Homework answers / question archive / ASSIGNMENT 01 This assignment covers the following units/modules/topics Review of basic algebra Exponents and logarithmic functions Economic application of functions and graphs The learning outcomes for this specific assignment is: To have students reinforce their understanding of algebraic concepts to prepare them for calculus

ASSIGNMENT 01 This assignment covers the following units/modules/topics Review of basic algebra Exponents and logarithmic functions Economic application of functions and graphs The learning outcomes for this specific assignment is: To have students reinforce their understanding of algebraic concepts to prepare them for calculus

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ASSIGNMENT 01 This assignment covers the following units/modules/topics Review of basic algebra Exponents and logarithmic functions Economic application of functions and graphs The learning outcomes for this specific assignment is: To have students reinforce their understanding of algebraic concepts to prepare them for calculus. Assignment one also aims to introduce functions and graphs and how they can be applied in Economics. Assignment instructions/requirements here • Answer all the questions in both assignments • Written work should be clear and legible • Credit will only be given for logical and systematic presentation of solutions • Correct answers without calculation steps will not attract full marks • Avoid copying work from others as this may lead to nullification of assignment marks TUTORIAL LETTER SEMESTER 1/2022 MATHEMATICS FOR ECONOMISTS 1A MFE511S 5 ASSIGNMENT 1 QUESTION 1 (Review of Basic Algebra) – (30 MARKS) 1.1 Simplify the following expressions (leave your answers in simplest form). 1.1.1 ( ?+? 2?−2? − ?−? 2?+2? − 2? 2 ?2−?2 ) ( 1 ? − 1 ? ) [8 marks] 1.1.2 [( ? ? ) 2 − ? ?2 ] ÷ ( ?−1 ? ) 2 [7 marks] 1.1.3 12? 4?+√? [5 marks] 1.1.4 log3 81−1 +log6 1 36 − log3 9 − 1 2 + log3 √3 3 3 [5 marks] 1.2 Solve for ?: log2 ? − log8 ? = 4 [5 marks] QUESTION 2 (Economic Applications of linear graphs and functions)- (25 MARKS) 2.1 Desired consumption, investment and government spending in a closed economy are given as follows: ? = 360 − 200? + 0.1? ? = 120 − 400? ? = 120 2.1.1 Find an equation for national saving (S) in terms of Y and the real interest rate (r)? [8 marks] 2.1.2 What value of real interest rate clears the good marked Y=540 and Y=650? [6 marks] 2.2 In a simple macroeconomic model, the value of national income ? may be found by solving the system: ? = 120 + 0.75?? TUTORIAL LETTER SEMESTER 1/2022 MATHEMATICS FOR ECONOMISTS 1A MFE511S 6 ? = 150 ? = 250 ? = 55 where ?? = ? − ?. 2.1.1 Calculate the equilibrium level of national income [6 marks] 2.1.2 Calculate the total increase in government expenditure and investments needed to increase the equilibrium of national income by N$ 250. [5 marks] QUESTION 3 (Economic Applications of non-linear graphs and functions) –(25 MARKS) 3.1 Consider the following production function for bus transportation in a city: Q L F K ? ? ? 1 2 ?3 = Where L = Labour input in worker hours F = Fuel input in gallons K = Capital input in number of busses Q = Output in millions of bus mileage and the estimate of the various parameters using historical data given as: ? = 0.012;?1 = 0.45;?2 = 0.35;?3 = 0.20 3.1.1 Define ? [2 marks] 3.1.2 State with reason what type of returns to scale are present in this production function. [4 marks] 3.1.3 Suppose that number of busses increases by 12%. By what percentage will output increase? [4 marks] 3.1.4 Assume that labour hours = 8 hours, the fuel input = 11500 litres, and number of buses = 60, what will be the total mileage output for these buses? [5 marks] TUTORIAL LETTER SEMESTER 1/2022 MATHEMATICS FOR ECONOMISTS 1A MFE511S 7 3.2 Grass Cutters CC produces and sells riding lawnmowers. The demand function for the lawnmowers is given as ? = − 1 10 ? + 1820 and the cost to produce the lawnmowers is given by the cost function C x x ( ) 80 8000 = + . 3.2.1 What is the domain of the demand function? [3 marks] 3.2.2 Determine the revenue function ?(?) [3 marks] 3.2.3 Express the profit ? as a function of ? [4 marks] TOTAL MARKS FOR ASSIGNMENT 01: 80 CONVERTED TO % END TO ASSIGNMENT 01 TUTORIAL LETTER SEMESTER 1/2022 MATHEMATICS FOR ECONOMISTS 1A MFE511S 8 ASSIGNMENT 02 This assignment covers the following units/modules/topics Differentiation and its application in Economics Basic integration and its application in Economics The learning outcomes for this specific assignment is: The student to have an ability to carry out differentiation and integration competently and be able to answer application problems. Assignment instructions/requirements here • Answer all the questions in both assignments • Written work should be clear and legible • Credit will only be given for logical and systematic presentation of solutions • Correct answers without calculation steps will not attract full marks • Avoid copying work from others as this may lead to nullification of assignment marks TUTORIAL LETTER SEMESTER 1/2022 MATHEMATICS FOR ECONOMISTS 1A MFE511S 9 ASSIGNMEN T2 QUESTION 1 (Derivatives and their applications) – (30 MARKS) 1.1 Differentiate the following functions with respect to the independent variable. Ensure to use proper notation and simplify your final answers. Wrong notation will be penalised 1.1.1 ?(?) = ? 2+3? √? 2+1 [4 marks] 1.1.2 ?(?) = ? √?+1 [4 marks] 1.2 Use the first principle to find the derivative of the following function: 1.2.1 ?(?) = 2 √? [3 marks] 1.2.2 ?(?) = 2? 2 − 2 ? [3 marks] 1.3 Evaluate the first and second-order derivatives of the cubic ? = ? 3 + 2? 2 − 6? + 7 at ? = 2, and hence state whether the function is increasing or decreasing and if the function convex or concave at this point? [5 marks] 1.4 A monopolist has a cost function, ?(?) = 30? and a demand curve given by ?(?) = 1000 − 15?. 1.4.1 Derive expressions for marginal cost (MC) and marginal revenue (MR) . [5 marks] 1.4.2 How much will the monopolist charge and how many units of output will it produce? [3 marks] 1.4.3 What is the socially efficient quantity (the quantity that maximises total surplus)? [3 marks] QUESTION 2 (Partial Derivatives and their applications) – (25 MARKS) 2.1 Find the marginal rate of technical substion for the following production function: [7 marks] ?(?, ?) = [?? −? + (1 − ?)? −? ] − 1 ? 2.2 Laplace Auto-Engineering company manufactures two types of gearbox - automatic (a) and manual (m). The revenue function of the company, in thousands, is given by: ?(?, ?) = 8? + 5? + 2?? − ? 2 − 2?2 + 20. TUTORIAL LETTER SEMESTER 1/2022 MATHEMATICS FOR ECONOMISTS 1A MFE511S 10 2.2.1 How many automatic and manual gearboxes, will Laplace Auto-Engineering company have to sell to maximise their revenue? [5 marks] 2.2.2 Calculate the maximum revenue? [2 marks] 2.3 Determine ? 3? ?????? , if ?(?, ?, ?) = ? ??? [6 marks] 2.4 Find ? 2? ?? 2 by implicit differentiation of 2? 2 − 3? 2 = 4 [5 marks] QUESTION 3 (Integration and its Applications) – (25 MARKS) 3.1 Determine the following integrals: 3.1.1 ∫ ( 2 ? 2 − 1 √? ) ?? [4 marks] 3.1.2 ∫ ?(? 2 + 2) 3 1 0 ?? [5 marks] 3.2 The supply functions of for bailes of vintage clothes from Angola is given (in N$) by ?(?) = ? 2 + 10?, and the demand function (in N$) by ?(?) = 900 − 20? − ? 2 . 3.2.1 Find the (Q, P) point at which supply and demand are in equilibrum [4 marks] 3.2.2 Find the consumer surplus [6 marks] 3.2.3 Find the producer surplus [6 marks] TOTAL MARKS FOR ASSIGNMENT 02: 80 CONVERTED TO % END OF TUTORIAL