Functions Questions and Answers

Question 8 of 16 DD Let f x 4x Macmillan Learning Simplify f x f a xa x a Express numbers in exact form Use symbolic notation and fractions where nee f x f a x a

roblem 2 The graph below represents the amount of water W in gallons in a small swimming pool after a certain number of hours t 1800 1600 1400 1200 1000 800 600 400 200 0 5 1 1 5 2 2 5 3 3 5 4 4 5 5 15 5 6 6 5 7 a Evaluate W 3 5 Interpret its meaning and include appropriate units b Determine the domain of the function W and explain what it represents 0 c Determine the range of the function W and explain what it represents d Compute the average rate of change of W t and the intervals 2 3 3 4 and 4 5 and interpret your answers Be sure to include appropriate units

This is a 2 page document Directions Determine whether the relations are one to one functions 3 8x0 2 1 Directions Determine if f x has an inverse if yes find f x State any restrictions in the domain 5 f x x 5 2 9 3 4 f x x 3 4 1 6 S x x 2 fix 8 f x 4 2x 1 7 f x x 7 1 9 f x 2x 5 x 7

7 a b For each of the following equations determine whether it defines y as a function of x If yes find the formula If not explain why x y xy x 1 y 2x 1

Directions Find the zero s and y intercept of each function algebraically 7 f x 2x 5 1 8 f x x 10x 24x to salva E51

a Consider the function g x x 9 Compute and simplify below g x g a x a g x h g x h Compute the quotient in part a at x 3 a 5 and the quotient in part b at x 3 h 2

4 a b Consider the function g t t2 5 Compute and simplify as much as possible the llowing 9 3 2 9 2t 1 2 1 2 1 24 47 411 g t h h 5 21 2

c In robotics a scheme known as the Minimum Distance Technique MDT is used to avoid line obstacles The MDT involves the calculation of the minimum distance from the robot to the line segment and the avoidance of the resultant point on the line segment Avoidance of the closest point on the line at any time 120 essentially results in the avoidance of the entire line segment Consider the Figure 3 Show that b b 2 x a e y b where c a a a a b b and f Hint 1 Determine the parametric equations of the line 2 Find the Euclidean distance from the robot to the line segment 3 Optimize the distance 2 0 2 1

Click the following link to view Personal Income Outlays data from FRED Then use that data to answer the following questions Real time data provided by Federal Reserve Economic Data FRED Federal Reserve Bank of Saint Louis For this exercise you will need to enter data from FRED for total personal consumption expenditures as well as personal consumption expenditures on durable good nondurable goods and services Using the data from FRED enter the values for the second quarter of 2023 shown as 2023 04 01 in FRED for the following series IDS Enter your responses exactly as they appear in FRED Title Personal Consumption Expenditures Personal Consumption Expenditures Durable Goods Personal Consumption Expenditures Nondurable Goods Personal Consumption Expenditures Services Series ID PCEC PCDG PCND PCESV Value billion billion billion billion

Name the solid and find its volume Use 3 14 as the approximate value of 1 What is the name of this given solid Hemisphere O Pyramid O Rectangular solid Sphere O Cylinder Cone The volume is 70 m

4 f x x 5x g x 3x 1 and h x 1 a x f x b h g x 5 f x x 6 g x 4x and h x 2x 5 a hof x b hog x Use the given functions to evalaute each function 3 6 f x x 4x g x 2 3x and h x X a f g 5 c goh 6 b g f 3 d hof 2 c fog x c gof x

6 20pts Let D20 be the dihedral group generated by a with z 20 and y with y 2 such that yxy x Simplify yxyx yx yx and leave the result in the form yx or x where 0 r 19

A gas is contained in a vertical frictionless piston cylinder device The piston has a mass of 3 2 kg and cross sectional area of 35 cm A compressed spring above the piston exerts a force of 160 N on the piston If the atmospheric pressure is 95 kPa the pressure of the gas inside the cylinder is mostly equal to which one below Please draw the free body diagram and show the work that supports your chosen answer A 140 kPa B 150 kPa C 160 kPa D 170 kPa A 35 cm Palm 95 kPa m 3 2 kg

7 g x Domain Range Discontinuities if xs 2 if 2 x 2 3 x 4 2 x 11 if x 2

N m Free Body Diagrams The figure below shows a mechanical assembly consisting of a horizontal piston cylinder that contains a pressurized gas with a pressure larger that the atmospheric pressure Patm 1 105 piston is locked with a pin such that it does not move On the outside of the assembly there is a spring of Initially the known elastic constant k 1 104 The piston has two circular surfaces of equal area A 0 025 m Initially as shown in the figure the spring is at its non deformed position neither stretched nor compressed Next the locking pin is removed and the piston moves to the right until it finds a new equilibrium position Please answer the following questions N m Gas O Feas O Fatm Piston A 0 025 m force due to gas force due to atmosphere Patm 1x10 N m Spring k 1x10 N m Locking pin a Indicate all the important forces you would include on a free body diagram FBD for the horizontal forces acting on the piston before the locking pin is removed Note that the friction between the wall of the cylinder and the piston is neglected O Rx OFs reaction force due to pin force due to spring

OPEN ENDED Sketch a graph of a function with each set of characteristics 69 infinite discontinuity at x 2 increasing on 2 increasing on 2 f 6 6

Complete the equation 5 a a0 5 2002

Minutes after injection 0 5 10 15 20 Penicillin concentration micrograms per milliliter 200 152 118 93 74 200 150 100 50 5 10 15 20 Time minutes When a patient receives a penicillin injection the kidneys begin removing the penicillin from the body The tab and graph above show the penicillin concentration in a patient s bloodstream at 5 minute intervals for the 20 minutes immediately following a one time penicillin injection The penicillin concentration in micrograms per milliliter in the patient s bloodstream t minutes after the penicillin injection is modeled by the function P defined by P t 200b If P approximates the values in the table to within 10 micrograms per milliliter what is the value of b rounded to the nearest tenth

If ax 2 bx 7 15x cx 14 for all values of x and a b 8 what are the two possible values for c 3 and 5 6 and 35 10 and 21 31 and 41

ollowing table illustrates the value added approach to calculating GDP Please complete the table Value of Product Value Added By The cotton farmer The textile mill The shirt company Firm Cotton farmer Textile mill Shirt company L L Bean Raw Cotton 2 Cotton woven into cotton fabric 4 Fabric made into a shirt 16 Shirt sold on L L Bean s CUS 51 M C website 36 L L Bean SEVE EI 1 Total Value Added Value Added Equals 2 2 12

An artist buys scrap metal from a local steel mill as a raw material for her sculptures Last year she purchased 5 000 worth of scrap metal With the steel she produced 10 authentic sculptures that she sold for 800 each to a local art dealer The art dealer then sold all of the sculptures to art collectors for an average of 1 000 each The total value added of the artist is Enter your response as an integer

State the domain of each 5x 2 5A f x 7x 12

Economists often are interested in percentage change from one period to the next The percentage rate of change of gross domestic product GDP is an important macroeconomic variable If in 2010 GDP was 11 150 billion dollars and GDP increased to 11 163 billion in 2011 what is the growth rate of the U S economy in 2011 6 Enter your response rounded to one decimal place

3 Show graphically the graph of the correspondence f R f x 0 5x x R wher

Domain dan range dari f x 3x 7 dari fung x 6 5

otplate is positio xy plane positive y direction The x 0 and x L edges are held at T 0 K with they 0 edge held at 500 K Jaholled sketch of the prem b State ve all general solutions of Lapiass Belime boandarditions for all 4 edges including a formula for T x 0 and hence explain which boundary concr s rule out which general solutions d Express the remaining solution as a series in e Hence find the necessary coefficients of this series f For a plate which is not of infinite length in the y direction show that sinh k A y is a general solution to the Laplace equation where A is the length in the y directio

K Identify which type of sampling is used random systematic convenience stratified or cluster To determine customer opinion of their musical variety Sony randomly selects 150 concerts during a certain week and surveys all concert goers Which type of sampling is used A Random B Cluster OC Convenience OD Stratified OE Systematic COLD

5 The average low temperature by month in Nashville is represented by the function f x 1 4x 19x 1 7 where x is the month Find the average rate of change from March to August

Anne and Nancy use a metal alloy that is 19 51 copper to make jewelry How many ounces of an alloy that is 19 copper must be mixed with an alloy that is 20 copper to form 145 ounces of the desit whole number A 79 ounces OB 73 ounces OC 71 ounces OD 74 ounces

This is a 2 page document Directions Determine if the function shown on the graph is continuous If not identify the type and location of discontinuity 1 4 7 4 100 2 5 8 LEAD Esi 3 6 9 Anx H N BE

Directions Describe the end behavior of each graph 11 10 13 0 16 Bas B PR K 14 17 ho fix 12 d 15 18 B END RAL 9H

15 The average low temperature by month in Nashville is represented by the function f x 1 4x 19x 1 7 where x is the month Find the average rate of change from March to August

Let R 1 4 Find the point P such that PR has components 1 1 P

Compute the dot product 10 9 3 3 8 7

6 Find the stanard matrices for A and A where T R R T x y x y y and T R R T x y z y z then find the composition of T T T

5 Use SCILAB to perform the necessary operations in order to decode the following message 3 marks 90 75 63 0 21 45 12 0 47 100 given the encoding matrix is 3 B 91 5 Consider the letters of the alphabet translate into numbers following the scheme showed below Copy and paste Scilab output Write the decoded message A BCDEFGHIJKLMN 012 3 4 5 6 7 6 7 8 9 10 11 12 13 14 O PQR S TU V W X Y Z 15 16 17 18 19 20 23 24 25 26 27 28

c Sketch the function f x 3 x 1 2 x 1 and its inverse in the same graph REMEMBER NO GRAPHING SOFTWARE ALL GRAPHS HOITADUMAA nar MUST BE DONE BY HAND Drawing tools are acceptable but be precise USE DIFFERENT COLOURS FOR EACH CURVE

In a lower one tailed hypothesis test chosen level of significance 0 10 sample size 55 and t test statistic 1 76 Which of the following is the conclusion O The null hypothesis is rejected There are many sampling errors O The null hypothesis cannot be rejected O This is a high probability and the conclusion is difficult

Using Chebyshev s theorem for standard deviation calculate the percentage of data that lie within five standard deviations of the mean 96 50 89 75

2 Sketch a possible graph of the function f x on the grid below such that all the conditions below are met T 1 4 D x ERI 3 x 3 R lyERI Osy 4 f 2 3 f 1 2 The x intercepts are 3 and 3 The y intercept is at 4 Make sure to clearly draw the points on the graph

en A 1 0 2 00 4 0 1 3 1 solve the system A x b where b 2 4

Use the graph to match the limit with the correct answer 0 6 lim f x x 2 lim f x x 2 lim f x x 2 ef 2 2 2 y f x 6 A 0 B DNE C 1

Find the derivative y x 4 x 4 ln x4 5 ay x 4 x 2 In x 5 Ob y 4x 5 2x In x 5 x 4 x 4 y 4x 5 2x 3 In x4 5 x 4 x2 4 y 4x 5 2x 3 4x3 x4 5 Od x 4 5 4x3 x 5

Evaluate S 1 a 108 Ob 6 5 3x x dx c 90 Od 112

oblem 2 The graph of z f x y is shown below In each part determine whether the given partial derivatives are positive negative or zero Note that the function is symmetric about 0 in both the x and y directions 15 10 5 0 5 10 4 2 0 a f 2 2 and fxx 2 2 b fy 2 2 and fyy 2 2 c fx 2 0 and fxx 2 0 d fy 2 0 and fyy 2 0 2 A positive negative B negative zero C zero zero D zero positive E positive positive F positive zero G negative negative H negative positive D zero negative

1 Find the dot product U V for U 4 1 V 6 3 8

2 Let two vector functions be defined by r t t t 3t r t t 1 4 5 t a Find the intersection point of the functions Is it a collision point b Find the equation of the line that goes through the intersection point that also orthogonal to both curves at that point

of the demand function for math anxiety pills is p Consumer surplus 49 X dollars D x 16 2x determine the consumer surplus at the market price of 2 dollars

A hypothetical computer keeps real numbers in the floating point format with 8 bits The first bit is used for the sign of the number the next four bits for the exponent and the rest of the bits for the fraction part of the mantissa Assume it works as same as the floating point numbers in R Which of the following is most nearly the machine epsilon O O 2 2

An inferior product with a large advertising budget sells well when it is introduced but sales fall as people discontinue use of the product Suppose that the weekly sales S are give 600t 0 t 2 2 where S is in millions of dollars and t is in weeks After how many weeks will sales be maximized weeks S