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Questions and Answers of Mathematics Economics Business

Sketch a graph of y = (1/2)4x, –2 ≤ x ≤ 2.
The data in Table 2 give the average weights of pike for certain lengths. Use a cubic regression polynomial to model the data. Estimate (to the nearest ounce) the weights of pike of lengths 39, 40,
Change each logarithmic form to an equivalent exponential form:(A) log3 9 = 2 (B) log4 2 = 1/2(C) log3(1/9) = –2
A function f is bounded if the entire graph of f lies between two horizontal lines. The only polynomials that are bounded are the constant functions, but there are many rational functions that are
Sketch the graph of each equation.(A) y = x2 – 4(B) y2 100 2 x + 1
To graph the equation y = –x3 + 3x, we use point-by-point plotting to obtain the graph in Figure 4.(A) Do you think this is the correct graph of the equation? Why or why not?(B) Add points on the
Sketch a graph of y = (1/2)4–x, –2 ≤ x ≤ 2.
Evaluate each basic elementary function at(A) x = 64 (B) x = –12.75Round any approximate values to four decimal places.
(A) Graph y = Ax2 for A = 1, 4, and 1/4 simultaneously in the same coordinate system.(B) Graph y = Ax2 for A = –1, –4, and -1/4 simultaneously in the same coordinate system.(C) Describe the
(A) How are the graphs of y = √x + 5 and y = √x – 4 related to the graph of y = √x? Confirm your answer by graphing all three functions simultaneously in the same coordinate system.(B) How
Change each exponential form to an equivalent logarithmic form:(A) 64 = 43 (B) 6 = √36 (C) 1/8 = 2–3
Determine which of the following equations specify functions with independent variable x.(A) 4y – 3x = 8, x a real number (B) y2 – x2 = 9, x a real number
Given the rational function(A) Find the domain.(B) Find the x and y intercepts.(C) Find the equations of all vertical asymptotes.(D) If there is a horizontal asymptote, find its equation.(E) Using
Given the quadratic function f(x) = –0.25x2 – 2x + 2(A) Find the vertex form for f.(B) Find the vertex and the maximum or minimum. State the range of f.(C) Describe how the graph of function f
Cholera, an intestinal disease, is caused by a cholera bacterium that multiplies exponentially. The number of bacteria grows continuously at relative growth rate 1.386, that is,where N is the number
Discuss how you could find y = log5 38.25 using either natural or common logarithms on a calculator.
Given the rational function(A) Find the domain.(B) Find the x and y intercepts.(C) Find the equations of all vertical asymptotes.(D) If there is a horizontal asymptote, find its equation.(E) Using
Change each exponential form to an equivalent logarithmic form:(A) 49 = 72 (B) 3 = √9 (C) 1/3 = 3–1
Given the quadratic function f(x) = 0.5x2 – 6x + 21(A) Find the vertex form for f.(B) Find the vertex and the maximum or minimum. State the range of f.(C) Describe how the graph of function f can
Suppose that $1,000 is deposited in a savings account at an annual rate of 5%. Guess the amount in the account at the end of 1 year if interest is compounded (1) Quarterly, (2) Monthly, (3)
Determine which of the following equations specify functions with independent variable x.(A) y2 – x4 = 9, x a real number (B) 3y – 2x = 3, x a real number
Refer to the exponential growth model for cholera in Example 2. If we start with 55 bacteria, how many bacteria (to the nearest unit) will be present(A) In 0.85 hour? (B) In 7.25 hours?Data from
Consider the set of students enrolled in a college and the set of faculty members at that college. Suppose we define a correspondence between the two sets by saying that a student corresponds to a
(A) How are the graphs of y = |x| + 4 and y = |x| – 5 related to the graph of y = |x|? Confirm your answer by graphing all three functions simultaneously in the same coordinate system.(B) How are
The graphs in Figure 5 are either horizontal or vertical shifts of the graph of f(x) = 3√x. Write appropriate equations for functions H, G, M, and N in terms of f. H G (A) Figure 5 Vertical and
Explain why applying any of the graph transformations in the summary box to a linear function produces another linear function.
Find the vertical and horizontal asymptotes of the rational function f(x) = 3x² + 3x - 6 2x² - 2
The financial department in Example 3, using statistical and analytical techniques, arrived at the cost functionwhere C(x) is the cost (in millions of dollars) for manufacturing and selling x million
Find the domain of the function specified by the equation y = √4 – x, assuming that x is the independent variable.
Find y, b, or x, as indicated.(A) Find y: y = log4 16 (B) Find x: log2 x = –3(C) Find b: logb 100 = 2
Cosmic-ray bombardment of the atmosphere produces neutrons, which in turn react with nitrogen to produce radioactive carbon-14 (14C). Radioactive 14C enters all living tissues through carbon dioxide,
This is a continuation of Example 7 in Section 2.1. Recall that the financial department in the company that produces a digital camera arrived at the following price–demand function and the
Find the vertical and horizontal asymptotes of the rational function f(x) x³ - 4x 2 r+5r Х
Find y, b, or x, as indicated.(A) Find y: y = log9 27 (B) Find x: log3 x = –1(C) Find b: logb 1,000 = 3
Refer to the exponential decay model in Example 3. How many milligrams of 14C would have to be present at the beginning in order to have 25 milligrams present after 18,000 years? Compute the answer
The graphs in Figure 4 are either horizontal or vertical shifts of the graph of f(x) = x2. Write appropriate equations for functions H, G, M, and N in terms of f. -5 24 HfG 5 X (A) Figure 4 Vertical
Find the domain of the function specified by the equation y = √x – 2, assuming x is the independent variable.
(A) How are the graphs of y = 2x and y = 0.5x related to the graph of y = x? Confirm your answer by graphing all three functions simultaneously in the same coordinate system.(B) How is the graph of y
Use logarithmic properties to write in simpler form:(A) logb wx/yz(B) logb (wx)3/5 (C) ex loge b(D) loge x/loge b
Consider the supply and demand equations:QSt = 2Pt + aPt-1 −20QDt = 100 − 8Ptfor some parameter, a.Assuming that the market is in equilibrium, show that Pt = 12 − 0.1aPt-1.If P0 = 10, find an
National income Y varies over time t according to the following model:dY/dt = 0.6(C + I - Y)C = 0.8Y + 600I = 800where C is consumption and I is planned investment. Initially, Y = 2000.(a) Find an
A two sector lagged-income model is defined byYt = Ct + ItCt = 0.8Yt-1 + 200It = 1000where Yt, Ct and It denote national income, consumption and planned investment in time period t. The initial
Government expenditure, G, satisfies the differential equationdG/dt = -0.05Gwhere G is measured in billions of dollars and time t is in years. The initial condition is G(0) = 500.(a) Solve this
(a) Solve the differential equation dy/dx 4/√x + 6x2 given the condition, y(4) = 3.(b) The population, N, of a small country is currently 5 million. It is expected that this population will grow
Find the equilibrium price and quantity given the following static supply and demandequations:QS = 6P − 10QD = −4P + 45The equations for the corresponding dynamic model are:Find an expression for
Oil reserves decrease at a constant proportional rate, k > 0, so thatdN/dt = -kNwhere N denotes the number of barrels of oil remaining in t years’ time.(a) Write down an expression for N(t) when
Consider the market model QS = 3P - 1QD = -2P + 9dP/dt = 0.5(QD - QS)Find expressions for P(t), QS(t) and QD(t) when P(0) = 1. Is this system stable or unstable?
Consider the two-sector modeldY/dt = 0.3(C + I - Y )C = 0.8Y + 300I = 0.7Y + 600Find an expression for Y(t) when Y(0) = 200. Is this system stable or unstable?
Consider the two-sector modeldY/dt = 0.5 (C + I - Y)C = 0.7Y + 500I = 0.2Y + 500Find an expression for Y(t) when Y(0) = 15 000. Is the system stable or unstable?
A principal of $60 is invested. The value, I(t), of the investment, t days later, satisfies the differential equationdI/dt = 0.002I + 5Find the value of the investment after 27 days, correct to 2
Solve the differential equation dy/dt = -3y + 180 in the case when the initial condition is(a) y(0) = 40 (b) y(0) = 80 (c) y(0) = 60Comment on the qualitative behaviour of the solution in
Use integration to solve each of the following differential equations subject to the given initial conditions: dy 21; y(0) = 7 (b) dt dy dt dy = t + 3t – 5; y(0) = 1 |(a) (c) = e-³; y(0) = 0 dt
The Harrod–Domar model of the growth of an economy is based on three assumptions. (1) Savings, St, in any time period are proportional to income, Yt, in that period, so thatSt = aYt (a > 0)(2)
Consider the two-sector model:Yt = Ct + ItCt = 0.75Yt-1 + 400It = 200Find the value of C2, given that, Y0 = 400.
Consider the supply and demand equationsQSt = 0.4Pt−1 − 12QDt = −0.8Pt + 60Assuming that equilibrium conditions prevail, find an expression for Pt when P0 = 70. Is the system stable or unstable?
Consider the two-sector model:Yt = Ct + ItCt = 0.7Yt-1 + 400It = 0.1Yt-1 + 100Given that Y0 = 3000, find an expression for Yt. Is this system stable or unstable?
Solve the following difference equations with the specified initial conditions:Comment on the qualitative behaviour of the solution as t increases. (b) Y, = -4Y,-1 + 5; Y, = 2 |(a) Y, = -Y,-1 + 6; Y,
Calculate the first four terms of the sequences defined by the following difference equations. Hence write down a formula for Yt in terms of t. Comment on the qualitative behaviour of the solution in
(a) Find the inverse of the matrixHence find the exact solution of the simultaneous equations4x + 3y = 9005x + 7y = 1750(b) A food supplier makes two different varieties of smoothie, ‘Exotic’ and
A fruit farmer supplies raspberries and blueberries to a leading supermarket. She is contracted to supply at least 100 kg of raspberries and 50 kg blueberries each week. The supermarket requires that
I decide to extend my investment portfolio by spending no more than $5000 on shares in companies A and B. The cost of each share is $5 and $10, respectively, and my financial advisor estimates that I
A specialist bakery makes two types of cakes, chocolate and fruit. Each chocolate cake uses 600 grams of flour, 100 grams of butter and 200 grams of sugar. The corresponding figures for a fruit cake
A designer clothing firm makes ladies’ scarves and blouses using a blend of wool and silk fibres. The quantity of fibre (in grams) required for a scarf and a blouse, and the profit (in $s) made
An American university has enough places for 9000 students. Government restrictions mean that at least 75% of the places must be given to US students but the remainder may be given to non-US
A food producer uses two processing plants, P1 and P2, that operate seven days a week. After processing, beef is graded into high-, medium- and low-quality foodstuffs. Highquality beef is sold to
A publisher decides to use one section of its plant to produce two textbooks called Microeconomics and Macroeconomics. The profit made on each copy is $12 for Microeconomics and $18 for
A small manufacturer produces two kinds of good, A and B, for which demand exceeds capacity. The production costs for each item of A and B are $6 and $3, respectively, and the corresponding selling
An Italian restaurant offers a choice of pasta or pizza meals. It costs $3 to make a pasta dish, which it sells for $13. The corresponding figures for pizzas are $2 and $10, respectively. The maximum
In a student’s diet a meal consists of beefburgers and chips. Beefburgers have 1 unit of nutrient N1, 4 units of N2 and 125 calories per ounce. The figures for chips are 1/2 unit of N1, 1 unit of
A small firm manufactures and sells litre cartons of non-alcoholic cocktails, ‘The Caribbean’ and ‘Mr Fruity’, which sell for $1 and $1.25, respectively. Each is made by mixing fresh orange,
A manufacturer produces two models of racing bike, B and C, each of which must be processed through two machine shops. Machine shop 1 is available for 120 hours per month and machine shop 2 for 180
What can you say about the solution of the linear programming problem specified in Question 6, if the(a) objective function is to be maximised instead of minimised?(b) second constraint is changed to
Find, if possible, the minimum value of the objective function 3x − 4y subject to the constraints,−2x + y ≤ 12, x − y ≤ 2, x ≥ 0 and y ≥ 0
What can you say about the solution to Question 4(c) if the problem is one of maximisation rather than minimisation? Explain your answer by superimposing the family of linesx + y = con the feasible
Use your answers to Question 3 to solve the following linear programming problems:(a) Maximise 4x + 9ysubject to5x + 3y ≤ 307x + 2y ≤ 28x ≥ 0y ≥ 0(b) Maximise 3x + 6ysubject to2x + 5y ≤ 20x
Sketch the feasible regions defined by the following sets of inequalities:(a) 5x + 3y ≤ 307x + 2y ≤ 28x ≥ 0 y ≥ 0(b) 2x + 5y ≤ 20 x − y ≤ 4x ≥ 0y ≥ 0(c) x − 2y ≤ 3x +
How many points with integer coordinates lie in the feasible region defined by 3x + 4y ≤ 12, x ≥ 0 and y ≥ 1?
Which of the following points satisfy the inequality2x − 3y > −5?(1, 1), (−1, 1), (1, −1), (−1, −1), (−2, 1), (2, −1), (−1, 2) and (−2, −1)
The trace of a 2 × 2 matrixis defined to be tr(A) = a + d.(a) Work out the trace of(b) Prove that if A and B are any 2 × 2 matrices, then tr(αA + bB) = a tr(A) + b tr(B) for any numbers a and
Consider the following macroeconomic model:Y = C + I* + G* + X* − M* (equilibrium of national income)C = aYd + b (consumption; 0 < a < 1; b > 0)T = tY (taxation; 0 < t < 1)where I*,
The supply and demand equations of two interdependent commodities arewhere t is the unit tax imposed on good 1.Show that the market equilibrium prices satisfy a matrix equation of the form Ax = b
(a) Find the inverse of the following matrix, in terms of a.For what value of a is this matrix singular?(b) Consider the three-commodity market model defined by:By making use of your answer to part
Airme and Blight are the only two airlines allowed to operate on the same route. The market share of regular business travellers changes from month to month. Airme retains four-fifths of its
(a) The demand and supply functions of two interdependent commodities are given byQD1 = 40 − 3P1 −P2QD2 = 30 − 4P1 − 0.5P2QS1 = −6 + 5P1QS2 = −5 + 2P2Show that the equilibrium prices
(a) Work out the following, where possible:(i) A + C (ii) 2A − CT (iii) AB (iv) BA(b) Solve each of the following equations for X:(i) 2X + AT = C (ii) BX = A 1 3 3 -1 0 6 4 -3 2
The demand and supply functions of a good are given byP + 4QD = 613P − QS = 14Find the equilibrium price and quantity using(a) the inverse matrix method;(b) Cramer’s rule.
If work out, where possible:(a) AT + 2B (b) AB (c) A−1 (d) B−1A 1 -2 4 6 -3 2 -4 and B =
total revenue function may be modelled by TR = aQ + bQ2.(a) If TR = 14 when Q = 2 and TR = 9 when Q = 3, write down a pair of simultaneous equations for the parameters a and b.(b) Use Cramer’s rule
Consider the two-sector macroeconomic modelY = C + I*C = aY + b(a) Express this system in the formAx = band A and b are 2 × 2 and 2 × 1 matrices to be stated.(b) Use Cramer’s rule to solve this
The demand and supply functions for two interdependent goods are given byQD1 = 400 − 5P1 − 3P2QD2 = 300 − 2P1 − 3P2QS1 = −60 + 3P1QS2 = −100 + 2P2(a) Show that the equilibrium prices
Use Cramer’s rule to solve the following sets of simultaneous equations:(a)4x + 3y = 1 2x + 5y = −3(b) 4x + 3y = 1 2x + 5y = 11(c)4x + 3y = −22x + 5y = −36
Use Cramer’s rule to find the value of y which satisfies each of the following pairs of simultaneous equations:(a) x + 3y = 9 2x − 4y = −2(b) 5x − 2y = 7 2x + 3y =
Use Cramer’s rule to find the value of x which satisfies each of the following pairs of simultaneous equations:(a)7x − 3y = 4 2x + 5y = 7(b)−3x + 4y = 5 2x + 5y = 12(c)x + 4y = 92x
(a) Evaluate each of these determinants.(b) Use your answers to part (a) to write down the solution of the simultaneous equations4x + 2x = −7x + 3y = 5 -7 2 (ii) (i) (ii) -7
If a, b and k are non-zero, show that(a) each of these 2 × 2 matrices is singular:(b) each of these 2 × 2 matrices is non-singular: b ов 8] (iї) ka kb |(i) b (ii) a (iї) a a -b a
The demand and supply functions for two interdependent goods are given byQD1 = 50 − 2P1 + P2QD2 = 10 + P1 − 4P2QS1 = −20 + P1QS2 = −10 + 5P2(a) Show that the equilibrium prices satisfy(b)
Use matrices to solve the following pairs of simultaneous equations:(a)3x + 4y = −15x − y = 6(b)x + 3y = 84x − y = 6
Evaluate the matrix product,Hence, or otherwise, write down the inverse of -3 5 -3 3 - 10 10 5 8. 3 10 5
If the matricesare singular, find the values of a and b. 2 b 3 -4 3 -4 and a
Let(1) Find(a)|A| (b) |B| (c) |AB|Do you notice any connection between u A u, u B u and u AB u?(2) Find(a) A−1 (b) B−1 (c) (AB)−1Do you notice any connection between A−1,

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