Mathematical Induction Questions and Answers

Suppose that out of 262 professional 119 of them owned a convertible 105 of them owned a giant screen TV 123 owned a sporting goods store 26 owned a convertible and a store 35 owned a TV and a store 64 owned a convertible and a TV 7 owned all three items Hint To answer the following questions draw and label a Venn diagram that numerically represents th information above 1 How many athletes did not own any of the three items 33 2 How many owned a convertible and a TV but not a store 57 3 How many athletes owned a convertible or a TV 160 4 How many athletes owned exactly one type of item in the survey 98 5 How many athletes owned at least one type of item in the survey 229 6 How many owned a TV or a store but not a convertible 74 X X

PERSONAL RESPONSE How do the reactions of the Titanic passengers affect your feelings such as sympathy about the collision How does reading these personal reactions help you better understand what happened Be sure to use evidence to support your response

Write the below expression in factored form by factoring out the greatest common factor or a negative common factor if the coefficient of the term of greatest degree is negative 22 2 8mn 40m n

Given the following demand data compute a simple exponential smoothing forecast for alpha values of 0 3 and 0 5 Use the actual value in Period 1 as your starting forecast in Period 2 Round all answers to two decimal places Use these rounded answers when computing subsequent answers Period Actual Demand 77784 OSAWNA 1 2 3 4 5 6 10 11 Period 1 2 4 5 6 7 8 9 10 11 32 30 2 mmmmmmm 29 32 33 33 35 32 32 30 30 Answer is complete but not entirely correct a 0 3 a 0 5 Forecast Forecast 32 00 31 40 30 68 31 08 31 66 32 06 32 94 32 66 32 46 31 72 32 00 31 20 X 30 32 X 30 99 31 79 32 27 X 33 36 32 82 X 32 49 X 31 49 X

Let U all soda pops A all diet soda pops B all cola soda pops C all soda pops in cans and D all caffeine free soda pops Describe the given set in words Anc OA All diet soda pops and all soda pops in cans OB All non diet soda pops in cans OC All non diet soda pops and all soda pops in cans O D All diet soda pops in cans

Prot 10 points CC Consider the following matrix A and it s inverse A 0 1 0 A 2 0 0 0 1 1 Let the entry wise 6 norm of a mx n matrix B or vector be defined as 1 6 A 0 1 m EE B 6 6 b 16 0 5 0 0 0 0 1 1 Note this norm is similar to the Euclidean norm just replacing the 2 with a 6 in the power and the root Provide your answers with at least 6 digits a By hand calculate A 6 6 b By hand calculate the condition number of A using the entry wise 6 norm

2 The following graph shows plots of the growth functions both polynomials for two algorithms A and B f n 10 16 12 10 14 6 A 8 10 B 12 14 16 18 20 22 24 hich algorithm would you recommend to use Answer A or B and then justify 26 N

Fill in the blanks below When multiplying a decimal number by powers of 10 such as 10 100 1000 and so on move the decimal point in the number to the of 10 When multiplying a decimal number by powers of 10 such as 10 100 1000 and so on move the decimal point in the number to the power of 10 the same number of places as there are the same number of places as there are in the power in the digits

A frame ABCD is supported by a roller at point D and by a pin at point A The structure is subjected to point loads shown in Figure 1 1 Calculate the internal loadings normal and shear forces bending moment at the cross sections through points G and H using the method of sections yk D 50 KN C 1 0 m G 1 5m H 1 0 m 3 0 m 40 KN 70 KN B 1 5 m Figure 1 1 Frame A 1 0 m 2 0 m For cuts on vertical structural members please use the following senses as the positive sign convention for internal reactions Z M M VN AN V

Option 1 The main social insurance programs like Social Security Medicare and unemployment compensation are funded by a payroll tax on the earnings of individuals who may receive benefits Do you think social insurance programs are effective How are social insurance programs affected by the socioeconomic and political forces Explain your answers I think the social insurance programs like Social Security Medicare and unemployment have been effective These programs function by requiring a contribution from the individual prior to using benefits Social Security and Medicare were developed to help reduce poverty and reduce inequality by providing financial assistance and insurance to the elderly Greenberg 2018 Because of this income elderly have a steady and reliable monthly income and insurance to get regular checkups The goal should be financial freedom and to build enough savings to not have to rely on Social Security but unfortunately it is easier said than done for many individuals Life happens every day from living above your means poor spending habits losing a spouse early in life illness and the list goes on and on as to reasons we struggle to save for retirement And some individuals believe that Social Security benefits should cover all their expenses when they retire Social insurance programs are affected by socioeconomic and political forces because the environment affects the amount of money needed to sustain life for the individual Political forces affect the social insurance programs because the policy makers are evaluating the programs to determine ways to sustain the success and longevity of them Cottle Caliendo 2022 Cottle Hunt E Caliendo F N 2022 Social security and risk sharing A survey of four decades of economic analysis Journal of Economic Surveys 36 5 1591 1609 https doi org 10 1111 joes 12492 B

15 10x 10x 7x x 1 17 2v 60v 44 2v 10 19 806 1806 1706 66 06 10 16 4x 6x 12x 12 4x 6 18 4x4 21x 39x 13x 84 4x 9 20 26 6 2 186 2b 1

A scientific study on mesothelioma caused by asbestos gives the following data table Micrograms of 67 70 Area of scar tissue cm2 162 189 188 215 184 Using technology it was determined that the total sum of squares SST was 1421 2 and the sum of squares due to error SSE 903 51 Calculate R and determine its meaning Round your answer to four decimal places Therefore 36 43 of the variation in the observed y values can be explained by the estimated regression If 0 3643 Therefore 0 3643 of the variation in the observed y values can be explained by the estimated regression 0 0357 0 6357 Talo 63 57 of the on the drved y values can be explained by the estimated regression

Use the cutting planes shown to draw the sectional views of the following figures Note Draw the appropriate top view with the section line through it and the corresponding section view for each figure Make sure the section view has appropriate annotations Figure 2 1 Figure 2 2

Problem 5 1 Algo A manufacturing shop is designed to operate most efficiently at an output of 1 090 units per day In the past month the plant averaged 870 units per day What was its capacity utilization rate last month Round your answer to 1 decimal place Capacity utilization rate

Height and time The height in feet of a baseball hit straight up from the ground with an initial velocity of 64 ft s is given by h f t 64t 16t2 where t is measured in seconds after the hit 1 a Is this function one to one on the interval 0 t 4 b Find the inverse function that gives the time t at which the ball is at height h as the ball travels upward Express your answer in the form t f h c Find the inverse function that gives the time t at which the ball is at height h as the ball travels downward Express your an swer in the form t f h d At what time is the ball at a height of 30 ft on the way up e At what time is the ball at a height of 10 ft on the way down

17 Which of the following ranks bonds from lowest energy to highest energy O A Triple bond between carbons single bond between carbons double bond between carbons OB Single bond between carbons triple bond between carbons double bond between carbons OC Single bond between carbons double bond between carbons triple bond between carbons OD Double bond between carbons single bond between carbons triple bond between carbons

20 Carbon fixation takes place during O A the Calvin Cycle OB absorption of light by chlorophyll O C photosystem II O D photosystem I

18 Which portion of a molecule of F O has partial positive charge OA The central O atom OB The partial charge on each atom is negative C The F atoms O D The partial charge on each atom is zero

2 For an ion the sum of the formal charges on each atom should equal O A the overall charge of the ion O B the electronegativity of each atom in the ion OC the opposite of the overall charge of the ion D zero

3 Set up for the PCR tubes for a Obtain 1 PCR tube containing Ready To Go PCR beads for each sample three in total and label with the sample type group color and tubulin b Add 22 5 uL of the tubulin primer loading dye mix to each tube Allow several min for bead to dissolve c Add 2 5 L of each DNA sample to its respective tube and label 4 Close each tube and mix by tapping the bottom of the tube Keep tubes on ice until all groups are ready for the next step 5 Give your tubes to your TA to place into the Thermocycler The amplification run setup is as follows PCR Cycle Program 1 cycle 95 C 5 min 36 cycles 94 C 10 sec 55 C 5 sec 72 C 5 sec 1 cycle 72 C 1 min The PCR cycle includes a denaturation step 94 C a reannealing step 55 C and an elongation step 72 C PCR Beads contain PuReTaq DNA Polymerase 10 mM Tris HCl 50 mM KCl 1 5 mM MgCl2 200 M of each dNTP deoxyribunucleated triphosphorylated DNA bases Preparing for Gel Electrophoresis Make the following solution 75 ml 10x TBE Tris base Boric acid EDTA 0 9 M Tris 0 889 M boric acid 20 mM EDTA This solution will be used next week for the gel electrophoresis in Part C of the protocol

10 Write a mail time story for 2 3 and indicate your answer bris 35 0 3 9 0 V dsdsrigls orit ni extiel Us to joe ori od A

0 0 0 Add 5 and 0 0 0 10 0 0 Carry bits 4 bit number 4 bit number

The Jacksons are saving up to go on a family vacation in 3 years They invest 2800 into an account with an annual interest rate of 1 27 compounded quarterly Answer the questions below Do not round any intermediate computations and round your final answers to the nearest cent If necessary refer to the list of financial formulas a b Assuming no withdrawals are made how much money is in the Jacksons account after 3 years How much interest is earned on the Jacksons investment after 3 years 0 5 X S

Calculate the absolute pressure P of the manometer shown in the Figure in kPa The local atmospheric pressure is 758 mmHg 12 cm 5 cm Fluid A 10 kN m 30 cm Atmospheric pressure 15 cm Fluid B 8 kN m Problem 2 A pressure cooker cooks a lot faster than an ordinary pan by maintaining a higher pressure and temperature inside The lid of a pressure cooker is well sealed and steam can escape only through an opening in the middle of the lid A separate metal piece the petcock sits on top of this opening and prevents steam from escaping until the pressure force overcomes the weight of the petcock The periodic escape of the steam in this manner prevents any potentially dangerous pressure buildup and keeps the pressure inside at a constant value Determine the mass of the petcock of a pressure cooker whose operation pressure is 100 kPa gage and has an opening cross sectional area of 4 mm Express your answer in grams with three significant figures Assume an atmospheric atm 101 kPa Pressure cooker Petcock A 4 mm

13 Dimitri is buying a car He chooses Option 1 to add a new sound system to his car What integer represents the change from the base price of the car to its final price Used Car DEALER Price Sticker 20 Base Price 700 00 1 400 00 Markdown 1 100 00 Sale Opt 1

Click the following link to view imports and exports data from FRED Then use that data to answer the following questions i Real time data provided by Federal Reserve Economic Data FRED Federal Reserve Bank of Saint Louis Using the data from FRED enter the values for real exports and real imports for the following two years for the following series IDs Enter your responses exactly as they appear in FRED Year Month Day 2021 01 01 2020 01 01 EXPGSCA billion billion IMPGSCA billion billion

The units of area are always OA cubic units OB square units OC units The units of volume are always OA square units OB cubic units O C units The units of perimeter and circumfrence are alway OA cubic units OB square units OC units

O Farm Security Administration Agricultural Adjustment Act AAA O Works Progress Administration PWA O Soil Conservation and Domestic Allotment Act F Que 1 5 9

The volume V of a liquid in a spherical tank of radius related to the depth h of the liquid by Th 3r h V f h 3 Determine h given r 1m and V 05m 05 17h 3 1 h 3 1 5 3 h Th 3 Th 1h 15

51 What is a suitable device to measure out 4 3 mL of a liquid S2 What is a suitable device to measure out 23 mL of a liquid 53 What is the SWQ in mg for an electronic balance with an SR of 0 01 g S4 What is the SWQ in mg for a torsion balance with an SR of 4 mg

Given the idea by Moses Nina s wondering if she should travel from her home in Edison N3 to Dustin s house in Dover DE to purchase the lawnmower there They re 144 miles from each other and she s got a hybrid vehicle that manages 54 miles per gallon is the roundtrip drive worth it in Nina s case Show your work

What are the primary dimensions of energy O watts ML Olb ft ML 7

The perimeters of the triangles shown are equal Find the side lengths of each triangle AB BC AC Type a whole number PQ QR PR 2x 4 B 2x 2x 3 C 2x 2x A P 3x 2

consequences of a limiting case Therefore the idea of limits is essential for discovering and developing important ideas definitions formulas and theorems in calculus Students must have a solid intuitive understanding of limits and be able to compute various limits including one sided limits limits at infinity the limit of a sequence and infinite limits They should be able to work with tables and graphs in order to estimate the limit of a function at a point Students should know the algebraic properties of limits and techniques for finding limits of indeterminate forms and they should be able to apply limits to understand the behavior of a function near a point Students must also understand how limits are used to determine continuity a fundamental property of functions BIG IDEA 2 DERIVATIVES Using derivatives to describe the rate of change of one variable with respect to another variable allows students to understand change in a variety of contexts In AP Calculus students build the derivative using the concept of limits and use the derivative primarily to compute the instantaneous rate of change of a function Applications of the derivative include finding the slope of a tangent line to a graph at a point analyzing the graph of a function for example determining whether a function is increasing or decreasing and finding concavity and extreme values and solving problems involving rectilinear motion Students should be able to use different definitions of the derivative estimate derivatives from tables and graphs and apply various derivative rules and properties In addition students should be able to solve separable differential equations understand and be able to apply the Mean Value Theorem and be familiar with a variety of real world applications including related rates optimization and growth and decay models BIG IDEA 3 INTEGRALS AND THE FUNDAMENTAL THEOREM OF CALCULUS Integrals are used in a wide variety of practical and theoretical applications AP Calculus students should understand the definition of a definite integral involving a Riemann sum be able to approximate a definite integral using different methods and be able to compute definite integrals using geometry They should be familiar with basic techniques of integration and properties of integrals The interpretation of a definite integral is an important skill and students should be familiar with area volume and motion applications as well as with the use of the definite integral as an accumulation function It is critical that students grasp the relationship between integration and differentiation as expressed in the Fundamental Theorem of Calculus a central idea in AP Calculus Students should be able to work with and analyze functions defined by an integral Your Assignment For each of these Big Ideas do the following 27 points total 1 Rate yourself 1 10 on how confident you are in your understanding of the Big Idea Pay particular attention to the statements about what students should be able to do 3 points for each Big Idea

Consider the truck tire shown to the right which is initially inflated to 50 psi when answering questions 1 3 below Problem 1 5 points If there are no air leaks from the tire and the valve is closed the air inside the tire is best analyzed as a A Isolated system B Closed system C Open system D All of the above

If the valve on the tire is opened and air is allowed to escape until the internal pressure is 0 psig the air inside the tire is best analyzed as a A Isolated system B Closed system C Open system D All of the above

A region s average movie theater ticket price in 2007 was 6 54 For 2017 it was estimated to be 8 16 Find the percent of increase in average movie theater ticket price fa this 10 year period The percent of increase in average movie theater ticket price is Round to the nearest tenth of a percent

find the Laplace transform of di iR E 1 L 2 dt where L R and E are constants and i 0 0 d x dq 2 dt dt t 2t

1 Statistician 2 Analyst 3 Instructor Director 5 Accountant 212 50 350 50 615 84 68 40 253 82 56 80 762 20 8 282 48 398 42 910 48 3 952 00 5 012 90 7 467 97 2 147 80 4 697 80 Commerce Bank hired Pete Drexel as its new branch office manager at an annual salary of 59 500 00 Advertising Costs Interview Expenses P Drexel L Lindsay Search Agency Fee 15 of first year s annual salary 421 27 85 25 94 60

Let Xn n 3 5n 6 and yn xn xn 1 Prove the following for all positive integers n i yn is an integer hint n n 1 is an even integer ii xn is an integer Hint use mathematical induction and part i

7 Find the eigenvalue s and eigenvectors for A 23 3 6

1 Each of the following sets X ARE NOT vector spaces over the given field F and with the given operations List with justification all vector space axioms failed by the sets a X is the open upper half plane in R X C 8 d and both vector addition and scalar multiplication are the same as in R b The set X R2 over the field F are given by R U 0 R where vector addition and scalar multiplication a d and C a ca c X Rover F Z2 where vector addition is the same as in R and scalar multiplication is given by 0 x 0 and 1 x x

Consider the poset H represented by the following Hasse diagram h g b Give each of the following If any do not exist explain why 14 1 All minimal elements of H 14 2 All maximal elements of H 14 3 All upper bounds of b c 14 4 All lower bounds of g h 14 5 The LUB of b c 14 6 The greatest element in H 14 7 The least element if H f a

In the linear trend equation Ft k at btk identify the term that signifies the trend O bt Ok 0 Ft k at

For problems 15 16 let X be from a binomial n 20 p distribution Let Y be from a geometric distribution with p 2 Assume Y is independent of X Let Z 4X 3Y Calculate the standard deviation of Z using the variance of X above O 28 10 O None of the above 260 O 260

Which of the following is not an example of feature leakage Using the Thank You page that follows a Purchase page as a feature when predicting purchases from web browsing logs Using previous searches to predict whether or not someone intends to buy a car When predicting whether or not a visitor to Netflix will start a subscription using account number as a feature When predicting whether or not a patient has a certain medical condition based on symptoms and other diagnostics using treatments for that disease as a feature Treatments are only applied after diagnosis is made so this it not a valid feature

Which of the following is not a benefit of writing reproducible code It requires you to spend more time writing and designing code It enables you to make design revisions with minimal additional work O It can help others audit and verify your results It makes it easier to embed the modeling flow in a production system

Which of the following best describes the goal of learning curve analysis O Learning curve analysis enables you to identify the impact of dataset size on model performance Learning curve analysis is an ideal solution for managing class imbalance Learning curve analysis enables you to quantify model estimation bias of a chosen algorithm O Learning curve analysis enables you to identify your most influential features

3 6 9 2 5 B The price of a home is 290 000 The bank requires a 10 down payment at the time of closing The mortgage on the house is a 20 year fixed rate mortgage at 7 Use this information to find the amount of the monthly payment Round your answer to the nearest cent Use the formula below to solve the problem P Payment 1 2 248 2 2 124 3 2 012

9 Assume that Cola Co has a share price of 43 59 The firm will pay a dividend of 1 22 in one year and you expect Cola Co to raise this dividend by approximately 6 2 per year in perpetuity a If Cola Co s equity cost of capital is 8 4 Cola Co s price per share should be the nearest cent b Given Cola Co s share price its dividend growth rate should be places Round to Round to two decimal