Questions and Answers of Business Economics And Finance

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)
Differentiate each of the following functions, using the power function rule. Continue to use the different notations. (a) y = 9x4 d (9x4)=36x (c) f(x)=7x-2 (d) y=-8x-3 (b) f(x)=-5x f' = -15x2
Use the rule for sums and differences to differentiate the following functions. Simply treat the dependent variable on the left as y and the independent variable on the right as x. (a) R 912 +71-4 dR
Given y = f(x) = 6x3(4x – 9), (a) use the product rule to find the derivative, (b) Simplify the original function first and then find the derivative, (c) Compare the two derivatives. (a) Recalling
Redo Problem 9.10, given y = f(x) = (x6 + 4)(x3 + 15). (a) Let g(x) = x6 +4 and h(x) = x3 + 15. Then g'(x) = 6x5 and h'(x) = 3x. Substituting these values in (9.3), y' = f'(x) = (x+4)(3x) + (x3 +
Differentiate each of the following functions, using the product rule.Note: The choice of problems is purposely kept simple in this and other sections of the book to enable students to see how
Differentiate each of the following functions by means of the quotient rule. Continue to apply the rules to the functions as given. Later, when all the rules have been mastered, the functions can be
Given f(x) = (6x + 7)2, (a) use the generalized power function rule to find the derivative. (b) Simplify the function first by squaring it and then taking the derivative. (c) Compare answers. (a)
Find the derivative for each of the following functions with the help of the generalized power function rule. (a) y=(3x+8)5 Here g(x)=3x3 +8, g'(x) = 9x, and n = 5. Substituting in the generalized
Use the chain rule to find the derivative dy/dx for each of the following functions of a function.Check each answer on your own with the generalized power function rule, noting that the generalized
Redo Problem 9.17, given (a) y (x+5x-8)6 Let y=u, u = x + 5x - 8, then dy/du = 6u5 and du/dx = 2x + 5. Substituting in (9.6), dy dx = 6u5 (2x+5)= (12x+30)u But u x+5x8. Substituting, therefore, = dy
Use whatever combination of rules is necessary to find the derivatives of the following functions. Do not simplify the original functions first. They are deliberately kept simple to facilitate the
Differentiate each of the following, using whatever rules are necessary: (a) y = (4x27)(6x+5) 3.x (x = 0) y' = Using the quotient rule and the product rule, 3x[(4x27)(6) + (6x+5) (8x)] - (4x -
For each of the following functions, (1) find the second-order derivative and (2) evaluate it at x = 3.Practice the use of the different second-order notations. (a) y 10x3 +8x + 19 (1) dy dx d y dx =
For each of the following functions, (1) investigate the successive derivatives and (2) evaluate them at x = 2. (a) f(x) = 2x +7x+9x-2 (1) f'(x)=6x + 14x+9 f"(x) = 12x + 14 f""(x) = 12 (4) (x) = 0
Given f(x) = g(x) + h(x), where g(x) and h(x) are both differentiable functions, prove the rule of sums by demonstrating that f′(x) = g′(x) + h′(x).From (9.2) the derivative of f(x) is
Given f(x) = g(x) · h(x), where g′(x) and h′(x) both exist, prove the product rule by demonstrating that f′(x) = g(x) · h′(x) + h′(x) · g′(x). From (9.2), the derivative of f(x) is:
Given f(x) = g(x)/h(x), where g′(x) and h′(x) both exist and h(x) ≠ 0, prove the quotient rule by demonstrating f'(x) = h(x) g'(x) g(x) h'(x) Starting with f(x) = g(x)/h(x) and solving for
Find the limits of the following functions: (a) lim (5x22x-18) (b) x-6 lim (4x2 +9x-5) 3x-4x+6 (c) lim x2x2+8x15 x-3 (d) lim x48x2+3x+4 7x212 (e) limx3+9x-1 x5 (f) lim 2.12 +5x+8 6-x
Find the limits of each of the following functions in which the limit of the denominator approaches zero: (a) lim x+12 x-12x2144 (c) lim x5 x-11x+30 x-5 X-7 (b) lim x-72x-98 (d) lim x 8 x+7x 120 - x
Find the limits of each of the following functions: 7.x-22 8.x-49 (a) lim (b) lim xx21x+8 9.x-4x+2 x3x3 +16 x4 -9 (c) lim (d) lim x842 +6x-7
Use the product rule to differentiate each of the following functions: (a) f(x)=(8x-9)(4x) (c) y = (6x+11) (9x3-4) (b) f(x)=12x (7x+3) (158x4) (6x6-5) (d) y
Differentiate each of the following functions, using the quotient rule: (a) y = (c) y = 22.x 15 - 8x1 5.x3 6x27x+2 (b) y = (d) y = 8x6 3.x +5 8.x +3.x - 7x2-4 9
Use the generalized power function rule to differentiate each of the following functions: (c) f(x) = (a) f(x)=(9x-4)5 1 (3x-11)2 (b) f(x)=(7x3+6)4 = (3x-11)-2 -50 (d) f(x) = =-50(6x-4x-9)-1 6x2-4x-9
Differentiate each of the following functions using whatever combination of rules is necessary. (a) y=5x(4x-9) (8.x-5)3 (c) f(x) = 3.x +2 (b) y = 6x(x+7) 4x+1 (d) f(x)=(7x-4) (3x+8)4
From the graphs in Fig. 10-8, indicate which graphs are (1) increasing for all x, (2) decreasing for all x, (3) convex for all x, (4) concave for all x, (5) which have relative maxima or minima, and
Indicate with respect to the graphs in Fig. 10-9 which functions have (1) positive first derivatives for all x, (2) negative first derivatives for all x, (3) positive second derivatives for all x,
Test to see whether the following functions are increasing, decreasing, or stationary at x = 3. (a) y = 5x212x+8 y' = 10x12 y' (3) 10(3) 12 = 18 > 0 (b) y=x-4x - 9x+19 = - y' = 3x - 8x-9 y' (3)=3(3)
Test to see if the following functions are concave or convex at x = 2. (a) y = -4x +5x+3x-25 y=-12x+10x +3 y" = -24x + 10 y" (2) 24(2) + 10 = -3 -38 < 0 concave
Find the relative extrema for the following functions by (1) finding the critical value(s) and (2)determining whether at the critical value(s) the function is at a relative maximum or minimum. (a)
For the function y = (x – 5)4, (1) find the critical values and (2) test to see if at the critical values the function is at a relative maximum, minimum, or possible inflection point.(1) Take the
Redo Problem 10.6, given y = (6 – x)3.Continuing to take successively higher-order derivatives and evaluating them at the critical value in search of the first higher-order derivative that does not
Redo Problem 10.6, given y = –3(x – 4)6Continuing on,With the first nonzero higher-order derivative an even-numbered derivative, y is at an extreme point;with y(6) (4) (1) (2) y=-18(x-4)=0 x=4
Redo Problem 10.6, given y = (x – 3)5Moving on to the third and higher-order derivatives,With the first nonzero higher-order derivative an odd-numbered derivative, y is at an inflection point. See
From the information below, describe and then draw a rough sketch of the function around the point indicated.(a) f (4) = 2, F(4) = 3, f “(4) = 5 With f(4) = 2, the function passes through the point
(a) Find the critical values, (b) test for concavity to determine relative maxima or minima, (c) check for inflection points, (d) evaluate the function at the critical values and inflection points,
Redo Problem 10.11, given f(x) = –2x3 12x2 72x – 70. (a) (b) f'(x)=-6x + 24x + 72=-6(x - 4x - 12) = 0 f'(x)=-6(x+2)(x-6)=0 x = -2 x=6 f"(x)=-12x+24 critical values f"(-2)=-12(-2) + 24 = 48 > 0
Redo Problem 10.11, given f(x) = (3 – x)4Continuing on, as explained in Section 10.7,With the first nonzero higher-order derivative even-numbered and greater than 0, f(x) is minimized at x = 3.(c)
Redo Problem 10.11, given f(x) = (2x – 8)1Continuing on to successively higher-order derivatives,(c) As explained in Section 10.7, with the first nonzero higher-order derivative an odd-numbered
Optimize the following quadratic and cubic functions by (1) finding the critical value(s) at which the function is optimized and (2) testing the second-order condition to distinguish between a
Find the successive derivatives of each of the following functions:(a) y = 3x4 – 5x3 + 8x2 – 7x – 13(b) y = (8x + 9)(10x – 3)(c) f(x) = (6x – 7)4(d) f(x) = (5 – 2x)4
Using the chain rule, find the first derivative of each of the following functions:(a) y = (6x4 – 35)8 (b) y = (27 – 8x3)5 (c) f(x) = (18x2 + 23)1/3(d) f(x) = (122x3 – 49)−4
Find the first derivative for each of the following functions:(a) y = 6x5 (b) f(x) = 2x + 9 (c) f(x) = 17 (d) y = 8x3 + 4x2 + 9x + 3 (e) y = 6x−4 (f) f(x) = −7x−2 (g) y = 18x1/3 (h) f(x) =

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