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Questions and Answers of Mathematics Economics Business

If $4000 is saved in an account offering a return of 4% compounded continuously, the future value, S, after t years is given by S = 4000e0.04t(1) Calculate the value of S when(a) t = 5 (b) t =
Write down the derivative of(a) y = e6x (b) y = e−342x (c) y = 2e−x + 4ex (d) y = 10e4x – 2x2 + 7
An electronic components firm launches a new product on 1 January. During the following year a rough estimate of the number of orders, S, received t days after the launch is given byS = t3 −
The average cost per person of hiring a tour guide on a week’s river cruise for a maximum party size of 30 people is given byAC = 3Q2 − 192Q + 3500 (0 < Q ≤ 30)Find the minimum average cost
A firm’s short-run production function is given byQ = 3√Lwhere L is the number of units of labour.If the price per unit sold is $50 and the price per unit of labour is $10, find the value of L
A manufacturer has fixed costs of $200 each week, and the variable costs per unit can be expressed by the function, VC = 2Q − 36.(a) Find an expression for the total cost function and deduce that
The supply and demand equations of a good are given by3P − QS = 3and2P + QD = 14respectively.The government decides to impose a tax, t, per unit. Find the value of t which maximises the
The demand and total cost functions of a good are4P + Q − 16 = 0andTC = 4 + 2Q - 3Q/10 + Q3/20(a) Find expressions for TR, π, MR and MC in terms of Q.(b) Solve the equationdπ/dQ = 0and hence
If the fixed costs are 13 and the variable costs are Q + 2 per unit, show that the average cost function isAC = 13/Q + Q + 2(a) Calculate the values of AC when Q = 1, 2, 3, . . . , 6. Plot these
A firm’s short-run production function is given byQ = 30L2 − 0.5L3Find the value of L which maximises APL and verify that MPL = APL at this point.
If the demand equation of a good isP = 40 − 2Qfind the level of output that maximises total revenue.
Find and classify the stationary points of the following functions. Hence give a roughsketch of their graphs.(a) y = −x2 + x + 1 (b) y = x2 − 4x + 4 (c) y = x2 − 20x + 105(d) y =
If the supply equation isQ = 7 + 0.1P + 0.004P2find the price elasticity of supply if the current price is 80.(a) Is supply elastic, inelastic or unit elastic at this price?(b) Estimate the
Consider the supply equationQ = 4 + 0.1P2(a) Write down an expression for dQ/dP.(b) Show that the supply equation can be rearranged asP = √(10Q - 40)Differentiate this to find an expression for
(a) Find the elasticity of demand in terms of Q for the demand function, P = 20 − 0.05Q.(b) For what value of Q is demand unit elastic?(c) Find an expression for MR and verify that MR = 0 when
The demand function of a good is given byQ = 1000/P2(a) Calculate the price elasticity of demand at P = 5 and hence estimate the percentage change in demand when P increases by 2%.(b) Comment on the
(a) If an airline increases prices for business class flights by 8%, demand falls by about 2.5%. Estimate the elasticity of demand. Is demand elastic, inelastic or unit elastic?(b) Explain whether
Find the price elasticity of demand at P = 6 for each of the following demand functions:(a) P = 30 − 2Q(b) P = 30 − 12Q(c) P 5 (100 2 2Q)
Find the price elasticity of demand at the point Q = 9 for the demand functionP = 500 − 4Q2and compare your answer with that of Question 1.
Given the demand functionP = 500 − 4Q2calculate the price elasticity of demand averaged along an arc joining Q = 8 and Q = 10.
If the consumption function isC = 300 + 2Y2/1 + Ycalculate MPC and MPS when Y = 36 and give an interpretation of these results.
Find expressions for marginal revenue in the case when the demand function is given by(a) P = (100 – Q)3 (b) P = 1000/Q + 4
Differentiatey = x5(x + 2)2(a) by using the product rule;(b) by first multiplying out the brackets and then differentiating term by term.
Differentiate y = (5x + 7)2(a) by using the chain rule;(b) by first multiplying out the brackets and then differentiating term by term.
Use the quotient rule to differentiate(a) y = x/x - 5(b) y = x/(x + 7)(c) y = x + 3/x - 2(d) y = 2x + 9/3x + 1(e) y = x/(5x + 6) (f) y = x + 4/3x - 7
Use the product rule to differentiate(a) y = x(3x + 4)2 (b) y = x2(x – 2)3(c) y = x (x + 2)(d) y = (x – 1)(x + 6)3 (e) y = (2x + 1)(x + 5)3 (f) y = x3(2x – 5)4
Use the chain rule to differentiate(a) y = (5x + 1)3 (b) y = (2x – 7)8 (c) y = (x + 9)5(d) y = (4x2 – 7)3 (e) y = (x2 + 4x – 3)4 (f) y = (2x + 1)(g) y = 1/3x + 1(h) y =
If the demand function is P = 3000 - 2√Q find expressions for TR and MR. Calculate the marginal revenue when Q = 9 and give an interpretation of this result.
The price of a company’s shares, P, recorded in dollars at midday is a function of time, t, measured in days since the beginning of the year. Give an interpretation of the statement:dP/dt =
If the consumption function is C = 0.02Y2 + 0.1Y + 25 find the value of Y when MPS = 0.38.
If the consumption function is C = 50 + 2√Y calculate MPC and MPS when Y = 36 and give an interpretation of these results.
A firm’s production function is Q = 50L − 0.01L2 where L denotes the size of the workforce. Find the value of MPL in the case when(a) L = 1(b) L = 10 (c) L = 100 (d) L = 1000Does
If the average cost function of a good isAC = 15/Q + 2Q + 9find an expression for TC. What are the fixed costs in this case? Write down an expression for the marginal cost function.
A monopolist’s demand function is given byP + Q = 100Write down expressions for TR and MR in terms of Q and sketch their graphs. Find the value of Q which gives a marginal revenue of zero and
If the demand function isP = 80 − 3Qshow thatMR = 2P − 80
If the demand function isP = 100 − 4Qfind expressions for TR and MR in terms of Q. Hence estimate the change in TR brought about by a 0.3-unit increase in output from a current level of 12 units.
Find expressions for(a) dQ/dP for the supply function Q = P2 + P + 1(b) d(TR)/dQ for the total revenue function TR = 50Q – 3Q2(c) d(AC)/dQ for the average cost function AC = 30/Q + 10(d) dC/dY for
By writing √4x = √4 x √x = 2√x, differentiate √4x. Use a similar approach to differentiate(a) √25x (b) 3√27x (c) 4√16x3 (d) 25/x
If f (x) = x2 – 6x + 8, evaluate f ′(3). What information does this provide about the graph of y = f (x) at x = 3?
Evaluate f ″(2) for the functionf (x) = x3 − 4x2 + 10x − 7
Find expressions for d2y/dx2 in the case when(a) y = 7x2 – x(b) y = 1/x2(c) y = ax + b
By writing x2 (x2 + 2x - 5/x2) = x4 + 2x3 - 5, differentiate x2 (x2 + 2x -5/2).Use a similar approach to differentiate(a) x2(3x – 4)(b) x(3x3 – 2x2 + 6x – 7)(c) (x + 1)(x – 6)(d) x2 -
Evaluate f ′(x) for each of the following functions at the given point:(a) f (x) = 3x9 at x = 1(b) f (x) = x2 – 2x at x = 3(c) f (x) = x3 – 4x2 + 2x – 8 at x = 0(d) f(x) = 5x4 - 4/x4 at x = -
Differentiate(a) y = 5x2 (b) y = 3/x(c) y = 2x + 3(d) y = x2 + x + 1 (e) y = x2 – 3x + 2 (f) y = 3x(g) y = 2x3 – 6x2 + 49x – 54(h) y = ax + b (i) y = ax2 + bx + c( j) y = 4x
Complete the following table of function values for the function, f(x) = x2 − 2x:Sketch the graph of this function and, by measuring the slope of the tangents, estimate(a) f ′(−0.5) (b) f
Differentiate the following functions, giving your answer in a similar form, without negative or fractional indices: (d) y = xVx (b) fx) = Jx (c) f(x) = (a) f(x) fx) =
Differentiate(a) y = x8 (b) y = x50 (c) y = x19 (d) y = x999
Differentiate the functionf (x) = x7Hence calculate the slope of the graph ofy = x7at the point x = 2.
Sketch the graph of the functionf (x) = 5Explain why it follows from this thatf ′(x) = 0
Verify that the points (0, 2) and (3, 0) lie on the line2x + 3y = 6Hence find the slope of this line. Is the line uphill, downhill or horizontal?
Find the slope of the straight line passing through(a) (2, 5) and (4, 9) (b) (3, –1) and (7, –5) (c) (7, 19) and (4, 19)
A proposed investment costs $130 000 today. The expected revenue flow is $40 000 at the end of year 1, and $140 000 at the end of year 2. Find the internal rate of return, correct to one decimal
An investor is given the opportunity to invest in one of two projects:Project A costs $10 000 now and pays back $15 000 at the end of four years.Project B costs $15 000 now and pays back $25 000 at
Determine the present value of an annuity that pays out $100 at the end of each year(a) For five years (b) In perpetuity if the interest rate is 10% compounded annually.
You are given the opportunity of investing in one of three projects. Projects A, B and C require initial outlays of $20 000, $30 000 and $100 000 and are guaranteed to return $25 000, $37 000 and
A company has the option of investing in a project and calculates the net present values shown in Table 3.31 at four different discount rates.(a) Estimate the internal rate of return of the
A financial company invests £250 000 now and receives £300 000 in three years’ time.Calculate the internal rate of return.
A builder is offered one of two methods of payment:Option 1: A single sum of $73 000 to be paid now.Option 2: Five equal payments of $15 000 to be paid quarterly with the first instalment to be paid
The revenue of a firm (in $100 000s) at the end of each year for the next five years is listed in Table 3.30. Calculate the present value of the revenue stream if the annual discount rate is 8%.
An investment company is considering one of two possible business ventures. Project 1 gives a return of $250 000 in four years’ time, whereas Project 2 gives a return of $350 000 in eight years’
A small business promises a profit of $8000 on an initial investment of $20 000 after five years.(a) Calculate the internal rate of return.(b) Would you advise someone to invest in this business if
Determine the present value of $7000 in two years’ time if the discount rate is 8% compounded (a) quarterly (b) continuously
A person wishes to save a regular amount at the beginning of each month in order to buy a car in 18 months’ time. An account offers a return of 4.8% compounded monthly.Work out the monthly savings
A person borrows $100 000 at the beginning of a year and agrees to repay the loan in ten equal instalments at the end of each year. Interest is charged at a rate of 6% compounded annually.(a) Find
A person invests $5000 at the beginning of a year in a savings account that offers a return of 4.5% compounded annually. At the beginning of each subsequent year an additional $1000 is invested in
The current extraction of a certain mineral is 12 million tonnes a year, and this is expected to fall at a constant rate of 6% each year. Estimate the current minimum level of world reserves if the
A prize fund is set up with a single investment of $5000 to provide an annual prize of $500. The fund is invested to earn interest at a rate of 7% compounded annually. If the first prize is awarded
Determine the monthly repayments needed to repay a $125 000 loan which is paid back over 20 years when the interest rate is 7% compounded annually. Round your answer to two decimal places.
An individual saves $5000 in a bank account at the beginning of each year for 10 years. No further savings or withdrawals are made from the account. Determine the total amount saved if the annual
Find the value of the geometric series1000 + 1000(1.03) + 1000(1.03)2 + . . . + 1000(1.03)9
Table 3.23 shows the depreciation in the value of two models of car.(a) Assuming that the depreciation of Car A is linear, estimate its value in Year 3.(b) Assuming that the depreciation of Car B is
The number of rail passenger journeys made between England and Scotland in 2004 was 5.015 million. In 2011 the figure was 7.419 million. Work out the yearly percentage rate of growth (assumed
The future value S of principal P invested for n years with an interest rate r% compounded annually may be calculated using the formulaRearrange this formula to express P in terms of S, r and n. S =
Find the APR of a loan if the monthly interest rate is 1.65%. Give your answer correct to two decimal places.
Find the value, in two years time, of $4000 invested at 5% compounded annually. In the following two years, the interest rate is expected to rise to 8%. Find the final value of the investment at the
Current annual consumption of energy is 78 billion units, and this is expected to rise at a fixed rate of 5.8% each year. The capacity of the industry to supply energy is currently 104 billion
Determine the APR if the nominal rate is 7% compounded continuously.
A department store has its own credit card facilities, for which it charges interest at a rate of 2% each month. Explain briefly why this is not the same as an annual rate of 24%. What is the annual
If a piece of machinery depreciates continuously at an annual rate of 4%, how many years will it take for the value of the machinery to halve?
How long will it take for a sum of money to triple in value if invested at an annual rate of 3% compounded continuously?
Find the future value of $100 compounded continuously at an annual rate of 6% for 12 years.
Which of the following savings accounts offers the greater return?Account A: an annual rate of 8.05% paid semi-annuallyAccount B: an annual rate of 7.95% paid monthly
A principal, $7000, is invested at 9% interest for eight years. Determine its future value if the interest is compounded (a) annually (b) semi-annually (c) monthly (d)
A piece of machinery depreciates in value by 5% a year. Determine its value in three years’ time if its current value is $50 000.
How long will it take for a sum of money to double if it is invested at 5% interest compounded annually?
The value of an asset, currently priced at $100 000, is expected to increase by 20% a year.(a) Find its value in 10 years’ time.(b) After how many years will it be worth $1 million?
Find the future value of $20 000 in two years’ time if compounded quarterly at 8% interest.
A bank offers a return of 7% interest compounded annually. Find the future value of a principal of $4500 after six years. What is the overall percentage rise over this period?
Table 3.16 shows the prices of a good for each year between Year 1 and Year 6.(a) Work out the index numbers, correct to one decimal place, taking Year 2 as the base year.(b) If the index number for
Table 3.15 shows the number of items (in thousands) produced from a factory production line during the course of a year. Taking the second quarter as the base quarter, calculate the associated index
Table 3.14 shows the index numbers associated with transport costs during a 20-year period. The public transport costs reflect changes to bus and train fares, whereas private transport costs include
Table 3.13 shows the monthly index of sales of a good during the first four months of the year.(a) Which month is chosen as the base year?(b) If sales in February are 3840, what are the sales in
The price of a good during the last five years is:$25 $30 $36 $43 $50Calculate the index numbers using the last year as the base year and hence comment on the rise in prices during this period.
Table 3.12 gives the annual rate of inflation during a five-year period.If a nominal house price at the end of Year 1 was $10.8 million, find the real house price adjusted to prices prevailing at the
Find the single percentage increase or decrease equivalent to(a) a 10% increase followed by a 25% increase(b) a 34% decrease followed by a 65% increase(c) a 25% increase followed by a 25%
An antiques dealer tries to sell a vase at 45% above the $18 000 which the dealer paid at auction.(a) What is the new sale price?(b) By what percentage can the dealer now reduce the price before
A TV costs $900 including 20% sales tax. Find the new price if tax is reduced to 15%.
A shop sells books at ‘20% below the recommended retail price (r.r.p.)’. If it sells a book for $12.40, find(a) The r.r.p.(b) The cost of the book after a further reduction of 15% in a sale(c)
A student discount card reduces a bill in a restaurant from $124 to $80.60. Work out the percentage discount.
Find the new quantities when(a) $16.25 is increased by 12%(b) the population of a town, currently at 113 566, rises by 5%(c) a good priced by a firm at $87.90 is subject to a sales tax of 15%(d) a

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