The marginal profit for a certain company is MP1(x) = -x2 + 14x - 24. The company expects the daily production level to rise from x = 6 to x = 8 units. The management is considering a plan that would have the effect of changing the marginal profit to M2(x) = -x2 + 12x - 20. Should the company adopt the plan? Determine the area between the graphs of the two marginal profit functions from x = 6 to x = 8. Interpret this area in economic terms.
> For the Riemann sum on an interval [a, b], determine n, b, and f (x). [(8.25)3 + (8.75)3 + (9.25)3 + (9.75)3](.5); a = 8
> Find the volume of the solid of revolution generated by revolving about the x-axis the region under each of the following curves. y = e-x from x = 0 to x = 1
> Find the volume of the solid of revolution generated by revolving about the x-axis the region under each of the following curves. y = 2x + 1 from x = 0 to x = 1 (The solid generated is called a truncated cone.)
> Find the volume of the solid of revolution generated by revolving about the x-axis the region under each of the following curves. y = 2x - x2 from x = 0 to x = 2
> Find the volume of the solid of revolution generated by revolving about the x-axis the region under each of the following curves. y = √x from x = 0 to x = 4 (The solid generated is called a paraboloid.)
> Find the volume of the solid of revolution generated by revolving about the x-axis the region under each of the following curves. y = kx from x = 0 to x = h (generates a cone)
> Find the volume of the solid of revolution generated by revolving about the x-axis the region under each of the following curves. y = x2 from x = 1 to x = 2
> Find the volume of the solid of revolution generated by revolving about the x-axis the region under each of the following curves. y = √ (r2 - x2) from x = -r to x = r (generates a sphere of radius r)
> Find the volume of the solid of revolution generated by revolving about the x-axis the region under each of the following curves. y = √ (4 - x2) from x = -2 to x = 2 (generates a sphere of radius 2)
> Determine the following: ∫ (-3e-x + 2x - e0.5x/2) dx
> Find the volume of the solid of revolution generated by revolving about the x-axis the region under each of the following curves. y = -x2 + 1 from x = 0 to x = 1.
> Find the volume of the solid of revolution generated by revolving about the x-axis the region under each of the following curves. y = x + 1 from x = 0 to x = 2.
> A savings account pays 4.25% interest compounded continuously. At what rate per year must money be deposited steadily in the account to accumulate a balance of $100,000 after 10 years?
> An investment pays 10% interest compounded continuously. If money is invested steadily so that $5000 is deposited each year, how much time is required until the value of the investment reaches $140,000?
> Suppose that money is deposited steadily in a savings account so that $14,000 is deposited each year. Determine the balance at the end of 6 years if the account pays 4.5% interest compounded continuously.
> Suppose that money is deposited steadily in a savings account so that $16,000 is deposited each year. Determine the balance at the end of 4 years if the account pays 8% interest compounded continuously.
> Suppose that money is deposited daily in a savings account at an annual rate of $2000. If the account pays 6% interest compounded continuously, approximately how much will be in the account at the end of 2 years?
> Suppose that money is deposited daily in a savings account at an annual rate of $1000. If the account pays 5% interest compounded continuously, estimate the balance in the account at the end of 3 years.
> For a particular commodity, the quantity produced and the unit price are given by the coordinates of the point where the supply and demand curves intersect. For the pair of supply and demand curves, determine the point of intersection (A, B) and the cons
> For a particular commodity, the quantity produced and the unit price are given by the coordinates of the point where the supply and demand curves intersect. For the pair of supply and demand curves, determine the point of intersection (A, B) and the cons
> Determine the following: ∫-2(e2x + 1) dx
> Figure 8 shows a supply curve for a commodity. It gives the relationship between the selling price of the commodity and the quantity that producers will manufacture. At a higher selling price, a greater quantity will be produced. Therefore, the curve is
> Figure 8 shows a supply curve for a commodity. It gives the relationship between the selling price of the commodity and the quantity that producers will manufacture. At a higher selling price, a greater quantity will be produced. Therefore, the curve is
> Figure 8 shows a supply curve for a commodity. It gives the relationship between the selling price of the commodity and the quantity that producers will manufacture. At a higher selling price, a greater quantity will be produced. Therefore, the curve is
> Figure 8 shows a supply curve for a commodity. It gives the relationship between the selling price of the commodity and the quantity that producers will manufacture. At a higher selling price, a greater quantity will be produced. Therefore, the curve is
> Find the consumers’ surplus for each of the following demand curves at the given sales level, x. p = √(16 - .02x); x = 350
> Find the consumers’ surplus for each of the following demand curves at the given sales level, x. p = 500/(x + 10) - 3; x = 40
> Find the consumers’ surplus for each of the following demand curves at the given sales level, x. p = x2/200 - x + 50; x = 20
> Find the consumers’ surplus for each of the following demand curves at the given sales level, x. p = 3 – x/10; x = 20
> One hundred dollars is deposited in the bank at 5% interest compounded continuously. What will be the average value of the money in the account during the next 20 years?
> One hundred grams of radioactive radium having a half-life of 1690 years is placed in a concrete vault. What will be the average amount of radium in the vault during the next 1000 years?
> Determine the following: ∫7/(2e2x) dx
> Assuming that a country’s population is now 3 million and is growing exponentially with growth constant .02, what will be the average population during the next 50 years?
> During a certain 12-hour period, the temperature at time t (measured in hours from the start of the period) was T(t) = 47 + 4t – 1/3 t2 degrees. What was the average temperature during that period?
> Determine the average value of f (x) over the interval from x = a to x = b, where f (x) = 1/√x; a = 1, b = 9.
> Determine the average value of f (x) over the interval from x = a to x = b, where f (x) = 1/x; a = 1/3, b = 3.
> Determine the average value of f (x) over the interval from x = a to x = b, where f (x) = 2; a = 0, b = 1.
> Shade the portion of Fig. 23 whose area is given by the integral 0∫2 [ f (x) - g (x)] dx + 2∫4 [h(x) - g (x)] dx. Figure 23: 3 1 y = h(x) y = g(x) 2 3 y = f(x) 4 5
> Write a definite integral or sum of definite integrals that gives the area of the shaded portions in Fig. 22. Figure 22: 2 1 U 0 -1 -2- 1 y = f(x) 不 3 4 y = g(r) M
> Write a definite integral or sum of definite integrals that gives the area of the shaded portions in Fig. 21. Figure 21: -1 0 y y = f(x) 2 3 4
> Use a graphing utility to find the intersection points of the curves, and then use the utility to find the area of the region bounded by the curves. y = 1/x, y = 3 - x
> Use a graphing utility to find the intersection points of the curves, and then use the utility to find the area of the region bounded by the curves. y = √(x + 1), y = (x - 1)2
> Determine the following: ∫e dx
> Use a graphing utility to find the intersection points of the curves, and then use the utility to find the area of the region bounded by the curves. y = 5 - (x - 2)2, y = ex
> Use a graphing utility to find the intersection points of the curves, and then use the utility to find the area of the region bounded by the curves. y = ex, y = 4x + 1
> The velocity of an object moving along a line is given by υ(t) = t2 + t - 2 feet per second. (a) Find the displacement of the object as t varies in the interval 0 ≤ t ≤ 3. Interpret this displacement using area under the graph of υ(t). (b) Find the total
> The velocity of an object moving along a line is given by υ(t) = 2t2 - 3t + 1 feet per second. (a) Find the displacement of the object as t varies in the interval 0 ≤ t ≤ 3. (b) Find the total distance traveled by the object during the interval of time 0
> Cars A and B start at the same place and travel in the same direction, with velocities after t hours given by the functions Ï…A(t) and Ï…B(t) in Fig. 29. (a) What does the area between the two curves from t = 0 to t = 1 represent? (b)
> Two rockets are fired simultaneously straight up into the air. Their velocities (in meters per second) are υ1(t) and υ2(t), and υ1(t) ≥ υ2(t) for t ≥ 0. Let A denote the area of the region between the graphs of y = υ1(t) and y = υ2(t) for 0 ≤ t ≤ 10. Wha
> After an advertising campaign, a company’s marginal profit was adjusted up from M1(x) = 2x2 - 3x + 11, before advertising, to M2(x) = 2x2 - 2.4x + 8, after advertising. Here x denotes the number of units produced, and M1(x) and M2(x) are measured in thou
> Refer to Exercise 39. The rate of new tree growth (in millions of cubic meters per year) in the Sudan t years after 1980 is given approximately by the function g(t) = 50 - 6.03e0.09t. Set up the definite integral giving the amount of depletion of the for
> Deforestation is one of the major problems facing sub-Saharan Africa. Although the clearing of land for farming has been the major cause, the steadily increasing demand for fuel wood has also become a significant factor. Figure 28 summarizes projections
> Determine the following: ∫ (7/2x3 – 3√x) dx
> Find all antiderivatives of each following function: f (x) = 9x8
> Suppose that the velocity of a car at time t is υ(t) = 40 + 8/(t + 1)2 kilometers per hour. (a) Compute the area under the velocity curve from t = 1 to t = 9. (b) What does the area in part (a) represent?
> Some food is placed in a freezer. After t hours, the temperature of the food is dropping at the rate of r(t) degrees Fahrenheit per hour, where r(t) = 12 + 4/(t + 3)2. (a) Compute the area under the graph of y = r(t) over the interval 0 ≤ t ≤ 2. (b) What
> Let M(x) be a company’s marginal cost at production level x. Give an economic interpretation of the number 0∫100 M(x)dx.
> Let M(x) be a company’s marginal profit at production level x. Give an economic interpretation of the number 44∫48 M(x)dx.
> Suppose that the marginal profit function for a company is P (x) = 100 + 50x - 3x2 at production level x. (a) Find the extra profit earned from the sale of 3 additional units if 5 units are currently being produced. (b) Describe the answer to part (a) as
> Suppose that the marginal cost function of a handbag manufacturer is C (x) = 3/32 x2 - x + 200 dollars per unit at production level x (where x is measured in units of 100 handbags). (a) Find the total cost of producing 6 additional units if 2 units are c
> After t hours of operation, an assembly line is producing lawn mowers at the rate of r(t) = 21 – 4/5 t mowers per hour. (a) How many mowers are produced during the time from t = 2 to t = 5 hours? (b) Represent the answer to part (a) as an area.
> A helicopter is rising straight up in the air. Its velocity at time t is υ(t) = 2t + 1 feet per second. (a) How high does the helicopter rise during the first 5 seconds? (b) Represent the answer to part (a) as an area.
> Find the area of the region bounded by y = 1/x, y = 4x, and y = x/2, for x ≥ 0. (The region resembles the shaded region in Exercise 29.) Figure 27: Y /y = 8x y y=x x
> Find the area in Fig. 27 of the region bounded by y = 1/x2, y = x, and y = 8x, for x ≥ 0. Figure 27: Y /y = 8x y y=x x
> Determine the following: ∫3e-2x dx
> Find the area of the region between y = x2 and y = 1/x2 (a) from x = 1 to x = 4, (b) from x = 1/2 to x = 4.
> Find the area of the region between y = x2 - 3x and the x-axis (a) from x = 0 to x = 3, (b) from x = 0 to x = 4, (c) from x = -2 to x = 3.
> Find the area of the region bounded by the curves. y = 4/x and y = 5 - x
> Find the area of the region bounded by the curves. y = 8x2 and y = √x
> Find the area of the region bounded by the curves. y = x3 and y = 2x2
> Find the area of the region bounded by the curves. y = x(x2 - 1) and the x-axis
> Find the area of the region bounded by the curves. y = x2 - 1 and y = 3
> Find the area of the region bounded by the curves. y = -x2 + 6x - 5 and y = 2x - 5
> Find the area of the region bounded by the curves. y = 4x(1 - x) and y = 3/4
> Find the area of the region bounded by the curves. y = x2 and y = x
> Determine the following: ∫ (7/2x3 – 3√x) dx
> Find the area of the region between the curves. y = e2x and y = 1 - x from x = 0 to x = 1
> Find the area of the region between the curves. y = ex and y = 1/x2 from x = 1 to x = 2
> Find the area of the region between the curves. y = x(2 - x) and y = 2 from x = 0 to x = 2
> Find the area of the region between the curves. y = x2 - 6x + 12 and y = 1 from x = 0 to x = 4
> Find the area of the region between the curves. y = x2 + 1 and y = -x2 - 1 from x = -1 to x = 1
> Find the area of the region between the curves. y = 2x2 and y = 8 (a horizontal line) from x = -2 to x = 2
> Find the area of the region between the curve and the x-axis. f (x) = e-x + 2 from -1 to 2
> Find the area of the region between the curve and the x-axis. f (x) = ex - 3 from 0 to ln 3
> Find the area of the region between the curve and the x-axis. f (x) = x2 + 6x + 5 from 0 to 1.
> Find the area of the region between the curve and the x-axis. f (x) = x2 - 2x - 3, from 0 to 2.
> Determine the following: ∫ (x - 2x2 + 1/3x) dx
> Find the area of the region between the curve and the x-axis. f (x) = x(x2 - 1), from -1 to 1.
> Find the area of the region between the curve and the x-axis. f (x) = 1 - x2, from -2 to 2
> Let g(x) be the function pictured in Fig. 26. Determine whether 1 > 0 g(x)dx is positive, negative, or zero. Figure 26: 2 1 -2 -3 y y = g(x) +x + 1 2 3 4 5/6 7 8
> Let f (x) be the function pictured in Fig. 25. Determine whether 1 > 0 f (x)dx is positive, negative, or zero. Figure 25: 3 2 1 -1 Y 12 y = f(x) 3 4 5 7 8 8
> Shade the portion of Fig. 24 whose area is given by the integral 0∫1L [ f (x) - g (x)] + 1∫2 [ g (x) - f (x)] dx. Figure 24: 2 y = f(x) 1 y = g(x) 0 У х а 1 2
> Compute the area of the shaded region in two different ways: (a) by using simple geometric formulas; (b) by applying Theorem I. Y f(x) = -x 2- -2 2 0
> Compute the area of the shaded region in two different ways: (a) by using simple geometric formulas; (b) by applying Theorem I. y 1 0 1 (x)=1 1 2 x
> Compute the area of the shaded region in two different ways: (a) by using simple geometric formulas; (b) by applying Theorem I. 2 Y 0 12 f(x) = 2 23 3 4
> Evaluate a Riemann sum to approximate the area under the graph of f (x) on the given interval, with points selected as specified. f (x) = x√(1 + x2); 1 ≤ x ≤ 3, n = 20, midpoints of subintervals f
> Evaluate a Riemann sum to approximate the area under the graph of f (x) on the given interval, with points selected as specified. f (x) = x√(1 + x2); 1 ≤ x ≤ 3, n = 20, midpoints of subintervals
> Determine the following: ∫ (2/√x + 2√x) dx
> Compute the area under the graph of y = 1 / (1 + x2) from 0 to 5.
> The area under the graph of the function e-x2 plays an important role in probability. Compute this area from -1 to 1.
