Questions and Answers of Business Economics And Finance

Simplify the following radicals: (a) 169x68 (b) 81x2426 (c) 108x3y5 (d) 450x4y526
Simplify the following radicals: (d) (a) 32 605 (b) 7.63 (e) 753 (c) 243 (f) 2886
Add or subtract the following as indicated: (a) (d) 13 x+9 x 17 - 6 x 15 6 x-14x+48 (b) x 8 2 + 3 41 15 (c) - x+4 x- 7 x-49
Add or subtract the following functions, finding the least common denominator where necessary: 225 2= 72 116 (b) 3 5 3 (e) 18x 16.x 4 + 38 3752 + 1 6699 $
Multiply or divide the following rational expressions: x+5 x 3 - (a) x x-8 x+9 (b) x+12 x-6 (c) x+9 x+2 (d) -4x+7 x-1x-6 2x-7 3x+2 x+4 x-9
Multiply or divide the following fractions as indicated: (a) 7x 4x 9y 3y 5x3y 8w2y5 12x (b) 3.x 11w2x4 3wx5 2wz4 7x2z3 (c) (d) 25y 5y 8yz3 2yz
Reduce the following fractions to their lowest terms: 14 (a) 22 35 (b) 8r5 110 (c) 56.x3 32x3 (d) x+x-30 614 (e) 2-8x+12 x+14x+48 (f) x-11x+18
Simplify the following: (a) 1.5 (b) (x3)1/2 (c) (x1/3)2 (d) (x4)-(1/5)
Simplify the following: (a) (x)4 (b) (x4)-2 (c) (xy4)2 (d) 3 3
Simplify the following exponential expressions: (a) (b) (d)
Use the laws of exponents to prove Rule 1 for radicals, namely,From Section 1.1,From Rule 3 of exponents, (Wx)" = x.
Given a polynomial in two variables, mx2 + nxy + py2, show that the factors (ax + cy)(bx + dy) are such that ab = m, ad + bc – n, and cd = p.Hence, ab = m, ad + bc = n, and cd = p. Multiplying, mx
Given the polynomial mx2 + nx + p, show that the factors are (ax + c)(bx +d) where ab = m, ad + bc= n, and cd = p.By reversing the process of multiplication, we can express the polynomialCarrying out
Use your calculator and the rules of exponents to solve for y. Note that all the radicals are expressed in exponential form. (a) y522-(1/2) +29-(1/2) 522 -(1/2) y = 29 = 18-(1/2) = 18-0.5 pr (Rule 5)
(a) Express the following radicals in their exponential forms, (b) Then use your calculator and the rules of exponents from Section 1.1 to solve for y. (1) y=3.17 (a) (b) To find 510.5, enter 51, hit
Use a calculator and the rules of radicals from Section 1.5 to estimate the value of y. Round all answers to five decimal places. (a) y = 17 y=3.17=51 (Rule 3) To find 51, enter 51, hit the INV key
Practice the use of a pocket calculator and the rules of exponents from Section 1.1 to solve for y.Round all answers to five decimal places. (a) y=45.43 y = 45+3 = 48 To find 48, enter 4 on your
Use the properties of radicals to solve for y in each of the following instances. (a) 3y=6x5 Squaring both sides of the equation, then using Rule 1, (3y) = (6.15)2 3y = 36x10 y = 12x10 (b) 7y=42x
Using the properties of radicals set forth in Section 1.5, simplify the following radicals. (a) 27 == +9 (Rule 3) (b) 5.45 (c) 245 3 27 3-27 = 81 = 5 45 5.45 = 225 = 15 = 55.4949 = 75 245= (d) 108 ==
Add or subtract the following fractions by finding the least common denominators. Reduce all answers to the lowest terms. (a) 9x (b) + 5.x 4 2 = 4 9x 5x 9.x Multiplying the first term by and the
Add or subtract the following fractions by finding the least common denominator.(a) To find the least common denominator, use the fundamental principle of arithmetic to decompose the denominators
Add or subtract the following fractions as in Rule 5:(a) To add or subtract two fractions, you must ensure that they have a common denominator.Since the product of the individual denominators will
Divide the following expressions by inverting the divisor and multiplying as in Rule 3 of Section 1.4. 6.xy 2x y (a) 65w4z4 11w4z2 6xy5 2xy 65w4z4 11w4z2 7x-14y 12x+32y (b) 12x-2y 18x-3y 7.x-14y
Multiply the following rational expressions involving quotients of binomials and reduce to lowest terms.(a) When multiplying rational expressions involving quotients of binomials, multiply numerator
Use Rule 2 of Section 1.4 to multiply the following rational expressions and reduce all answers to the lowest terms.(a) When multiplying rational expressions involving quotients of monomials,
Use Rule 1 to raise the following fractions to higher terms with a common denominator of 48: (a) 689 11 16 -im 66 33 16 113 113 112 112 (c) - 21% 22 2 = 22 4 1 1 4 28 88 || 8180 91 -16 911 (d)
According to the fundamental principle of fractions expressed in Rule 1 of Section 1.4, when the numerator and denominator of a rational number are multiplied or divided by the same nonzero
Using Rule 2 from Section 1.3 and the techniques developed in Problems 1.5 to 1.11, factor the following polynomials:(a) x2 + 11xy + 28y2(1) c · d = 28 [1, 28; 2, 14; 4, 7](2) c + d = 11 [4 + 7 =
Redo Problem 1.11 to factor the following expressions in which the coefficient of the x2 term now has multiple factors.(a) 6x2 + 23x + 20(1) a · b = 6. Factors are [1, 6; 2, 3; 3, 2; 6, 1].(2) c ·
Use the techniques and procedures developed in Problems 1.5 to 1.10 to factor the following expressions in which the coefficient of the x2 term is no longer limited to 1.Of all the possible
Factor each of the following, noting that there is no x term and the constant term is negative.(a) x2 – 81 Here (c ·d) is negative and (c +d) = 0. For this to be true, the factors must be of
Factor each of the following expressions, in which both the coefficient of the x term and the constant term are now negative.(a) x2 − 8x − 48 With (c ·d) and (c +d) both negative, the factors
Factor each of the following, noting that the coefficient of the x term is now positive and the constant term is negative.(a) x2 + 19x – 42 With (c ·d) negative, the two integer factors must be of
Factor each of the following, noting that the coefficient of the x term is negative while the constant term is positive.(a) x2 – 13x + 30 With (c ·d) positive and (c +d) negative, the two integer
Factor each of the following using integer coefficients:(a) x2 + 10x + 21 Here, using the notation from Rule 1 in Section 1.3, m = 1, n = 10, and p = 21. For simplicity we limit our search to
Simplify each of the following polynomials by factoring out the greatest common factor: 18x + 27x 18x2 +27x9x(2x+3) 32x-8 (a) 32x - 88(4x-1) (b) (c) (d) 14x5 - 35x4 14x5 35x47x4(2x-5) - 45x2y5 -
Perform the indicated operations, recalling that each term in the first polynomial must be multiplied by each term in the second and their products summed. (a) (2x+7)(4x-5) (b) (5x 6y) (4x-3y) - (c)
Add or subtract the following polynomials as indicated. Note that in subtraction the sign of every term within the parentheses must be changed before corresponding elements are added.(a) (25x – 9y)
Perform the indicated arithmetic operations on the following polynomials: (a) 35xy + 52xy 35xy+52xy=87xy (b) (c) 22yz - 46yz 22yz2-46yz2=-24yz 79x2y3-46x2y3 79x2y3-46x2y3=33xy3 (d) 16x1x2 +62x1 x2
Simplify the following expressions using the rules of exponents from Section 1.1. pt-pt (0) (b) x-x-3 x-x4x34x7 2* = (6-1+5=-*- (c) x-2-x-4 [-] x-2.xx =(- xxx (d) x/2.x 9 10 S 712 XXXXX-Y I -]
Find the equilibrium level of income Ye, given(a) Y = C + I, C = 150 + 0.75Y, I = 35(b) Y = C + I, C = 275 + 0.75Y, I = 40 + 0.15Y(c) Y = C + I + G, C = 320 + 0.65Y, I = 65 + 0.25Y, G = 150(d) Y = C
Indicate what types of solutions are possible for the different systems of equations in Problem 4.1.(a) An underconstrained system may have an unlimited number of possible solutions or no solution,
Use a calculator to multiply or divide the following:(a) 5236 · 0.015(b) 0.065 · 3.75(c) 145 ÷ 0.25(d) 0.675 ÷ 0.045
Factor each of the following quadratic equations in which the coefficient of the x2 is no longer 1:(a) 3x2 + 17x + 10(b) 5x2 − 19x + 12(c) 7x2 − 22x − 24(d) 4x2 + 16x + 15(e) 6x2 – 23x +
Factor each of the following quadratic equations:(a) x2 + 10x + 24(b) x2 − 10x + 21(c) x2 + 9x − 22(d) x2 − 7x − 60(e) x2 − 49(f) x2 – 23x + 120
Simplify each of the following polynomials by factoring out the greatest common factor:(a) 12 – 15x(b) 42x3 + 28x(c) 15x3y4 + 55x2y5(d) 18x2y6z4 + 22x3y4z3 – 26x5y4z6
Multiply each of the following polynomial expressions:(a) (5x + 2y)(8x + 3y)(b) (6x – 15y)(10x – 3y)(c) (3x2 – 5xy + 4y2)(6x + 2y)(d) (20x – 5y)(8x2 + 2xy + 3y2)
Add or subtract the following polynomials as indicated:(a) (6x + 9y) + (3x – 8y)(b) (20x + 13y) – (4x + 7y)(c) (12x – 17y) – (9x – 8y)(d) (5x2 – 12xy + 3y2) – (2x2 + 9xy + 7y2)
Perform the indicated arithmetical operations for each of the following polynomial expressions:(a) 76xy – 22xy(b) 14x1x2 + 31x1x2(c) 32xy2 – 48xy2(d) 50xyz – 43xyz
Use the rules of exponents to simplify the following expressions:(a) x4 · x3(b) x5 · x1/2(c) x6 · x−4(d) x−2 · x−7
Find the producers’ surplus at x0 = 6 and p0 = 83.4, given the supply function p = 0.15x2 + 8x + 30.
Find the consumers’ surplus at x0 = 5 and p0 = 125, given the demand function x = –6x2 + 275.
A car worth $6000 depreciates over the years at the rate V’(t) = 250(t – 6) for 0 ≤ t ≤ 6. Find the total amount by which the car depreciates in (a) the first 3 years and x) the last three
Estimate the additional total revenue received from increasing sales from 10 to 12 units, given MR= 104 - 6x.
Find a firm’s total revenue TR function, given the marginal revenue function MR = –0.24x2 – 1.5x+ 660.
Find a firm’s total cost of producing 3 extra units as a firm moves from a production level of 4 units to 7 units, given MC = 3x2 – 4x + 525.
A firm’s marginal cost function is MC = x2 – 11x + 385. Its fixed costs are $450. Find the total cost TC function.
What will a machine worth $3.8 million today be worth in 3 years if it depreciates at a continuous rate of 15 percent a year?
Using an exponential function a consultant suggests that a newspaper’s circulation will increase from 1.25 million at present to 1.73 million in 5 years. What growth rate is the consultant assuming?
What is the projected level of revenue in 3 years for a company with current revenues of $48.25 million if estimated growth under continuous compounding is 8.6 percent?
A country’s population of 44.35 million is growing at a rate of 2.65 percent a year. Find the expected population in 5 years.
A company with 1993 sales of $8.9 million estimates sales of $13.82 million in 1997. (a) Determine the exponential function used for the estimate and (b) indicate the projected growth rate.
A firm’s total revenue TR, growing consistently over time, has increased from $345 million in 1985 to $616.18 million in 1993. (a) Find the exponential function and (b) indicate the growth rate of
A company’s sales have been growing consistently over time from $23.2 million in 1987 to $32.27 million in 1993. (a) Express sales 5 as a natural exponential function of time t. (b) Indicate the
Given the current interest rate of 7 percent compounded semiannually, find the present value of$10,000 to be paid in (a) I year, (b) 3 years, (c) 5 years, and (d) 10 years.
Given the current interest rate of 6 percent compounded annually, find the present value of $10,000 to be paid in (a) 1 year, (b) 3 years, (c) 5 years, and (d) 10 years.
Repeat Problem 11.58 for $8000 to be paid in 4 years when the current interest rate is 5 percent.
Determine the present value of $5000 to be paid in 8 years time if current interest of 10 percent is compounded (a) annually, (b) semiannually, (c) quarterly, and (d) continuously.
How long will it take money to treble at 9 percent interest when compounded annually?For each of the following questions, determine the future value of the given principals when compounded (a)
How long will it take money to double at 8 percent interest when compounded semiannually?For each of the following questions, determine the future value of the given principals when compounded (a)
$10,000 at 9 percent for 8 years.For each of the following questions, determine the future value of the given principals when compounded (a) annually, (b) semiannually, (c) quarterly, and (d)
$500 at 12 percent for 6 years.For each of the following questions, determine the future value of the given principals when compounded (a) annually, (b) semiannually, (c) quarterly, and (d)
$2500 at 6 percent for 4 years.For each of the following questions, determine the future value of the given principals when compounded (a) annually, (b) semiannually, (c) quarterly, and (d)
$1000 at 8 percent for 5 years.For each of the following questions, determine the future value of the given principals when compounded (a) annually, (b) semiannually, (c) quarterly, and (d)
For each set of the following total revenue and total cost functions, first express profit π as a function of output x and then determine the maximum level of profit by finding the vertex of the
Determine the total revenue TR function for each of the firms confronted with the following linear demand functions:(a) P = −17.5Q + 2675(b) P = −9.8Q + 860
For each of the following quadratic functions, indicate the shape of the parabola by determining (1)whether the parabola opens up or down, (2) the coordinates of the vertex, and (3) the x intercepts,
Use the quadratic formula to solve each of the following quadratic equations:(a) 6x2 + 31x + 40(b) 8x2 – 50x + 33(c) 4x2 − 31x − 45(d) 9x2 – 67.5x + 126
Solve the following quadratic equations by factoring:(a) x2 – 11x + 28(b) x2 + 5x − 24(c) x2 + 11x + 18(d) x2 – 8x – 48
A company in a purely competitive market receives $95 in revenue for each item sold. If the company has fixed costs of $8800 and a marginal cost of $67.50 per item, express the company’s profit π
A new car bought today for $13,500 depreciates by $2250 at the end of each calendar year. Express the value V of the car as a function of years t.
A firm has a fixed cost of $125,000 and variable costs per item manufactured of $685. Express the firm’s total cost TC as a function of output x.
Given F(x) = (9x – 2)/4x and G(x) = x5, find (a) F[G(x)] and (b) G[F(x)], also called functions of functions.
Given f(x) = x3 — 3x + 4 and g(x) = 5x2, find the composite functions (a) f[g(x)] and (b) g[f(x)].
Given f(x) = (x – 7)/(x + 2) and g(x) = (x + 3)/(x – 8), find (a) (f – g)(x) and (b) (f ÷ g)(x).
Given G(x) = 6/(x + 3) and H(x) = 11/x2, find (a) (G + H)(x) and (b) (G · H)(x).
Given f(x) = 30x2 – x – 99 and h(x) = 5x + 9, find (a) (f – h)(x) and (b) (f ÷ h)(x).
Given F(x) = 3x2 – 7x + 8 and G(x) = 9x – 4, find (a) (F + G)(x) and (b) (F · G)(x).
Given g(x) = 4x – 9 and h(x) = 12 – 5x, find (a) (g – h)(x) and (b) (g ÷ h)(x).
Given f(x) = 7x – 2 and g(x) = 3x + 8, find (a) (f + g)(x) and (b) (f · g)(x).
Evaluate the following functions at the given parametric values of x:(a) f(x) = 7x3 – 12x2 – 38x + 115 at x = a(b) g(x) = 8x2 + 5x – 13 at x = a + 2(c) h(x) = 4x2 – 6x + 7 at x = b – 4(d)
Evaluate the following functions at the given numerical values of x:(a) f(x) = x2 – 9x + 42 at x = 3(b) g(x) = 2x1 + 5x – 9 at x = −4(c) h(x) = x3 – 3x2 + 6x – 7 at x = 2(d) F(x) = (5x2 –
Which of the following equations are functions and why?(a) y = −3x + 8 The equation y = − 3x + 8 is a function because for each value of the independent variable x there is one and only one value
Use the rules of limits to find the limits for the following functions: (a) lim[x(x+5)] x-3 lim [x4(x+5)] = lim x4 lim (x+5) x-3 x-3 x3 = (3) (3+5)= 81.8 = 648 (b) lim 7x-9x t-5 x+8 (Rule 5) 7x-9x
Find the limits for the following polynomial and rational functions. (a) lim (3x25x+9) x-4 From the properties of limits it can be shown that for all polynomial functions and all rational functions,
Find the limits of the following rational functions, aware that if the limit of the denominator equals zero, neither Rule 6 nor the generalized rule for rational functions used above apply.The limit
Graph the following functions and explain the significance of the graphs in terms of Problem 9.3 (c):From the graphs in Fig. 9-7(a) and (b) it is clear that f(x) and g(x) are the same everywhere that
Find the limits of the following functions, noting the role that infinity plays. (a) 2 lim - x0x (x = 0) As seen in Fig. 3-2, as x approaches zero from the right [x 0+], f(x) approaches positive
Indicate whether the following functions are continuous at the specified points by determining whether at the given point all of the following conditions from Section 9.2 hold: (1) f(a) is
Differentiate each of the following functions and practice the use of the different notations for a derivative. (a) f(x) = 29 f'(x)=0 (constant rule) (c) y=6x+13 y' = 6 (linear function rule) (b)

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