Question: If the demand equation for a monopolist

> Let P(x) be the profit from producing (and selling) x units of goods. Match each question with the proper solution. Questions A. What is the profit from producing 1000 units of goods? B. At what level of production will the marginal profit be 1000 dollar

> The revenue from producing (and selling) x units of a product is given by R(x) = 3x - .01x2 dollars. (a) Find the marginal revenue at a production level of 20. (b) Find the production levels where the revenue is $200.

> Estimate the cost of manufacturing 51 bicycles per day in Exercise 37. Exercise 37: Let C(x) be the cost (in dollars) of manufacturing x bicycles per day in a certain factory. Interpret C (50) = 5000 and C’ (50) = 45.

> Let C(x) be the cost (in dollars) of manufacturing x bicycles per day in a certain factory. Interpret C (50) = 5000 and C’ (50) = 45.

> If s = 7x2y√z, find: (a) d2s/dx2 (b) d2s/dy2 (c) ds/dz

> If s = Tx2 + 3xP + T2, find: (a) ds/dx (b) ds/dP (c) ds/dT

> Differentiate. y = (2x + 4)3

> If s = P2T, find (a) d2s/dP2, (b) d2s/dT2.

> If s = PT, find (a) ds/dP, (b) ds/dT.

> A supermarket finds that its average daily volume of business, V (in thousands of dollars), and the number of hours, t, that the store is open for business each day are approximately related by the formula Find dV/dt |t=10. V = 20 1 100 100+ 1², t

> A company finds that the revenue R generated by spending x dollars on advertising is given by Find dR/dx |x=1500. R = 1000 + 80x-.02x2, for 0 ≤ x ≤ 2000.

> Compute the following. d/dt (dy/dt), where υ = 2t2 + 1/t + 1

> Compute d/dt (dυ/dt) |t=2, where υ(t) = 3t3 +4/t

> Compute the following. g’(0) and g’’(0), when g(T ) = (T + 2)3

> Compute the following. f ‘(1) and f ’’(1), when f (t) = 1/2 + t

> Compute the following. d/dx (dy/dx) |x=1, where y = x3 + 2x - 11

> Compute the following. d2/dx2 (3x3 - x2 + 7x - 1) |x=2

> Differentiate. y = 4x3 - 2x2 + x + 1

> Compute the following. d2/dx2 (3x4 + 4x2) |x=2

> Compute the following. d/dz (z2 + 2z + 1)7 |z= -1

> Compute the following. d/dt (t2 + 1/t + 1) |t=0

> Compute the following. d/dx (2x + 7)2 |x=1

> Find the first and second derivatives. T = (1 + 2t)2 + t3

> Find the first and second derivatives. f (P) = (3P + 1)5

> Find the first and second derivatives. y = π2 + 3x2

> Find the first and second derivatives. f (r) = πr2

> Find the first and second derivatives. υ = 2t2 + 3t + 11

> Find the first and second derivatives. y = √(x + 1)

> Differentiate. f (x) = x4 + x3 + x

> Find the first and second derivatives. y = 100

> Find the first and second derivatives. y = √x

> Find the first and second derivatives. y = (x + 12)3

> Find the first and second derivatives. y = x + 1

> Find d/dP (T2 + 3P)3.

> Find d/dt (a2t2 + b2t + c2).

> Find d/dP √(s2 + 1).

> Find d/dP (3P2 – 12P + 1).

> Find the first derivatives. x = 16t2 + 45t + 10

> Find the first derivatives. y = T5 - 4T4 + 3T2 - T - 1

> Differentiate. f (x) = 12 + 1/73

> The straight line in the figure is tangent to the parabola. Find the value of b. Y = 2x² – 4x + 10 (0, b) + 6

> Draw the graph of a function f (x) for which the function and its first derivative have the stated property for all x. f (x) increasing and f (x) decreasing

> Draw the graph of a function f (x) for which the function and its first derivative have the stated property for all x. f (x) and f (x) decreasing

> Draw the graph of a function f (x) for which the function and its first derivative have the stated property for all x. f (x) and f (x) increasing

> The straight line in the figure is tangent to the graph of f (x). Find f (4) and f ‘(4). 5 (0, 3) ม 4 y = f(x) X

> Figure 2 shows the graph of the function f (x) and its tangent line at x = 3. Find f (3), f ‘(3), and f’’ (3). Figure 2: y 5 4 3 2 1 1 2 y = f(x) 3 4 5

> Figure 1 contains the graph of f ‘(x), the derivative of f (x). Use the graph to answer the following questions about the graph of f (x). (a) For what values of x is the graph of f (x) increasing? Decreasing? (b) For what values of x is

> A travel agency offers a boat tour of several Caribbean islands for 3 days and 2 nights. For a group of 12 people, the cost per person is $800. For each additional person above the 12-person minimum, the cost per person is reduced by $20 for each person

> Jane wants to drive her tractor from point A on one side of her 5-mile-wide field to a point, B, on the opposite side of the field, as shown in Fig. 7. Jane could drive her tractor directly across the field to point C and then drive 15 miles down a road

> A publishing company sells 400,000 copies of a certain book each year. Ordering the entire amount printed at the beginning of the year ties up valuable storage space and capital. However, printing the copies in several partial runs throughout the year re

> A small orchard yields 25 bushels of fruit per tree when planted with 40 trees. Because of overcrowding, the yield per tree (for each tree in the orchard) is reduced by 12 bushel for each additional tree that is planted. How many trees should be planted

> A long rectangular sheet of metal 30 inches wide is to be made into a gutter by turning up strips vertically along the two sides (Fig. 6). How many inches should be turned up on each side to maximize the amount of water that the gutter can carry? Figure

> A closed rectangular box with a square base is to be constructed using two different types of wood. The top is made of wood costing $3 per square foot and the remainder is made of wood costing $1 per square foot. If $48 is available to spend, find the di

> An open rectangular box is to be 4 feet long and have a volume of 200 cubic feet. Find the dimensions for which the amount of material needed to construct the box is as small as possible.

> The tangent line to the curve y = x3 - 6x2 - 34x - 9 has slope 2 at two points on the curve. Find the two points.

> Find the minimum value of the function g(t) = t2 - 6t + 9, 1 ≤ t ≤ 6.

> Find the maximum value of the function f (x) = 2 - 6x - x2, 0 ≤ x ≤ 5, and give the value of x where this maximum occurs.

> For what x does the function f (x) = 1/4 x2 - x + 2, 0 ≤ x ≤ 8, have its maximum value?

> Let f (x) be the number of people living within x miles of the center of New York City. (a) What does f (10 + h) - f (10) represent? (b) Explain why f ‘(10) cannot be negative.

> The water level in a reservoir varies during the year. Let h(t) be the depth (in feet) of the water at time t days, where t = 0 at the beginning of the year. Match each set of information about h(t) and its derivatives with the corresponding description

> A car is traveling on a straight road and s(t) is the distance traveled after t hours. Match each set of information about s(t) and its derivatives with the corresponding description of the car’s motion. Information A. s(t) is a constant function. B. s’(

> Let f (x) be a function whose derivative is f ‘(x) = √(5x2 + 1). Show that the graph of f (x) has an inflection point at x = 0.

> Let f (x) be a function whose derivative is f ‘(x) = 1/(1 + x2). Note that f ‘(x) is always positive. Show that the graph of f (x) has an inflection point at x = 0.

> Show that the function f (x) = (2x2 + 3) 3/2 is decreasing for x < 0 and increasing for x > 0.

> Let f (x) = (x2 + 2)3/2. Show that the graph of f (x) has a possible relative extreme point at x = 0.

> The tangent line to the curve y = 1/3 x3 - 4x2 + 18x + 22 is parallel to the line 6x - 2y = 1 at two points on the curve. Find the two points.

> Sketch the following curves. y = 1/2x + 2x + 1 (x > 0)

> Sketch the following curves. y = x/5 + 20/x + 3 (x > 0)

> Sketch the following curves. y = x4 - 4x3

> Sketch the following curves. y = x4 - 2x2

> Sketch the following curves. y = x3 - 6x2 - 15x + 50

> Sketch the following curves. y = - 1/3 x3 - 2x2 - 5x

> Sketch the following curves. y = x3 - 3x2 - 9x + 7

> Sketch the following curves. y = 11/3 + 3x - x2 – 1/3 x3

> Sketch the following curves. y = 100 + 36x - 6x2 - x3

> Sketch the following curves. y = x3 - 3x2 + 3x - 2

> If h(x) = [f (x)]2 + 1/g(x), determine h(1) and h’(1), given that f (1) = 1, g(1) = 4, f ‘(1) = -1, and g’(1) = 4.

> Sketch the following curves. y = x3 – 3/2 x2 - 6x

> Sketch the following curves. y = 2x3 + 3x2 + 1

> Sketch the following parabolas. Include their x- and y-intercepts. y = 2x2 + x - 1

> Sketch the following parabolas. Include their x- and y-intercepts. y = -x2 + 20x - 90

> Sketch the following parabolas. Include their x- and y-intercepts. y = -x2 + 8x - 13

> Sketch the following parabolas. Include their x- and y-intercepts. y = x2 + 3x + 2

> Sketch the following parabolas. Include their x- and y-intercepts. y = x2 - 9x + 19

> Sketch the following parabolas. Include their x- and y-intercepts. y = -2x2 + 10x - 10

> Sketch the following parabolas. Include their x- and y-intercepts. y = 4 + 3x - x2

> Sketch the following parabolas. Include their x- and y-intercepts. y = x2 + 3x - 10

> If g(1) = 4 and g’(1) = 3, find f (1) and f ‘(1), where f (x) = 5 * √g(x).

> Sketch the following parabolas. Include their x- and y-intercepts. y = 7 + 6x - x2

> Sketch the following parabolas. Include their x- and y-intercepts. y = 3 - x2

> United States electrical energy production (in trillions of kilowatt-hours) in year t (with 1900 corresponding to t = 0) is given by f (t), where f and its derivatives are graphed in Figs. 5(a) and 5(b). (a) How much electrical energy was produced in 195

> In Figs. 4(a) and 4(b), the t-axis represents time in hours. (a) When is f (t) = 1? (b) Find f (5). (c) When is f (t) changing at the rate of -.08 unit per hour? (d) How fast is f (t) changing after 8 hours? Figure 4: 2.4 2.0 1.6 1.2 .8 4 fi 4 y =

> Property of functions is described next. Draw some conclusion about the graph of the function. H(0) = 0, H’(0) = 0, H’’(0) = 1

> Property of functions is described next. Draw some conclusion about the graph of the function. g(5) = -1, g’(5) = -2, g’’(5) = 0

> Property of functions is described next. Draw some conclusion about the graph of the function. f (4) = -2, f ‘(4) > 0, f ’’(4) = -1

> Property of functions is described next. Draw some conclusion about the graph of the function. G (10) = 2, G’(10) = 0, G’’(10) > 0

> Property of functions is described next. Draw some conclusion about the graph of the function. F ‘(2) = -1, F ’’(2) 6 0

> Property of functions is described next. Draw some conclusion about the graph of the function. h’(3) = 4, h’’(3) = 1

> If g(3) = 2 and g’(3) = 4, find f (3) and f ‘(3), where f (x) = 2 * [g(x)]3.

> Property of functions is described next. Draw some conclusion about the graph of the function. g(1) = 5, g’(1) = -1