BEE1004 / BEE1005 UNIVERSITY OF EXETER FACULTY OF UNDERGRADUATE STUDIES SCHOOL OF BUSINESS AND ECONOMICS. January 2002

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1 BEE / BEE5 UNIVERSITY OF EXETER FACULTY OF UNDERGRADUATE STUDIES SCHOOL OF BUSINESS AND ECONOMICS January Basic Mathematics for Economists Introduction to Mathematical Economics Duration : TWO HOURS Please ll in your candidate number: Submit this eamination paper together with your answer book. Only one answer book is required. You can obtain up to up to marks on this eamination and further 6 marks on the eamination in June. This eamination consists of three parts, A, B and C. You can gain up to marks on Part A, up to marks on Part B and up to 5 marks in Part C. Please follow the instructions for each individual part carefully. This is an \open book eamination". You may consult any written material you have brought into the eamination room. Graphic calculators are permitted, but full work must be shown.

2 Part A In this part we test your understanding of some basic ideas of algebra and calculus. You can gain up to marks, but no more marks andnolessthanzero marks for this part. This part is multiple choice. No calculations on your part are necessary. Only one answer for each question is correct. If you mark the correct answer with \ p ", you gain points as indicated. If you mark a false answer, you lose marks as indicated. Questions with no marks \ p " or with several answers marked or questions which are crossed out are not counted. Question : (+ marks or - mark) Which of the following statements is correct for a function y () with the above graph? i) The function is strictly concave... ii) The function is di erentiable... iii) The function is continuous... p Question : (+ marks or - mark) You have to decide which of the following statements is false for a function y () with the above graph. i) false: The function is not continuous... ii) false: The function is non-decreasing... iii) false: The function is increasing... (BEE{BEE5 January Please turn over.)

3 Question : (+ marks or - mark) Consider a function y () with the curved graph given in the above gure. Which of the following statements is true? i) The rst derivative y () at = isy ( ) =... p ii) The rst derivative y () at = isy ( ) = 9... iii) The rst derivative y () at = isy ( ) =... 9 iv) The rst derivative y () at = isy ( )=5... Question : (+ marks or - mark) Which of the following statements is incorrect for a function y () withtheabove graph? i) The rst derivative y () of the function is positive for >... ii) The function is concave for >... iii) The function has two in ection points... iv) The second derivative y () of the function is negative for <<... v) = is an absolute minimum of the function for (BEE{BEE5 January )

4 Question 5 : (+ marks or - mark) You are given the following information about the signs of the rst and second derivative of a function y (): 7 y () + + y () + + Which of the following three graphs is consistent with this information? Graph Graph Graph i) Graph... ii) Graph... p iii) Graph... Question 6 : (+ marks or - mark) contains errors: y () = 5( ) ( ) ( ) The following sign diagram for the function < = <c< = << = < Row : 5( ) + Row : ( ) Row : Row : y () + + Find the errors in rows -. How many entries are incorrect? i) less than 6... p ii) more then 6... iii) eactly 6... Correct the errors in rows - of the table. How many of the four \+" and \ " signs in row are correct, in spite of the errors in the previous lines? i)... ii)... iii)... p iv)... v)... (BEE{BEE5 January Please turn over.)

5 Question 7 : (+ marks or - marks) All but one of the following calculations contain errors: A)(Astudentisaskedtoepandanalgebraicepression.) + 8 = = B) (A student is asked to rewrite an epression as a power with rational inde.) p ( p ) = ³ = 5 = 5 = = 9 C) (A student is asked to di erentiate using the product rule.) ³ ³ y () = 5 + µ y () = 5 ³ ³ µ D) (A student is asked to di erentiate using the general power rule.) y () = 5 = 5 5 = 5 Which calculation is correct? i) Calculation A... ii) Calculation B... iii) Calculation C... p iv) Calculation D... Question 8 : (+ marks or - mark) Which of the following arguments is correct? i) A total cost function is always an invertible function. This is so because total costs are always increasing and therefore any horizontal line intersects its graph at most once. Consequently, when we interchange the horizontal and the vertical ais, any vertical line intersects the inverted graph at most once... p ii) A total cost function is not necessarily increasing. This is so because they are often assumed to be U-shaped. A horizontal line can intersect a U-shaped function twice. Therefore, when the horizontal and the vertical ais are interchanged, a vertical line may intersect it twice. Consequently, a total cost function does not have to be invertible, because the inverted graph may fail the vertical line test... (BEE{BEE5 January )

6 Part B : In this part we test your ability to correctly perform basic algebraic operations, to di erentiate and to nd the optima of simple functions. You can gain up to, but no more than, marks on this part. Pleasewriteyouranswerinyouranswerbook.Pleasetakecare { to clearly mark your answer with the number of the question or subquestion, { to include all relevant intermediate calculations { and to underline or frame clearly the nal result of your calculations. Question 9 : Solve the following equations for : a) ( mark) b) ( mark) (6 +)+5 = (5 ) += Question : a) ( marks) b) ( mark) c) ( marks) Simplify 6p p p + + ³ + µ 5 5 Question : a) ( marks) b) ( mark) Question : a) ( mark) b) ( marks) Find all solutions to the following quadratic equations: 6 + = 5 +5= Find the rst and the second derivative of y () = y () = p + p (BEE{BEE5 January Please turn over.) 5

7 Question : Di erentiate the function y () = in two di erent ways: a) Epand rst and then di erentiate. ( mark) b) Di erentiate rst using the product rule and then epand. ( marks) Hint: Check that both results coincide. Question : Use the quotient rule to di erentiate ( marks) y () = Question 5 : Use the general power rule to di erentiate ( marks) y () =5 + p 5 Question 6 : Use the quotient rule in conjunction with the general power rule to di erentiate. Then simplify. ( marks) y () = ( + ) Question 7 : a) Use the product rule in conjunction with the general power rule to di erentiate the function ( marks) y () =( +) ( ) b) Factorize the rst derivative and use a sign diagram to determine which critical points are relative maima (peaks) or relative minima (troughs). ( marks) Question 8 : ( mark) a) Calculate the rst, the second and the third derivative of the function y () = 5 b) Find all critical points of the function and determine whether they are relative maima (peaks) or relative minima (troughs). ( marks) c) Find the absolute maimum and the absolute minimum of the function in the interval 6. ( marks) d) Use the third derivative to determine whether the roots of the second derivative are in ection points. ( mark) (BEE{BEE5 January ) 6

8 Question 9 : a) Calculate the @y of the function ( marks) z (; y) = y y b) Find the critical points of the function. ( marks) c) Calculate the second partial z z. d) Di with respect to. ( e) Di with respect to y. ( Part C : This part tests your ability to apply optimization techniques to economic problems. You can gain up to, but no more than, 5 marks on this part. Please writeyouranswerinyouranswerbook.pleasetakecare { to clearly mark your answer with number of the question or subquestion, { to include all relevant intermediate calculations { and to underline or frame your nal answer to a subquestion. The statement or the relevant formulae should make sense on their own. Question : A British farmer estimates that if 6 tomato plants are planted, the average yield per tree will be tomatoes. The average yield will decrease by. tomatoes per plant for each additional tree planted on the same acreage. How many tomato plants should the farmer plant to maimize the total yield? (6 marks) Question : A manufacturer estimates that when Q units of a particular commodity are produced each month, the total costs will be in Pounds Sterling and all units can be sold at a price of TC(Q) =8+6Q +8Q P (Q) = 8 Q Pounds Sterling per unit. i) Determine the level of production that results in the maimum pro t. What is the maimum pro t? ( marks) ii) At what level of production is the total average cost per unit TAC(Q) = TC(Q) Q minimized? What are the marginal costs at this quantity? What are the minimal total average costs? ( marks) (BEE{BEE5 January Please turn over.) 7

9 Question : An isolated farmhouse (point C =(; ) in the graph) is to be connected to the main road by a straight path. The main road itself runs from points A =(; ) to B =(; ) in the graph along a straight line. B 8 y 6 C A a) Describe the line representing the road in slope-intercept form y = y (). ( mark) b) What is the distance D () of the farmhouse to any given point E =(; y ()) on the road? (Hint: Use the law of Pythagoras.) ( marks) c) What would be the length of the shortest path to the main road? At which point E =( ;y( )) would it connect to the main road? (Hint: It su±ces to minimize the square of the distance D ().) ( marks) Question : cost function A producer operating in a perfectly competitive market has the total TC(Q) =Q 8Q +9Q + where costs are given in Pounds Sterling. a) Calculate the marginal cost function MC (Q) and the average variable cost function AV C (Q). ( mark) b) At what quantity are average variable costs minimized? What are the minimum average variable costs? ( marks) c) What quantity maimizes pro ts when the market price is P =? ( marks) d) What quantity maimizes pro ts when the market price is P =8? ( mark) (BEE5 January End of paper.) 8