exact answer OB There are no critical numbers c What conclusion can be made from your answer to part b Select the correct choice below and if necessary fill in the answer box within your choice OA At the critical number x B At the critical number x OC At the critical number x D At the critical number x E No conclusion can be made using the second derivative test at this critical number CHE the function f is concave down and corresponds to a relative minimum of f x the function f is concave down and corresponds to a relative maximum of f x the function f is concave up and corresponds to a relative maximum of f x the function f is concave up and corresponds to a relative minimum of f x d Set the formula for the derivative f x 0 to find any possible inflection points 1 1 f x 0 at x Use the table feature for the graph of f x to examine the concavity of the graph of f x What can you conclude Select the correct choice below and if necessary fill in the answer boxes within your choice OA There is a point of inflection at x B There is a point of inflection at x where the graph of f changes from concave down to concave up The slope of the graph of f at this inflection point is where the graph of f changes from concave up to concave down The slope of the graph of f at this inflection point is

For the following function a give the coordinates of any critical points and classify each point as a relative maximum a relative minimum or neither b identify intervals where the function is increasing or decreasing c give the coordinates of any points of inflection d identify intervals where the function is concave up or concave down and e sketch the graph h x 5x 15x CD There are no live mum points and there are no live maximum points b On what interval s is h increasing or decreasing Select the correct choice below and if necessary fill in the answer box es to complete your choice A The function is increasing on 1 1 00 The function is decreasing on 1 1 Simplify your answers Type your answers in interval notation Use a comma to separate answers as needed OB The function is decreasing on The function is never increasing Simplify your answer Type your answer in interval notation Use a comma to separate answers as needed OC The function is increasing on The function is never decreasing Simplify your answer Type your answer in interval notation Use a comma to separate answers as needed OD The function is never increasing or decreasing c What are the coordinates of the inflection point s Select the correct choice below and if necessary fill in the answer box to complete your choice 4 A The coordinates of the inflection point s are 0 0 Simplify your answer Type an ordered pair Use integers or fractions for any numbers in the expression Use a comma to separate answers as needed OB There are no inflection points d On what interval s is h concave up or concave down Select the correct choice below and fill in the answer box es to complete your choice Simplify your answer Type your answer in interval notation Use a comma to separate answers as needed OA The function is concave up on and concave down on OB The function is concave up on and is never concave down din nover concave up

12 223 At the instant shown cars A and B are traveling at speeds of 55 mi h and 40 mi h respectively If B is increasing its speed by 1200 mi h2 while A maintains a constant speed determine the velocity and acceleration of B with respect to A Car B moves along a curve having a radius of curvature of 0 5 mi 28 40 mi h ROXO 30 V 55 mi h Prob 12 223

graph that possesses the characteristics listed below f 7 0 f 7 0 f 7 0 f 3 0 f 3 0 f 3 6 f 5 0 f 5 3 Choose the correct graph below O A Q Q OB A Q O C O D X

ch the graph that possesses the characteristics listed f is concave down at 1 5 concave up at 5 3 and has an inflection point at 3 1 Choose the correct graph below OA 5 B Q C Q OD

Find any relative extrema of the function List each extremum along with the x value at which it occurs Identify intervals over which the function is increasing and over which it is decreasing Then sketch a graph of the function g x 4 2x x HUB Describe any relative extrema Select the correct choice below and if necessary fill in the answer box es to within your choice A The relative maximum point s is are 1 5 and there are no relative minimum points Simplify your answer Type anfordered pair using integers or fractions Use a comma to separate answers as needed OB The relative minimum point s is are Simplify your answer Type an ordered and there are no relative maximum points pair using integers or fractions Use a comma to separate answers as needed OC The relative minimum point s is are and the relative maximum point s is are Simplify your answers Type ordered pairs using integers or fractions Use a comma to separate answers as needed OD There are no relative minimum points and there are no relative maximum points Identify any intervals over which the function is increasing or decreasing Select the correct choice below and fill in the answer box es within your choice OA The function g x is decreasing over the interval s and is not increasing anywhere Type your answer in interval notation Use integers or fractions for any numbers in the expression Use a comma to separate answers as needed and is not decreasing anywhere OB The function g x is increasing over the interval s Type your answer in interval notation Use integers or fractions for any numbers in the expression Use a comma to separate answers as needed and decreasing over the interval s OC The function g x is increasing over the interval s Type your answers in interval notation Use integers or fractions for any numbers in the expressions Use a comma to separate answers as needed

The temperature of a person during a certain illness is given by the following equation where T is the temperature F at time t in days Find the relative extreme points and sketch a graph of th function T t 0 11 0 8t 98 6 0 t 8 What are the relative extreme points Select the correct choice below and fill in the answer box to complete your choice Simplify your answer Type an ordered pair Use integers or decimals for any numbers in the expression Use a comma to separate answers as needed OA The relative maximum point s is are OB The relative minimum point s is are OC The relative minimum point s is are and the relative maximum point s is are

Find a any f x x 6x 10 a Select the correct choice below and if necessary fill in the answer box within your choice A The critical value s of the function is are 3 Use a comma to separate answers as needed OB The function has no critical values b Select the correct choice below and if necessary fill in the answer box es within your choice OA The relative maximum point s is are and there are no relative minimum points Simplify your answer Type an ordered pair using integers or fractions Use a comma to separate answers as needed OB The relativ minimum point s is are Simplify your answer Type an ordered as needed and there are no relative maximum points pair using integers or fractions Use a comma to separate answers OC The relative minimum point s is are and the relative maximum point s is are Simplify your answers Type ordered pairs using integers or fractions Use a comma to separate answers as needed OD There are no relative minimum points and there are no relative maximum points

THE CHAIN RULE I Composite Functions 1 f x x g x 2x 2 If f x x and g x x 5 II The Chain Rule C

Laura plans to purchase a new SUV The dealer requires a 10 down payment on the 38 000 vehicle Laura will finance the rest of the cost with a fixed rate amortized auto loan at 5 annual interest with monthly payments over 6 years Complete the parts below Do not round any intermediate computations Round your final answers to the nearest cent if necessary If necessary refer to the list of financial formulas a Find the required down payment 0 b Find the amount of the auto loan 0 c Find the mont S payment X

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Complete parts a and b a Write the inverse of y 2 in logarithmic form b Graph y 25 and its inverse and discuss the symmetry of their graphs GETES a The inverse of y 2 in logarithmic form is y log x b Use the graphing tool to graph y 2 and its inverse Click to enlarge graph KILD

a i What is 10110110 in Hexadecimal Show your work

a i Evaluate sin 105 using the formula sin A B show your work ii Find the solutions to sin x 0 9 for the interval 0 x 360 4

f Find the following i How many 3 digits numbers can be formed by using the digits 1 3 6 and 8 when the digits may be repeated any number of times ii In how many different ways can the letters of the word OFFICES be arranged Show your work

Is f RR defined by f x 2x 3 a bijective function O it is an injective function but not surjective O it is not an injective function and not a surjective O it is not an injective function but it is a surjective O it is a bijective function

Choose two of the three figures below For each figure you choose draw a scaled copy using the given scale factor so that corresponding side lengths form equivalent ratios Figure M Scale Factor 2 Figure N Scale Factor 1 W Figure P Scale Factor M 2

Use this information to answer Parts A and B In a random sample of 60 seniors at a high school the mean weekly time spent on homework is 8 5 hours The population standard deviation for weekly homework time is 0 75 hour Part A What is a 90 confidence interval for the mean weekly time u spent on homework for all seniors at the high school Round your answers to two decima places as needed Enter the correct values in the boxes to complete the confidence interval Show Hints

The influence of the Russian Revolution of 1917 There was a striking similarity in the position of Russia and China In the early twentieth century Both countries had recently been defeated by Japan both were trying to come to terms with the need for economic and political modernisation and both were pour relations when compared with the advanced wealth producing nations of Western Europe and the USA The was therefore much about Russia that appealed to Chinese revolutionaries rekemers This attraction was intensified when the Chinese learned of the Bolshevik success in the October Revolution Chinese intellectuals wrote admitingly the achievements of Lenin and the Bolsheviks in taking power and establie a workers state Admiration increased when the Chinese learned of the Bolsheviks defest of the foreign interventionists in Russia in the period 1918 20 Here was a living example of the overthrow of Western imperiali made mare impressive by the fact that the nations which the Bolsheviks hi repelled were the very same as those currently occupying China The revoluticeary ideas of Karl Marx had been known in China since the beginning of the century but what gave them special appeal was the appa failure of the 1911 Revolution and of the republic that folkwed to advanc China s cause Disallusioned Chinese radicals turned impatiently away fro what they regarded as the failure of democracy in China They were draw instead to another Western philosophy but this time one that had been r by the West The fear with which the imperialist nations regarded Marxis it an added attraction for Chinese Nationalists To the young intellectuals who became drawn to Marxist ideas the great inspiration therefore was the successful October Revolution in Russia in They could now observe Marxism in action in anti imperialist and anti Russia The rejection of Western values impact in the Bolshevik Revolut appealed greatly to Chinese revolutionaries for whom the main attract of Marxism Leninism was its explanation of the imperialist phase of apitalism the mess which had led to China s current humiliation a

R VE 2 7 8 ty P q 20 16 12 8 5 0 6 15 3 5 10 5 8 X 0 2 In the graph to the right identify a pair of

e An investor deposits 50 on the first day of each month in an account that pays 3 interest compounded monthly What is the balance at the end of 2 years Show your working

5 f x 3x Domain x int Increasing Interval Decreasing Interval Symmetry Discontinuities End Behavior Range y int

A city s population is 1100 and growing at 3 a year a Find a formula for the population P at time t years from now assuming that the 3 per year is an annual rate or a continuous annual rate i An annual rate P t ii A continuous annual rate P t b In each case in part a estimate the population of the city in 25 years NOTE Round your answers to the nearest integer i The population of the city will be ii The population of the city will be

What is the image of 12 8 after a dilation by a scale factor of centered at the origin

78 of the questions on a student s test were correct There were 50 questions How many of the questions were correct questions

4 Kara enlisted her friend Frannie to help shovel all the driveways in her grandma s condominium community It takes Kara 40 minutes to shovel a driveway and it takes Frannie 45 minutes to shovel the driveway Write an equation for a function d that gives the number of driveways Kara and Frannie car shovel by working together for hours

If void ratio of a soil is 1 11 then its porosity will be

Determine the coefficient q and exponents m and n when the following expression is simplified q m n 4x 4y xm qyr

Determine the exponent m when the following expression is simplified m x x

3 56 ints Dexter Company uses the direct write off method March 11 Dexter determines that it cannot collect 8 200 of its accounts receivable from Leer Company March 29 Leer Company unexpectedly pays its account in full to Dexter Company Dexter records its recovery of this bad debt Prepare journal entries to record the above transactions No 1 2 3 Date March 11 March 29 March 29 Answer is complete but not entirely correct Bad debts expense Accounts receivable Leer Company Cash General Journal Accounts receivable Leer Company Bad debts expense Merchandise inventory X Debit 8 200 8 200 8 200 Credit 8 200 8 200 8 200

The function f x 8x arctan 5x is represented as a power series f x Find the first few coefficients in the power series CO C1 C2 C3 C4 IL Find the radius of convergence R of the series R Cn x n 0

Part 1 of 2 a What value is 1 5 standard deviations below the mean The value that is 1 5 standard deviations below the mean is 28 Part 1 2 Part 2 of 2 b What values are 1 standard deviations from the mean The values that are standan aviations from 3 C X

a Find an equation of the plane that passes through 3 2 1 1 2 3 and 9 5 4

21 Distance between Two Ships A ship leaves its home port and sails on a bearing of S 61 50 E Another ship leaves the same port at the same time and sails on a bear ing of N 28 10 E If the first ship sails at 24 0 mph and the second sails at 28 0 mph find the distance between the two ships after 4 hr WA N 28 10 61 50 1 I E

Hence from 1 the d c s of the line are a b C a b c a b c a b c where depending on the desired sign of k either a positive or a negative sign is to be taken for 1 m and n For any line if a b c are direction ratios of a line then ka kb kc k 0 is also a set of direction ratios So any two sets of direction ratios of a line are also proportional Also for any line there are infinitely many sets of direction ratios 1 t 11 2 1 Direction cosines of a line passing through two points Since one and only one line passes through two given points we can determine the direction cosines of a line passing through the given points P x y z and Q x Z as follows Fig 11 2 a J Therefore Let 1 m n be the direction cosines with the x y and z axis respectively 380 where m t cos y Draw perpendiculars from P and Q to XY plane to meet at R and S Draw a perpendicular from P to QS to meet at N Now in right angle triangle PNQ ZPQN Y Fig 11 2 b cos MATHEMATICS 200 CERT NO PQ n t 1 17 PQ PQ Similarly and cos Hence the direction cosines of the line segment joining the points P x y z and Q x y z are b of the line PQ and let it makes angles a and y N X PQ be repy fish Rationalised 2023 24 Q X X 3 3 2 2 PQ PQ PQ PQ x x 1 3 Note The direction ratios of the line segment joining P x y z and Q x may be taken as X X Y Y 2 z or x x 2 2 Example 1 If a line makes angle 90 60 and 30 with the positive direction of x y and z axis respectively find its direction cosines I Solution Let the d c s of the lines be 1 m n Then cos 90 0 m cos 60 2 3

A closed piston cylinder device contains 0 2 kg of air initially at 300 K and 100 kPa The air is now slowly compressed in an isothermal process to a final pressure of 250 kPa Assume the air is an ideal gas with constant specific heat Cp 1 004 kJ kg K Cv 0 717 kJ kg K The molecular weight for air is 29 kg kmol A Draw a schematic and list the necessary assumptions B Determine the volume in m of the air at the initial state C Determine the temperature and volume of the air at the final state D Determine the work in kJ during the process Is the work done by the air or on the air F Determine the heat transfer in k I during the process is a heat input or output

Hydrogen is contained in a rigid tank 5 m at an initial pressure of 150 kPa Hea transfers from the tank causing the temperature to decrease from 327 C to 27 C Assume constant specific heats at the average temperature a The final pressure in the tank is A 7 kPa B 12 kPa C 75 kPa D 137 kPa b The amount of heat transfer is A 943 6 kJ B 1318 6 kJ C 1731 4 kJ D 3113 1 kJ

Evaluate the following statement and determine whether it is True or False The specific heats of ideal gases are dependent on both temperature and pressure A True B False