Good luck! School of Business and Economics. Business Mathematics. Date: 25 October, Time: 12:00. Calculator allowed:

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1 School of Business and Economics Exam: Code: Examinator: Co-reader: Business Mathematics E_IBA1_BUSM dr. R. Heijungs dr. G.J. Franx Date: 25 October, 2017 Time: 12:00 Duration: Calculator allowed: Graphical calculator allowed: 2 hours Yes No Number of questions: 3 Type of questions: Answer in: Open / multiple choice English Remarks: (1) You will receive a special answer sheet for question 1 (2) You will receive normal empty paper for questions 2 and 3 (3) Please write your name and student number (7 digits) on paper (1) and (2) (4) You may keep the questions Credit score: Start Question 1 Question 2 Question Grades: 8 November, Inspection: Number of pages: Will be announced on Canvas. 5 (including front page) Good luck!

2 Question 1 (42 points) Question 1 consists of 14 short subquestions. Each subquestion counts for 3 points. You must give an answer only, on a separate special answer sheet. Note the following in answering the subquestions: The indication exact means that you have to fill in an exact number, such as 12, 2 3 and 3e 2. The indication 1 decimal means you have to fill in a number at the specified accuracy, such as In addition, you may have to specify additional text, such as euro. The indication 2 significant digits means you have to fill in a number at the specified accuracy, such as In addition, you may have to specify additional text, such as euro. The indication text, means you have to supply a phrase, such as There is no stationary point. The indication mathematical formulation means that you have to fill in a mathematical expression, such as 0 x 1 or a The indication choose one means that you have to choose one option, such as (B). The indication choose one or more means that you have to choose one or more options, such as (B) and (D) and (F). No indication means you must decide for yourself. Given is the function H(x, y) = xe 4xy2 + 3x. Find H. Simplify where possible. y 2 (b) Given is x i = i + 1. Find i=0 x i. Give all solutions of ln ( 1 4 x2 ) = 0. Note that the answer may be no solutions. Given is a matrix A of order 5 3. What is the order of the product A A? Note that the answer may be impossible. (e) Evaluate the following indefinite integral for x > 0: ( 1 2x + 3) dx. (f) For a cheese-producing company, profit (π, in /yr) is modeled as π = αq + βq + γp 2 Q 2, where P is price (in /kg) and Q is sales (in kg/yr). What is the unit of coefficient γ? If the coefficient is dimensionless, write a clear dash ( ). (g) (h) (i) Give all solutions of e (x2 3) + 5 = 0. Note that the answer may be no solutions. Given is f(x) = 10 3x. Find df dx. Given is a continuous differentiable function f(x) on a domain D = [ 1,1] for which it is known that df < 0 on D, and f(1) < 0. What can you conclude from this? (choose one or dx more) A) f(x) < 0 for x D C) f(x) 0 for x D E) none of these B) f(x) > 0 for x D D) f(x) 0 for x D 2

3 q 1 q 2 (j) Given is a vector q = ( ). Express the result of the inner product (1, q) in -notation. q n Simplify as much as possible. (k) Years of education (x) and monthly salary (y) of 100 persons were measured. The correlation coefficient (r xy, defined as s xy s x s y ) was found to be What can you say about the correlation coefficient (r xz ) between the years of education (x) and the annual (=yearly) salary (z)? Choose one. A) r xz = B) r xz = C) r xz = 0.83 D) r xz has another value than those given in A) B) C) E) Not enough information (l) (m) A function y = f(x) is defined implicitly by 5xy 6y 2 = 10. Find dy dx. Solve the following matrix equation for X: AX B 2 = I. Remove all unnecessary terms. 5 λ 5 2μ (n) Line L is specified as {( λ ) λ R} and line M as {( 2μ ) μ R}. Which 3 + λ 2μ + 3 statement(s) is/are true? Choose one or more. A) Both lines pass through (8,3,6) B) The lines fully overlap C) At least one of the lines pass through (5,0,3) D) The lines are parallel but not overlapping E) None of the above Bonus question: if you miss one of the above questions, you may still obtain maximum score by correctly answering the question below. (o) Which Greek letters have the names kappa and omega? Answer like F+B, E+A, etc. A) ψ B) χ C) ζ D) ω E) θ F) ρ G) κ H) φ I) ν J) ξ 3

4 Question 2 (26 points) Question 2 must be answered on the empty exam sheets. Please start at the top of a page. You must specify all steps you take and use good notation principles. Streaming services (Netflix, Spotify) have been challenging more traditional media services. Costs (C) for streaming services are modelled as C = C 0 + β N where C 0 > 0 is fixed costs, β is a coefficient and N > 0 is the number of customers. Benefits (B) are B = φn where φ is the subscription fee. What should be the subscription fee, given that we have N customers and that we don t make any profit, so B = C? For this case, find φ = f(n; C 0, β). Simplify as much as possible. You may assume N to be a continuous variable. (4 points) (b) See. In a more refined model, costs are C = C 0 + β NH + αh Here, H > 0 is the number of hours that an average consumer uses the streaming service. C is supposed to increase if N increases. Investigate what this implies for α and β. The final answer looks like α [0,1], β 0, α < N, no conclusion for β, etc. (4 points) (e) Indicate the time a consumer spends on streaming services by t s and the time spent on other services by t o (both in hours per day). The total daily time spent using either of these services is 4 hours. The price of 1 hour of streaming service is 0.1 euro, the price of 1 hour of other services is 0.2 euro. The total daily expense is m euro. Condense these facts in a system of linear equations where t s and t o are the variables. Rewrite the system of linear equations as a matrix equation At = b, where t = ( t s t o ). (4 points) Streaming requires a lot of bytes. The pattern of streaming services, R in gigabyte/hour, is modelled as R = t 2, where t [0,24] is a continuous variable indicating the time of the day. Calculate the total number of gigabytes streamed in a full day. (4 points) The number of customers watching action movies (N A, in millions) and the number watching romantic movies (N R, in millions) is restricted by 2N A + N R = 3. Costs for offering these movies is given by C = ln(2 + 5N A N R ) Determine the values of N A and N R at which costs are maximal. Include a proof that the value is indeed a maximum. You may treat the numbers N A and N R as continuous variables. (10 points) 4

5 Question 3 (22 points) Question 3 must be answered on the empty exam sheets. Please start at the top of a page. You must specify all steps you take and use good notation principles. Running a dating site can be a profitable business. The company Thunder runs such a site, exclusively for heterosexual dates. The monthly fee for registration fee (in $/month) for men is M, for women it is W. Profit is as follows: π = 4M 2 8W 2 2MW + 36M + 40W Determine the values of M and W such that π is a maximum, and give the value of π. Include a proof that it is a maximum indeed. (8 points) (b) In a modified set-up, a constraint is introduced which is further analyzed with the Lagrangian L(M, W, λ) = 4M 2 8W 2 2MW + 36M + 40W λ(3mw M 5) What is the equation that apparently has been introduced as a constraint? And what is the objective function? (4 points) Use the profit function of. Suppose the fee has been set at (M, W) = (M, W ), with profit π = π. (This is not necessarily the optimum value). There is a shortage of registered women, so the management considers lowering W from W to a new value W. Give a firstorder approximation of the function π = f(m, W ). Hint: the expression looks like a function of only M, W, π and W. You may use the type of formula for a first-order approximation of f(x) around a, namely f(x) f + df (x a), but now adapted for dx a function of two variables. (6 points) In week i, the number of registered persons is denoted as N i. At start (week 0), there are 10,000 registered persons, equally divided over both sexes. After a successful date both candidates will unregister. The number of successful dates in a week is a constant (α) fraction of the number of registered persons in that week. No new members register after week 0. Give an expression for the number of registered persons in week n > 0. (4 points) 5

STAT:2020 Probability and Statistics for Engineers Exam 2 Mock-up. 100 possible points

STAT:2020 Probability and Statistics for Engineers Exam 2 Mock-up 100 possible points Student Name Section [letter/#] Section [day/time] Instructions: 1) Make sure you have the correct number of pages.