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mathematics
calculus with applications
Calculus With Applications 12th Edition Margaret L. Lial - Solutions
Each of the functions in Exercises represents the rate of flow of money in dollars per year. Assume a 10-year period at 8% compounded continuously and find the following: (a) The present value; (b) The accumulated amount of money flow at T = 10.ƒ(t) = 0.05t + 500
Find the volume of the solid of revolution formed by rotating about the x-axis each region bounded by the given curves.ƒ(x) = 2ex, y = 0, x = -2, x = 1
Find the area between y = (x + 1) ln x and the x-axis from x = 1 to x = e.
Determine whether each improper integral converges or diverges, and find the value of each that converges. S 0 8e-8x dx
Find each integral, using techniques from this or the previous chapter. (3x + 6)e-3x dx
Find the volume of the solid of revolution formed by rotating about the x-axis each region bounded by the given curves. f(x) = 2 √x y = 0, x = 1, x = 3
Exercises are mixed—some require integration by parts, while others can be integrated by using techniques discussed in the chapter on Integration. 1 x² dx 2x³ + 1
Determine whether each improper integral converges or diverges, and find the value of each that converges. S 50e-50x dx
Exercises are mixed—some require integration by parts, while others can be integrated by using techniques discussed in the chapter on Integration. x²e²x dx
Each of the functions in Exercises represents the rate of flow of money in dollars per year. Assume a 10-year period at 8% compounded continuously and find the following: (a) The present value; (b) The accumulated amount of money flow at T = 10.ƒ(t) = 1000t - 100t2
Exercises are mixed—some require integration by parts, while others can be integrated by using techniques discussed in the chapter on Integration. [x³²√x + 4 dx
Find each integral, using techniques from this or the previous chapter. In 4x + 5 dx
Find each integral, using techniques from this or the previous chapter. 20 (x - 1) In x dx
Find the volume of the solid of revolution formed by rotating about the x-axis each region bounded by the given curves. f(x) = 2 √x + 2 y = 0, x= -1, x = 2
Exercises are mixed—some require integration by parts, while others can be integrated by using techniques discussed in the chapter on Integration. (2x - 1) In (3x) dx
Determine whether each improper integral converges or diverges, and find the value of each that converges. -0 -00 1000e* dx
Determine whether each improper integral converges or diverges, and find the value of each that converges. 1. -00 5e60x dx
Each of the functions in Exercises represents the rate of flow of money in dollars per year. Assume a 10-year period at 8% compounded continuously and find the following: (a) The present value; (b) The accumulated amount of money flow at T = 10.ƒ(t) = 2000t - 150t2
Determine whether each improper integral converges or diverges, and find the value of each that converges. .-1 -00 In |x| dx
Find each integral, using techniques from this or the previous chapter. X 25 - 9x² dx
Find the volume of the solid of revolution formed by rotating about the x-axis each region bounded by the given curves. x² f(x) == ₁ 2² y = 0, x = 0, x = 4
An investment is expected to yield a uniform continuous rate of money flow of $20,000 per year for 3 years. Find the accumulated amount at an interest rate of 4% compounded continuously.
Exercises are mixed—some require integration by parts, while others can be integrated by using techniques discussed in the chapter on Integration. [(8x (8x + 10) In(5x) dx
Find the volume of the solid of revolution formed by rotating about the x-axis each region bounded by the given curves.ƒ(x) = x2, y = 0, x = 1, x = 5
Find each integral, using techniques from this or the previous chapter. X V16 + 8r2 dx
A real estate investment is expected to produce a uniform continuous rate of money flow of $8000 per year for 6 years. Find the present value at the following rates, compounded continuously.(a) 2% (b) 5% (c) 8%
The rate of a continuous flow of money starts at $5000 and decreases exponentially at 1% per year for 8 years. Find the present value and final amount, as well as the total interest earned, at an interest rate of 8% compounded continuously.
Find each integral, using techniques from this or the previous chapter. f₁ x³ In x dx
Determine whether each improper integral converges or diverges, and find the value of each that converges. xp |x|u[ J1 00
Exercises are mixed—some require integration by parts, while others can be integrated by using techniques discussed in the chapter on Integration. frem dx
The rate of a continuous money flow starts at $1000 and increases exponentially at 5% per year for 4 years. Find the present value and accumulated amount, as well as the total interest earned, if interest earned is 3.5% compounded continuously.
Find the volume of the solid of revolution formed by rotating about the x-axis each region bounded by the given curves.ƒ(x) = 1 - x2, y = 0
Determine whether each improper integral converges or diverges, and find the value of each that converges. S dx (4x + 1)³
Exercises are mixed—some require integration by parts, while others can be integrated by using techniques discussed in the chapter on Integration. 1 x² dx 3 2x³ + 1
Find the amount of a continuous money flow in 3 years if the rate is given by ƒ(t) = 1000 - t2 and if interest is 5% compounded continuously.
Find the area between y = (3 + x2)e2x and the x-axis from x = 0 to x = 1.
The function defined by y = √r2 - x2 has as its graph a semicircle of radius r with center at (0, 0) (see the figure on the next page). In Exercises, find the volume that results when each semicircle is rotated about the x-axis. ƒ(x) = √1 - x2
A money market fund has a continuous flow of money at a rate of ƒ(t) = 1500 - 60t2, reaching 0 in 5 years. Find the present value of this flow if interest is 5% compounded continuously.
Find each integral, using techniques from this or the previous chapter. S₁ 0 x²ex/2 dx
Determine whether each improper integral converges or diverges, and find the value of each that converges. 00 S dx (x + 1)²
Exercises are mixed—some require integration by parts, while others can be integrated by using techniques discussed in the chapter on Integration. -2 La- (1 - x²)e²x dx
Find the volume of the solid of revolution formed by rotating about the x-axis each region bounded by the given curves.ƒ(x) = 2 - x2, y = 0
Determine whether each improper integral converges or diverges, and find the value of each that converges. -1 -00 2x x² 2 1 xp. X
Exercises are mixed—some require integration by parts, while others can be integrated by using techniques discussed in the chapter on Integration. x³ dx V3 + x² 2
Find the area between y = x3(x2 - 1)1/3 and the x-axis from x = 1 to x = 3.
The function defined by y = √r2 - x2 has as its graph a semicircle of radius r with center at (0, 0) (see the figure on the next page). In Exercises, find the volume that results when each semicircle is rotated about the x-axis. ƒ(x) = √36 - x2
The function defined by y = √r2 - x2 has as its graph a semicircle of radius r with center at (0, 0) (see the figure on the next page). In Exercises, find the volume that results when each semicircle is rotated about the x-axis.ƒ(x) = √r2 - x2 (-r, 0) y (0,r) 0 y = √√₁²_ (r, 0) x
Exercises are mixed—some require integration by parts, while others can be integrated by using techniques discussed in the chapter on Integration. -5 xVx² + 2 dx 2
Determine whether each improper integral converges or diverges, and find the value of each that converges. 00 4x + 6 x2 + 3x dx
Suppose that u and v are differentiable functions of x withand the following functional values.Use this information to determine -1 0 v du = 2
Use the table of integrals, or a computer or calculator with symbolic integration capabilities, to find each indefinite integral. 16 √x² + 16 -2 X dx
Determine whether each improper integral converges or diverges, and find the value of each that converges. .00 J2 1 x ln x dx
Use the table of integrals, or a computer or calculator with symbolic integration capabilities, to find each indefinite integral. 10 x² - 25 2 dx
Determine whether each improper integral converges or diverges, and find the value of each that converges. 00 1 1₂ 2 x(In x)² dx
Suppose that u and v are differentiable functions of x withand the following functional values.Use this information to determine -2 v du = -11 0
Determine whether each improper integral converges or diverges, and find the value of each that converges. хр хәх X J 00
Use the table of integrals, or a computer or calculator with symbolic integration capabilities, to find each indefinite integral. 3 xV121 - x² 2 dx
Determine whether each improper integral converges or diverges, and find the value of each that converges. -0 ₂0.2x dx xe
Find the volume of the solid of revolution formed by rotating each bounded region about the x-axis. f(x)=√x - 4, y = 0, x = 13
Use the table of integrals, or a computer or calculator with symbolic integration capabilities, to find each indefinite integral. 2 3x(3x - 5) dx
Use the table of integrals, or a computer or calculator with symbolic integration capabilities, to find each indefinite integral. -6 x(4x + 6)²° 2 dx
Find the volume of the solid of revolution formed by rotating each bounded region about the x-axis.ƒ(x) = 3x - 1, y = 0, x = 2
Determine whether each improper integral converges or diverges, and find the value of each that converges. -00 x³ex dx
Find the average value of each function on the given interval.ƒ(x) = 2 - 3x2; [1, 3]
Explain why the two methods of solving Example 2 are equivalent.
Use the table of integrals, or a computer or calculator with symbolic integration capabilities, to find each indefinite integral. [V.₁² √x² + 15 dx
Determine whether each improper integral converges or diverges, and find the value of each that converges. -00 хр а e т
Find the average value of each function on the given interval.ƒ(x) = x2 - 4; [0, 5]
Find the volume of the solid of revolution formed by rotating each bounded region about the x-axis. f(x) = 1 √x - 1' y = 0, x = 2, x = 4
Find the volume of the solid of revolution formed by rotating each bounded region about the x-axis.ƒ(x) = e-x, y = 0, x = -2, x = 1
Determine whether each improper integral converges or diverges, and find the value of each that converges. -0 X 2 x² + 1 dx
Find the average value of each function on the given interval.ƒ(x) = (2x - 1)1/2; [1, 13]
Find the volume of the solid of revolution formed by rotating each bounded region about the x-axis. (x) = 1/²/² 4 y = 0, x = 4
Find the average value of each function on the given interval.ƒ(x) = √x + 1; [3, 8]
Find the volume of the solid of revolution formed by rotating each bounded region about the x-axis.ƒ(x) = 4 - x2, y = 0, x = -1, x = 1
Find the average value of each function on the given interval.ƒ(x) = e0.1x; [0, 10]
What rule of differentiation is related to integration by parts?
Find the average value of each function on the given interval.ƒ(x) = ex/7; [0, 7]
Determine whether each improper integral converges or diverges, and find the value of each that converges. 00 -00 2x + 4 x² + 4x + 5 dx
Use integration by parts to derive the following formula from the table of integrals. [x²eax dx = xneax a # x² Lex eax dx + C, a # 0 a
The function W(t) = -3.75t2 + 30t + 40 describes a typist’s speed (in words per minute) over a time interval 30, 54.(a) Find W(0).(b) Find the maximum W value and the time t when it occurs.(c) Find the average speed over [0, 5].
Example 1(b) leads to a paradox. On the one hand, the unbounded region in that example has an area of 1/2, so theoretically it could be colored with ink. On the other hand, the boundary of that region is infinite, so it cannot be drawn with a finite amount of ink. This seems impossible, because
Find the area between the graph of the given function and the x-axis over the given interval, if possible. f(x) 3 (x - 1)³¹ for (-∞0, 0]
Find the value of each integral that converges. 00 dx (3x + 1)²
The Yasuko Okada Fragrance Company (YOFC) receives a shipment of 400 cases of specialty perfume early Monday morning of every week. YOFC sells the perfume to retail outlets in California at a rate of about 80 cases per day during each business day (Monday through Friday). What is the average daily
The DeMarco Pasta Company receives 600 cases of imported San Marzano tomato sauce every 30 days. The number of cases of sauce in inventory t days after the shipment arrives is N(t) = 600 - 20√30t. Find the average daily inventory.
Find the area between the graph of each function and the x-axis over the given interval, if possible. 5 ƒ(x) = (x - 2)²² - for (-∞0, 1]
The figure shows the blood flow in a small artery of the body. The flow of blood is laminar (in layers), with the velocity very low near the artery walls and highest in the center of the artery. In this model of blood flow, we calculate the total flow in the artery by thinking of the flow as being
Recall from the previous chapter that the consumers’ surplus is defined bywhere D(q) is the demand function, q0 is the equilibrium quantity, and p0 = D(q0) is the equilibrium price. Find the consumers’ surplus for each of the following demand functions and equilibrium quantities.D(q) = 10 -
The rate of change of revenue (in dollars per calculator) from the sale of x calculators is R′(x) = (x + 1) ln(x + 1). Find the total revenue from the sale of the first 12 calculators.
Recall from the previous chapter that the producers’ surplus is defined bywhere S(q) is the supply function, q0 is the equilibrium quantity, and p0 S(q0) is the equilibrium price. Find the producers’ surplus for each of the following supply functions and equilibrium quantities.S(q) = 5q - q
Each function in Exercises represents the rate of flow of money (in dollars per year) over the given time period, compounded continuously at the given annual interest rate. Find the present value in each case.ƒ(t) = 5000, 8 years, 9%
Most people assume that the earth has a spherical shape. It is actually more of an ellipsoid shape, but not an exact ellipsoid, since there are numerous mountains and valleys. Researchers have found that a datum, or a reference ellipsoid, that is offset from the center of the earth can be used to
Each function in Exercises represents the rate of flow of money (in dollars per year) over the given time period, compounded continuously at the given annual interest rate. Find the present value in each case.ƒ(t) = 25,000, 12 years, 10%
That if an object falls from rest under the constant acceleration of gravity g, its velocity at time t is given by v(t) = gt and the distance that it has fallen is given by s(t) = 1/2gt2.(a) Show that the average velocity of an object falling from rest under the constant acceleration of gravity g
Each function in Exercises represents the rate of flow of money (in dollars per year) over the given time period, compounded continuously at the given annual interest rate. Find the present value in each case.ƒ(t) = 150e0.04t, 5 years, 6%
An investment produces a perpetual stream of income with a flow rate of R(t) = 1200e0.03t. Find the capital value at an interest rate of 7% compounded continuously.
Each function in Exercises represents the rate of flow of money (in dollars per year) over the given time period, compounded continuously at the given annual interest rate. Find the present value in each case.ƒ(t) = 15t, 18 months, 8%
As we saw in an earlier chapter, a person’s metabolic rate tends to go up after eating a meal and then, after some time has passed, it returns to a resting metabolic rate. This phenomenon is known as the thermic effect of food, and the effect (in kJ per hour) for one individual is F(t) = -10.28 +
The United States has been steadily reducing its production of veal in recent decades. Annual U.S. veal production (in millions of pounds) can be approximated by p(t) = 663e-0.042t, where t is the number of years since 1970. Assuming this trend continues, estimate the total amount of veal produced
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