Questions and Answers of Business Economics And Finance

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) =
Optimize the following higher-order polynomial functions, using the same procedure as in Problem 10.15.(a) y = 2x4 – 8x3 – 40x2 + 79(b) y = –5x4 + 20x3 + 280x2 – 19(c) y = (7 –
Find (1) the marginal and (2) the average functions for each of the following total functions.Evaluate them at Q = 2 and Q = 4. (a) TC Q+9Q+16 dTC (1) MC = = 20 +9 dQ MC(2)2(2)+9 = 13 MC(4)=2(4)+9=17
Find the MR functions associated with each of the following demand functions and evaluate them at Q = 20 and Q = 40. (a) (b) (c) P = -0.1Q+25 To find the MR function, given simply a demand function,
For each of the following consumption functions, use the derivative to find the marginal propensity to consume MPC = dC/dY. (a) CbY + Co MPC dC/dY = b (b) C=0.85Y +1250 MPC dC/dY = 0.85 =
Maximize the following total revenue TR and total profit π functions by (1) finding the critical value(s), (2) testing the second-order conditions, and (3) calculating the maximum TR or π. (a) TR =
From each of the following total cost TC functions, (1) find the average cost AC function, (2) the critical value at which AC is minimized, and (3) the minimum average cost. TC (1) AC = = (a) TC2Q
Given the following total revenue TR and total cost TC functions for different firms, maximize profit r for the firms as follows: (1) set up the profit function π = TR — TC, (2) find the critical
Show that marginal revenue (MR) must equal marginal cost (MC) at the profit-maximizing level of output. By definition, = TR - TC. Taking the derivative and setting it equal to zero, since d/dQ must
Use the MR = MC method to (a) maximize profit Tr, and (b) check the second-order conditions, given TR 800Q 7Q2 8000-702 (a) MR TR'800 - 14Q Equating MRMC to maximize profits, TC 20 Q + 800 + 150 = -
Redo Problem 10.24, given (a) TR = MR Equating MR MC, = = 500Q TR = 11Q 500 - 22Q TC 3Q3 2Q +680 +175 = MC = - TC' 9Q - 40+68 = 500-220-902-40+68
A total product curve TP of an input is a production function which allows the amounts of one input(say, labor) to vary while holding the other inputs (capital, land) constant. Given TP — 562.5L 2
Given y = ax2 + bx +c, prove that the coordinates of the vertex of a parabola are the ordered pair[–b/2a, (4ac — b2)/4a], as was asserted in Section 3.6.The vertex is the maximum or minimum point
Indicate whether the following functions are increasing or decreasing at the indicated points: (a) f(x)=5x-4x-89 at x = 3 (c) f(x) = -4x3 + 2x27x+9 at x = 5 (b) f(x)=8x+6x - 15x at x = 4 (d) f(x)=2x-
Determine whether the following functions are concave or convex at x = 3. (a) f(x)=7x + 19x-24 (c) f(x)=(4-9x)3 (b) f(x)=5x-81x + 11x+97 (d) f(x)=(3x4-7)
Find all relative extrema and inflection points for each of the following functions and sketch the graphs on your own. (a) f(x)=4x-48x2-240x +29 (c) f(x) = -2x3 + 24x + 288x - 35 (e) f(x)=-5x + 135x2
Optimize the following functions and test the second-order conditions at the critical points to distinguish between a relative maximum and a relative minimum. (a) y=8x2-208x + 73 (c) y = -2x3 + 69x +
Find the marginal and average functions for each of the following total functions: (a) TR=Q2 +960 (c) TCQ2 +30 +55 = (b) TR-3Q2+ 198Q (d) TC 0.502 +20+69- =
Find the marginal revenue functions associated with each of the following demand functions: (a) P=0.3Q+228 (c) P-2.5Q + 145 (b) P=-8Q+ 1465- (d) P-40+875
Find the critical points at which each of the following total revenue TR functions and profit 7r functions is maximized. Check the second-order conditions on your own. (a) TR3Q2 +2100 = (c) = -3Q
Find the critical points at which each of the following average cost AC functions is minimized.Check the second-order conditions on your own. (a) AC = 302 180+ 585 - (b) AC = 2.25Q2-27Q+768
Find the critical points at which profit zr is maximized for each of the following firms given the total revenue TR and total cost TC functions. Check the second-order conditions on your own. (a) TR
Using the original matrices above, find the following product matrices: (a) AB, (b) CE, (c) FD, (d)GH, (e) EA, (f) GF, (g) CH, and (h) BD.
Find the following differences: (a) A – B (b) C – D (c) E – F (d) G – H.
Find the following sums: (a) A + B (b) C + D (c) E + F (d) G + H.
Construct the initial simplex tableau.(a) Add slack variables to the inequalities to make them equations.(b) Express the constraint equations in matrix form.(c) Form the initial simplex tableau
Redo Problem 8.1 for the equation and inequalities specified below: Maximize subject to 2x1 + x2 14 5x15x240 =50x130x2 X1+3x218 X1, X20
Redo Problem 8.1, given the data below: Maximize subject to 1. Set up the initial tableau: 6x1 + 2x2 36 3x15x230 = 250x1 + 200x2 x1 + 4x2
For the following primal problem, (a) formulate the dual, (b) solve the dual graphically. Then use the dual solution to find the optimal values of (c) the primal objective function and (d) the primal
Redo Problem 8.4 for the following primal problem: Minimize subject to (a) The dual is Maximize c=40y+60y2 + 48y3 5y 3y2+4y3 7 2y+12y2+8y3 21 1, 2, 0 = 7x1 +21x2 subject to 5x1 + 2x2 40 3x+12x260
Redo Problem 8.4 for the following primal problem: Maximize subject to =30x120x2 + 24x3 2x1 +5x2+2x3 80 6x1+x2+3x3 160 X1, X2, X3 0
Redo Problem 8.4 for the following primal problem: Maximize subject to (a) The dual is Minimize subject to = 36x+84x2 + 16x3 3x16x2+x3 20 2x16x2+4x3 15 X1, X2, X30 c = 20y1 + 15 y2 3y1 +2y2 36 y
(a) Use the duals in Problems 8.4 to 8.7 to determine the shadow prices or marginal values (MVs) of the resources in the primal constraints, (b) Using A and B for the resources in the constraints,
For the following problem (a) formulate the dual, (b) Solve the dual, using the simplex algorithm,(c) Use the final dual tableau to determine the optimal values of the primal objective function and
Redo Problem 8.9 for the following linear programming problem: Minimize subject to c = 240y1 + 120y2 4y + 8y2 56 2y +2y2 24 3y+y2 18 Y1, y20 (a) Maximize subject to = 56x1 +24x2 + 18x3 4x12x2+3x3
Redo Problem 8.9, given Minimize subject to c = 36y1 +30y2 + 20y3 6y13y2 y3 250 2y+5y2+4y3 200 Y1, y2, y3 0 0 (a) Maximize subject to = 250x1+200x2 6x1 + 2x2 36 3x1+5x230 x1 + 4x2 20 X1, X20 (b)
Use the simplex algorithm to solve each of the following linear programming problems. Maximize 20x18x2 subject to 4x15x2200- 6x13x2 180 8x12x2160- X1, X2 0
Use the simplex algorithm to solve each of the following linear programming problems. Maximize subject to = 4x1 + 3x2 3x1+9x2207 6x14x2 120 15x15x2225 X1, X2 0
Use the simplex algorithm to solve each of the following linear programming problems. Maximize subject to =8x+2x2 5x14x2 216 6x13x2 180 12x12x2 312 X1, X2 0
Use the simplex algorithm to solve each of the following linear programming problems. Maximize = 9x1 +5x2 subject to 2x1 +4x2 280 6x15x2450 15x16x2720 x1, x2 > 0
Use the simplex algorithm to solve each of the following linear programming problems. Maximize subject to = 39x170x2 + 16x3 x1+ 2x2+5x3 180 5x1 +2.5x2 x3 300 X1, X2, Xx3 0
Use the simplex algorithm to solve each of the following linear programming problems. Maximize = 168x1+222x2 + 60x3 subject to 7x114x2+2x390 8x18x2+4x3 120- X1, X2, X3 0
Find the dual of Problem 8.12,
Find the dual of Problem 8.16,
Solve each of the following problems by first finding the dual and then using the simplex algorithm. Minimize subject to c=225y1 +180y2 8y1 y232 7y1 +4y2 112 y+6y2 54 yi, y2,0
Solve each of the following problems by first finding the dual and then using the simplex algorithm. Minimize subject to c = 540y +900y2 3y 15y2195 4y+5y2140 10y 2y2 80 y1 y20
Solve each of the following problems by first finding the dual and then using the simplex algorithm. Minimize subject to c24y1 +61 y2 + 60y3 2y1+2y2+6y3 60 y13y2 y3 15 yi, y2 y30
Solve each of the following problems by first finding the dual and then using the simplex algorithm. Minimize subject to c48y1 +168y2 + 145y3 2y1 +4y28y3 48 3y 12y2+6y3 96 y1 y2 y30
Redo Problem 7.9 for the following data: Maximize = 8x+6x2 subject to 2x+5x240 (constraint A) 3x + 3x2 30 (constraint B) 8x14x264 (constraint C) X1, X20
Redo Problem 7.9 for the following data:See Fig. 7-6(a) for the graphed constraints and Fig. 7-6(b) for the feasible region.In Fig. 7-6(b) the point of tangency occurs at x1 = 5 and x2 = 4. π =
Redo Problem 7.9 for the following data: Maximize =25x1+50x2 subject to 9x112x2 144 (constraint A) 10x16x2 120 (constraint B) X2 9 (constraint C) X1, X20 See Fig. 7-7. From the critical values, x =
Using the following data from Problem 7.6, graph the inequality constraints by first solving each for y2 in terms of y1. Regraph and darken in the feasible region. Compute the slope of the objective
Read the critical values for y1 and y2 at the point of tangency, and evaluate the objective function at these values. Minimize c = 20y1 + 15y2 subject to 3y12y236 (constraint A) 6y16y284 (constraint
Redo Problem 7.14, using the data derived from Problem 7.7.The constraints are graphed in Fig. 7-9(a) and the feasible region in Fig. 7-9(b). In Fig. 7-9(b) the slope of the isocost line is . Hence
Redo Problem 7.14, using the following data: Minimize c = 6y1 + 3y2 subject to y +2y2 14 (constraint A) y+ y2 12 (constraint B) 3y1 y2 18 (constraint C) Yi, y2 > 0 See Fig. 7-10. From Fig. 7-10(b),
Redo Problem 7.14, using the following data:.See Fig. 7-11. From Fig. 7-11(b), y1 = 8 and y2 = 3. Hence c = 15(8) + 12(3) = 156. Minimize c = 15y1 +12y2 subject to 4y + 8y2 56 (constraint A)
Redo Problem 7.14, using the following data:See Fig. 7-12. From Fig. 7-12(b), y1 = 6 and y2 = 2. Hence c = 7(6) + 28(2) = 98. Minimize c = 7y + 28y2 subject to 3y + 3y2 24 (constraint A) 5y + y2 20
Redo Problem 7.14, using the following data:In Fig. 7-13, with the isocost line tangent to constraint A, there is no unique optimal feasible solution. Any point on the line between (5, 8) and (10, 4)
Minimize c = 3y1 + 2.5y2 subject to the same constraints in Problem Maximize subject to =5x+4x2 7.5x15x2150 3x+4x2108 8x12x2 120 X1, X20

Showing 1300 - 1400 of 1467