4 Total Question 4. Intro to Financial Maths: Functions & Annuities Page 8 of 17

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1 Intro to Financial Maths: Functions & Annuities Page 8 of 17 4 Total Question 4. /3 marks 4(a). Explain why the polynomial g(x) = x 3 + 2x 2 2 has a zero between x = 1 and x = 1. Apply the Bisection Method four times to find an approximation for this zero. That is, if x 1 = 0 is the first approximation, find x 4. /1 mark 4(b). Consider the function g(x) = x3 +2. Can you use the same argument x+4 as in part (a) to conclude that g(x) = 0 for some value of x between 2 and 2? Give reasons for your answer. Please turn over for page 9.

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4 Intro to Financial Maths: Algebra Page 16 of 17 4 Total Question 6. /4 marks An esky body is made from a box and two handles. To make an esky, a lid is secured to the esky body by two hinges. If an order y and gross production vector x are given by y = eskies bodies boxes handles lids hinges x 1 x 2 and x = x 3 x 4 = x 5 x 6 total eskies total bodies total boxes total handles total lids total hinges Find the appropriate parts-listing matrix Q and a production vector x such that (I Q)x = y. End of examination questions. Please turn over for page 17.

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7 Intro to Financial Maths: Functions & Annuities Page 12 of 17 5 Total Question 6. /3 marks 6(a). For an interest rate of 8% per annum compounded quarterly find the equivalent rate i per annum compounded two monthly and the equivalent rate j compounded annually. /2 marks 6(b). A deposit of $10000 is made into an account in which interest is paid at a rate of 6% per annum compounded continuously. How long does it take for the account to triple? Please turn over for page 13.

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10 Introduction to Financial Mathematics I: Functions and Annuities Paper Page 11 of Show all steps in your working for each part of this question. (a) For an interest rate of 4% per annum compounded monthly find the equivalent rate i per annum if (i) interest is compounded 6 monthly; (ii) interest is compounded continually. marks: (b) How much money must be deposited now to provide a perpetuity with payments of $20,000 each 3 months with an interest rate of 8% per annum compounded 3 monthly starting (i) in 3 months time? 4 (ii) now? marks: 2 Please turn over for pages 12 and 13

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14 (c) Minimise x + 2y with respect to the restrictions in 1(c). (d) Maximise 3x + y with respect to the restrictions in 1(d). 3. Find the maximum and minimum of the linear function f(x, y) = x y/2 in the region of Question 1(c), if possible. If they exist, where do they occur? If they do not exist, why not? 4. A bakery makes two kinds of pies, apple pie and cherry pie. One apple pie requires 0.3 kg of pastry, 0.5 kg of apples and kg of spice. One cherry pie requires 0.3 kg of pastry, 0.1 kg of cherries. The fruit supplier can provide up to 8 kg of cherries and 20 kg of apples. The bakery makes its own pastry and will make a minimum of 12 kg and a maximum of 30 kg. The bakery does not want to waste any pastry. The supply of spice is unlimited. The profit on an apple pie is $2 and the profit on a cherry pie is $1.50. (a) Write an equation to describe the bakery s profit from apple and cherry pies. (b) Write equations to describe all constraints on the bakery s pie making. (c) Graph the feasible region and find all vertices. (d) What is the maximum possible profit? How many of each type of pie should be made to achieve this profit? 5. Suppose that a bounded closed convex region has vertices (0, 2), (1, 0), (8, 4), (0, 0). Find the maximum and minimum value of f(x) = 3x 5y in this region. 6. A pet food manufacturer produces two types of food, regular and premium. A 10 kg bag of regular food requires 3 hours to prepare and 2 hours to cook; a 10 kg bag of premium food requires 5 hours to prepare and 2 hours to cook. The materials used to prepare the food are available 12 hours per day and the oven used to cook the food is available 6 hours per day. The profit on a 10 kg bag of regular food is $25 and on a 10 kg bag of premium food it is $30. (a) Write the inequalities that describe the constraints. (b) Graph the solution to the system of inequalities and find the vertices of the region. (c) Find how many bags of each type of food should be made to maximise the profit. (Fractional bags can be produced, they just save the leftovers for the next day.)

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18 Intro to Financial Maths: Functions & Annuities Page 2 of 17 5 Total Question 1. /3 marks 1(a). A company purchases a machine for $10,000. The machine depreciates linearly so that its value after ten (10) years is $5,000. (i) Give a formula for the value V (t) of the machine as a function of time and draw its graph. (ii) What is the value of the machine after 16 years? /2 marks 1(b). Sketch the graph of a cubic polynomial ax 3 + bx 2 + cx + d which has zeros at x = 2, x = 1 and x = 3 and with a < 0. Please turn over for page 3.

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20 Intro to Financial Maths: Functions & Annuities Page 4 of 17 4 Total Question 2. 2(a). Evaluate, if possible, the following limits. If not possible, state so clearly. /2 marks (i) lim x 4 x 2 4x x 4 (ii) lim x 3 x 2 2x + 1 x 3 /2 marks 2(b). By calculating values of the function close to the limit point, estimate the limit 5 x 1 lim. x 0 x Please turn over for page 5.

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