Mathematics Level A. Højere handelseksamen. Friday May 19, 2017 At hhx171-mat/a

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1 Mathematics Level A Højere handelseksamen hhx171-mat/a Friday May 19, 2017 At

2 Mathematics A The paper consists of two partial tests. The partial test without Tools consist of the exercises 1 to 5, i.e. all in all 5 questions. The answers to this partial test must be handed in by 10am. The partial test with Tools consists of the exercises 6 to 12, i.e. all in all 18 questions. The 23 questions each carry a weight of 5 points in the assessment of the total paper. You are only to hand in one of the assignments 12A, 12B and 12C. If you hand in more of these assignments, only the first assignment will be assessed. During the first hour of the test tools apart from writing and drawing tools are not allowed. In the last 3 hours of the test all tools are allowed. In the assessment of the answers to the individual questions and the overall impression emphasis will be put on whether the examinee s train of thought is clearly reflected in the answers. The paper must account for the method used and documentation in the form of a sufficient number of intermediate results and/or a mathematical explanation of the use of different facilities offered by an IT program. If graphs and illustrations are used there must be a clear relationship between text and illustration. The following two data files belong to this set of exam questions: hotels washing machine

3 Friday May 19, 2017 page 1 of 9 The partial test without Tools At Exercise 1 f 4 2 A function is given by the equation f( x) x 4x 7x 19. a) Find f ( x) and check whether the graph of f has a tangent with a positive slope when x 1. Exercise 2 The graph shows the cumulative frequency of the length of Danish trawlers. 100 relative cumulative frequency in percent length in meters a) Find the 25th percentile and find the proportion of Danish trawlers with a length of more than 40 meters. You may use attachment 1. Source: dst.dk/fisk1

4 Friday May 19, 2017 page 2 of 9 Exercise 3 Graphs of three functions y f( x) 6 x 0.5 gx ( ) hx ( ) 6 0.5x x 6 4 C have been drawn in the same coordinate system a) Account for which graph belongs to, and h, respectively. f g 2 A B x Exercise 4 R C The revenue (in DKK 1000) and the costs (in DKK 1000) of the production and sales are found by the functions with the equations Rx x x x 1 2 ( ) 7, Cx ( ) 2x 9, 0 x 20 where x is the produced and sold quantity of the commodity in tons. DKK 1000 R C a) Find the interval on which the revenue is greater than the costs. quantity in tons Exercise 5 A function f is given by the equation f x x x 3 2 ( ) a) Find the antiderivative F of f, going through P(1, 7). The answers to the test without tools must be turned in by 10.00

5 Friday May 19, 2017 page 3 of 9 The partial test with Tools At Exercise 6 Below the point of intersection between the functions f( x) x and gx ( ) x has been found. a) You must give explanations to the calculations. You may use attachment 2. f() x gx () x x x x 2 y The point of intersection is ( xy, ) (2,28.80). Exercise 7 The function f( x) x ln( x) 1, x 0 can be described by different characteristics, e.g.: The zeros/roots The sign variation The monotonicity The extrema The curvature a) Of the characteristics mentioned above find three for the function f and mark the results found on a graph of f.

6 Friday May 19, 2017 page 4 of 9 Exercise 8 In a survey of 1854 representative chosen Danes made by Kompas Kommunikation for Hotels.com the following question among others was asked: Have you ever booked a vacation (e.g. flights, hotel, package deals) using an app or a homepage on your cell phone/smartphone? The table below shows an excerpt of data found in the file hotels. Booked via cell phone? Age group Yes 60+ years No 60+ years No years : : a) Make a table like the one below containing data from the survey years years years 60+ years Total Yes No Total 1854 Hotels.com wishes to check whether there is a correlation between the Danes answers to the question and which age group they belong to. b) State a hypothesis that can be used to test this correlation and test the hypothesis at a 5% significance level. From the survey an estimate of the proportion among Danes in the age group years answering Yes to the question can be calculated to p c) Find a 95% confidence interval for this proportion. Assume that the proportion of Danes in the age group years that answers Yes to the question is p 32.9%. d) Find the probability of a maximum of 25 out of 100 Danes between the ages of answering Yes to the question. Source: hotels.com

7 Friday May 19, 2017 page 5 of 9 Exercise 9 A consumer group examines the prices of washing machines that can be bought online. The group has taken a sample of 30 front-loading washing machines from different online stores. The table below shows an excerpt of data found in the file washing machine. Yearly energy consumption (kwh) Price in DKK : : a) Make a graphical presentation of the prices of the washing machines. b) Find the mean, the median and the 25th percentile of the prices. c) Make a linear regression model px ( ) a x b, describing the correlation between the yearly energy consumption x and the price y. d) Based on your answers to questions a), b) and c) write a short comment to the consumer group s magazine Exercise 10 The following three functions have been drawn in the same coordinate system. Sources: wupti.com, skousen.com, elgiganten.com, power.com 2 f( x) x 9x 2 gx ( ) x 5x hx ( ) x y The points of intersection A, B, and C between the graph of h and the graphs of f and g have been marked on the figure. a) Find the coordinates of points A, B, and C. A B g C f h x b) Find the area of the grey field in the figure.

8 Friday May 19, 2017 page 6 of 9 Exercise 11 For a commodity A the relationship between demand and price is found by the function Dx ( ) x 2400, 0 x where x denotes the quantity demanded and Dx () denotes the corresponding price. The relationship between supply and price for the same commodity A is found by the function Sx ( ) x 700, 0 x where x denotes the quantity supplied and Sx ( ) denotes the corresponding price. The equilibrium quantity is defined as the quantity making supply and demand equal, i.e. Dx () Sx (). a) Find the equilibrium quantity and the corresponding price of commodity A. price 2000 E U quantity A unit duty of DKK T is now added to the commodity. The relationship between demand and price can still be described by the function Dx (), whereas the relationship between supply and price now can be described by the function S ( x) x 700 T, 0 x T The unit duty gives a new equilibrium quantity m. The equilibrium quantity m is a solution to the equation S( x) Dx ( ). b) Find the equilibrium quantity m in terms of T i.e. isolate m in the equation m 700 T m 2400 T You may use a CAS-tool.

9 The tax revenue SP at a unit duty of DKK can be found as the equilibrium quantity m times the unit duty, i.e. T SP m T T Friday May 19, 2017 page 7 of 9 c) Find m at a unit duty of DKK T 500 and find the corresponding tax revenue SP. d) Account for the tax revenue SP as a function of being found by 2 SP( T) 25T 42500T T and find the unit duty that gives the highest tax revenue. price 3000 E UT U SP m quantity

10 Friday May 19, 2017 page 8 of 9 You may only turn in one of the exercises 12A, 12B, and 12C for assessment. If you turn in more of these exercises, only the first exercise will be assessed. Exercise 12A A student wishes to take a student loan for the rest of her expected study time. From September 1, 2016 and until July 1, 2018 the student can borrow DKK 3020 per month. During the education the interest rate is 0.33% per month. a) Find the size of the debt right after the last payment of DKK The repayment of the loan starts on January 1, The loan accrues interest for 18 periods at an interest rate of 0,33% until the repayment starts. b) Find the size of the debt on January , and find the monthly payment when the loan is being repaid over 6 years at an interest rate of 0.29% per year. Student loans (SU) My SU apply online Rates for loans Deadlines Amortizing Loan plan Increased loan Final loan Types of loans and interest rates Student loans since 1991 Student loans Student loans since 1991 Student loans paid since 1991 a loan 6 For a loan 6 the interest rate is four percent per year during your studies. After you have graduated the debt accrues interest according to the discount rate plus an extra fee (of maximum one percent) or minus a deduction. The extra fee/the deduction is determined in the yearly Finance Bill. You must start amortizing a loan 6 from January 1st one year after the end of the year in which you graduated (unless you also have a loan 4 and/or a loan 5). If for instance you graduate on June 30th 2009 you must start amortizing the loan on January 1st The final loan paid since 1994 We administer a final loan as a loan 6 unless you also have a loan 4 and/or a loan 5. Source:

11 Friday May 19, 2017 page 9 of 9 Exercise 12B The productivity of a newly hired employee for a given production company can be measured by the production time function. Tn ( ) denotes the employee s production time of the n th unit of the commodity. The production time fulfills the following differential equation T T ( n) 0.08 ( Tn ( ) 25), n 1 It is given that the employee s production time of the first unit of the commodity is 42 minutes, i.e. T(1) 42. a) Find an equation for T and find how many units must be produced for the production time to be under 30 minutes per unit. b) Find T (10) and account for what this number shows about the productivity. Exercise 12C A company produces and sells two kinds of protein powder FIT100 and MUSCLE-P. The production includes the mixing of different chemical substances under heating after which the substance is purified. To produce one kg of FIT minutes are needed for the mixing and 10 minutes for the purification. To produce one kg of MUSCLE-P 20 minutes are needed for the mixing and 12 minutes for the purification. There are a maximum of minutes per week available for the mixing and a maximum of 8400 minutes per week available for the purification. The company earns a profit of DKK 40 per kg FIT100 and DKK 60 per kg MUSCLE-P. FIT100 MUSCLE-P Maximum capacity Time consumption mixing Time consumption purifying Profit per kg in DKK The function f(, xy) a x b y denotes the total weekly profit. a) Draw the polygon area defined by the constraints above and find the quantity of FIT100 and the quantity of MUSCLE-P, which gives the company the biggest total weekly profit. b) Find out by how much the profit of MUSCLE-P can decrease without changing the optimal combination of FIT100 and MUSCLE-P found in question a).

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