Determine if the given expression approaches a limit as b → ∞, and find that number when it does. 5 - (b - 1)-1
> Determine if the given expression approaches a limit as b → ∞, and find that number when it does. b2
> Determine if the given expression approaches a limit as b → ∞, and find that number when it does. 5/b
> The capital value of an asset such as a machine is sometimes defined as the present value of all future net earnings. (See Section 9.5.) The actual lifetime of the asset may not be known, and since some assets may last indefinitely, the capital value of
> The capital value of an asset such as a machine is sometimes defined as the present value of all future net earnings. (See Section 9.5.) The actual lifetime of the asset may not be known, and since some assets may last indefinitely, the capital value of
> The capital value of an asset such as a machine is sometimes defined as the present value of all future net earnings. (See Section 9.5.) The actual lifetime of the asset may not be known, and since some assets may last indefinitely, the capital value of
> If k > 0, show that ∫e ∞ k / x(ln x)k+1 dx = 1.
> If k > 0, show that ∫1 ∞ k / xk+1 dx = 1.
> If k > 0, show that ∫0 ∞ k e-kx dx = 1.
> Evaluate the following improper integrals whenever they are convergent. ∫-∞ ∞ e-x/(e-x + 2)2 dx
> Evaluate the following improper integrals whenever they are convergent. ∫0 ∞ e-x / (e-x + 2)2 dx
> Determine the integrals by making appropriate substitutions. ∫ x2/(3 - x3) dx
> Evaluate the following improper integrals whenever they are convergent. ∫-∞ 0 1/(24 – x) dx
> Evaluate the following improper integrals whenever they are convergent. ∫-∞ 0 6/(1 - 3x)2 dx
> Evaluate the following improper integrals whenever they are convergent. ∫-∞ 0 8/(x - 5)2 dx
> Evaluate the following improper integrals whenever they are convergent. ∫-∞ 0 e4x dx
> Evaluate the following improper integrals whenever they are convergent. ∫1 ∞ (5x + 1)-4 dx
> Evaluate the following improper integrals whenever they are convergent. ∫0 ∞ 2x(x2 + 1)-3/2 dx
> Evaluate the following improper integrals whenever they are convergent. ∫0 ∞ x/(x2 + 1) dx
> Evaluate the following improper integrals whenever they are convergent. ∫0 ∞ xe-x2 dx
> Evaluate the following improper integrals whenever they are convergent. ∫2 ∞ 1 / x ln x dx
> Evaluate the following improper integrals whenever they are convergent. ∫3 ∞ x2/ √(x3 – 1) dx
> Determine the integrals by making appropriate substitutions. ∫ ln√x / x dx
> Determine the integrals by making appropriate substitutions. ∫ 2(2x - 1)7 dx
> Evaluate the following improper integrals whenever they are convergent. ∫1 ∞ e-0.2x dx
> Evaluate the following improper integrals whenever they are convergent. ∫0 ∞ 6e1-3x dx
> Evaluate the following improper integrals whenever they are convergent. ∫0 ∞ 4/(2x + 1)3 dx
> Evaluate the following improper integrals whenever they are convergent. ∫0 ∞.01e- 0.01x dx
> Evaluate the following improper integrals whenever they are convergent. ∫2 ∞ e2-x dx
> Evaluate the following improper integrals whenever they are convergent. ∫2 ∞ 1/(x - 1)5/2 dx
> Evaluate the following improper integrals whenever they are convergent. ∫0 ∞ (x2 + 1) dx
> Evaluate the following improper integrals whenever they are convergent. ∫1 ∞ e2x dx
> Evaluate the following improper integrals whenever they are convergent. ∫0 ∞ e-3x dx
> Evaluate the following improper integrals whenever they are convergent. ∫0 1 1/ (2x + 3)2 dx
> Determine the integrals by making appropriate substitutions. ∫ x-2 (1/x + 2)5 dx
> Evaluate the following improper integrals whenever they are convergent. ∫1 ∞ 2/ x3/2 dx
> Evaluate the following improper integrals whenever they are convergent. ∫1 ∞ x3 dx
> Show that the region under the graph of y = (x - 1)-1/3 for x ≥ 2 cannot be assigned any finite number as its area.
> Show that the region under the graph of y = (14x + 18)-4/5 for x ≥ 1 cannot be assigned any finite number as its area. (See Fig. 9.) Figure 9: 16
> Find the area under the graph of y = (2x + 6)-4/3 for x ≥ 1. (See Fig. 8.) Figure 8: 1 16 Y
> Find the area under the graph of y = (x + 1)-3/2 for x ≥ 3.
> Find the area under the graph of y = 4 e-4x for x ≥ 0.
> Find the area under the graph of y = e-x/2 for x ≥ 0.
> Find the area under the graph of y = (x + 1)-2 for x ≥ 0.
> Find the area under the graph of y = 1/x2 for x ≥ 2.
> Determine the integrals by making appropriate substitutions. ∫ (x – 3)/(1 - 6x + x2)2 dx
> Determine if the given expression approaches a limit as b → ∞, and find that number when it does. 2 - e-3b
> Determine if the given expression approaches a limit as b → ∞, and find that number when it does. e-b/2 + 5
> Determine if the given expression approaches a limit as b → ∞, and find that number when it does. 4 (1 - b-3/4)
> Determine if the given expression approaches a limit as b → ∞, and find that number when it does. 5 (b2 + 3)-1
> Find the present value of a continuous stream of income over 4 years if the rate of income is 25 e-0.02t thousand dollars per year at time t and the interest rate is 8%.
> Find the present value of a continuous stream of income over the time from t = 1 to t = 5 years when the interest rate is 10% and the income is produced at the rate of $12,000 per year.
> A continuous stream of income is being produced at the constant rate of $60,000 per year. Find the present value of the income generated during the time from t = 2 to t = 6 years, with a 6% interest rate.
> Find the present value of a continuous stream of income over 5 years when the rate of income is constant at $35,000 per year and the interest rate is 7%.
> Suppose that the population density function for a city is 40 e-0.5t thousand people per square mile. Let P(t) be the total population that lives within t miles of the city center, and let t be a small positive number. (a) Consider the ring about the cit
> Determine the following indefinite integrals: ∫ x2 ln x2 dx
> A volcano erupts and spreads lava in all directions. The density of the deposits at a distance t kilometers from the center is D (t) thousand tons per square kilometer, where D (t) = 11(t2 + 10)-2. Find the tonnage of lava deposited between the distances
> The population density of Philadelphia in 1940 was given by the function 60 e-0.4t. Calculate the number of people who lived within 5 miles of the city center. Sketch the graphs of the population densities for 1900 and 1940 (see Exercise 9) on a common g
> Use the population density from Exercise 9 to calculate the number of people who lived between 3 and 5 miles from the city center. Exercise 9: In 2012, the population density of a city t miles from the city center was 120 e-0.65t thousand people per squ
> In 2012, the population density of a city t miles from the city center was 120 e-0.65t thousand people per square mile. (a) Write a definite integral whose value equals the number of people (in thousands) who lived within 5 miles of the city center. (b)
> Find the present value of a stream of earnings generated over the next 2 years at the rate of 50 + 7t thousand dollars per year at time t assuming a 10% interest rate.
> A growth company is one whose net earnings tend to increase each year. Suppose that the net earnings of a company at time t are being generated at the rate of 30 + 5t million dollars per year. (a) Write a definite integral that gives the present value of
> A continuous stream of income is produced at the rate of 20 e1-0.09 t thousand dollars per year at time t, and invested money earns 6% interest. (a) Write a definite integral that gives the present value of this stream of income over the time from t = 2
> Find the present value of a continuous stream of income over 3 years if the rate of income is 80 e-0.08t thousand dollars per year at time t and the interest rate is 11%.
> Divide the given interval into n subintervals and list the value of Δx and the endpoints a0, a1, …. , an of the subintervals. -1 ≤ x ≤ 2; n = 5
> Divide the given interval into n subintervals and list the value of Δx and the endpoints a0, a1, …. , an of the subintervals. 3 ≤ x ≤ 5; n = 5
> Determine the integrals by making appropriate substitutions. ∫ x/√(x2 + 1) dx
> Consider the definite integral ∫0 1 4/ (1 + x2) dx, which has the value π. Suppose the trapezoidal rule with n = 15 is used to estimate π. Graph the second derivative of the function in the window [0, 1] by [-10, 10], and then use the graph to obtain a
> Consider the definite integral ∫0 1 4/ (1 + x2) dx, which has the value π. Suppose the midpoint rule with n = 20 is used to estimate π. Graph the second derivative of the function in the window [0, 1] by [-10, 10], and then use the graph to obtain a bo
> Approximate the integrals by the midpoint rule, the trapezoidal rule, and Simpson’s rule with n = 10. Then, find the exact value by integration and give the error for each approximation. Express your answers to the full accuracy given by the calculator o
> Approximate the integrals by the midpoint rule, the trapezoidal rule, and Simpson’s rule with n = 10. Then, find the exact value by integration and give the error for each approximation. Express your answers to the full accuracy given by the calculator o
> Approximate the integrals by the midpoint rule, the trapezoidal rule, and Simpson’s rule with n = 10. Then, find the exact value by integration and give the error for each approximation. Express your answers to the full accuracy given by the calculator o
> Approximate the integrals by the midpoint rule, the trapezoidal rule, and Simpson’s rule with n = 10. Then, find the exact value by integration and give the error for each approximation. Express your answers to the full accuracy given by the calculator o
> In Fig. 17 a definite integral of the form ∫a b f (x)dx is approximated by the trapezoidal rule. Determine f (x), a, b, and n. Figure 17: NORMAL FLOAT AUTO REAL RADIAN CL (sin(22)+sum (sea(2sin(X²), X.2.5.5.5..5))+sin(62)).25 Ans
> In Fig. 16 a definite integral of the form ∫a b f (x)dx is approximated by the midpoint rule. Determine f (x), a, b, and n. Figure 16: NORMAL FLOAT AUTO REAL RADIAN CL sum(sea(1/(25-X2), X.3.1.4. 4489645238 4489645238 9..2))*.2 An
> Subdivide the interval 0 ≤ t ≤ 22 into n subintervals of length Δt = 22/n seconds. Let ti be a point in the ith subinterval. (a) Show that (R/60)Δt ≈ [number of liters of blood flowing past the monitoring point during the ith time interval]. (b) Show tha
> (a) Suppose that the graph of f (x) is above the x-axis and concave down on the interval a0 ≤ x ≤ a1. Let x1 be the midpoint of this interval, let Δx = a1 - a0, and construct the line tangent to the gra
> Determine the integrals by making appropriate substitutions. ∫ x4/(x5 + 1) dx
> Approximate the value of ∫a b f (x)dx, where f (x) ≥ 0, by dividing the interval a ≤ x ≤ b into four subintervals and constructing five rectangles. (See Fig. 14.) Note that the w
> (a) Show that the area of the trapezoid in Fig. 13(a) is ½ (h + k) l. (b) Show that the area of the first trapezoid on the left in Fig. 13(b) is ½ [ f (a0) + f (a1)]Δx. (c) Derive the trapezoidal rule for the case
> Consider ∫1 2 f (x)dx, where f (x) = 3 ln x. (a) Make a rough sketch of the graph of the fourth derivative of f (x) for 1 ≤ x ≤ 2. (b) Find a number A such that | f ’’ ’’ (x) | ≤ A for all x satisfying 1 ≤ x ≤ 2. (c) Obtain a bound on the error of using
> Consider ∫0 2 f (x)dx, where f (x) = 1/12 x4 + 3x2. (a) Make a rough sketch of the graph of f ’’(x) for 0 ≤ x ≤ 2. (b) Find a number A such that | f ’’(x) | ≤ A for all x satisfying 0 ≤ x ≤ 2. (c) Obtain a bound on the error of using the midpoint rule wi
> In a drive along a country road, the speedometer readings are recorded each minute during a 5-minute interval. Use the trapezoidal rule to estimate the distance traveled during the 5 minutes. Time (minutes) Velocity (mph) 0 1 2 33 32 28 3 30 4 5 32
> Upon takeoff, the velocity readings of a rocket noted every second for 10 seconds were 0, 30, 75, 115, 155, 200, 250, 300, 360, 420, and 490 feet per second. Use the trapezoidal rule to estimate the distance the rocket traveled during the first 10 second
> To determine the amount of water flowing down a certain 100-yard-wide river, engineers need to know the area of a vertical cross section of the river. Measurements of the depth of the river were made every 20 yards from one bank to the other. The reading
> In a survey of a piece of oceanfront property, measurements of the distance to the water were made every 50 feet along a 200-foot side. (See Fig. 12.) Use the trapezoidal rule to estimate the area of the property. Figure 12: 200' 100' 90' 125' 150'
> The following integrals cannot be evaluated in terms of elementary antiderivatives. Find an approximate value by Simpson’s rule. Express your answers to five decimal places. ∫-1 2 √(1 + x4) dx; n = 4
> The following integrals cannot be evaluated in terms of elementary antiderivatives. Find an approximate value by Simpson’s rule. Express your answers to five decimal places. ∫0 2 √(sin x) dx; n = 5
> Determine the integrals by making appropriate substitutions. ∫ √(ln x)/x dx
> The following integrals cannot be evaluated in terms of elementary antiderivatives. Find an approximate value by Simpson’s rule. Express your answers to five decimal places. ∫0 1 1 / (x3 + 1) dx; n = 2
> The following integrals cannot be evaluated in terms of elementary antiderivatives. Find an approximate value by Simpson’s rule. Express your answers to five decimal places. ∫0 2 √(1 + x3) dx; n = 4
> Approximate the following integrals by the midpoint rule, the trapezoidal rule, and Simpson’s rule. Then, find the exact value by integration. Express your answers to five decimal places. ∫1 5 (4x3 - 3x2) dx; n = 2
> Approximate the following integrals by the midpoint rule, the trapezoidal rule, and Simpson’s rule. Then, find the exact value by integration. Express your answers to five decimal places. ∫2 5 x ex dx; n = 5
> Approximate the following integrals by the midpoint rule, the trapezoidal rule, and Simpson’s rule. Then, find the exact value by integration. Express your answers to five decimal places. ∫0 3 x √(4 – x) dx; n = 5
> Approximate the following integrals by the midpoint rule, the trapezoidal rule, and Simpson’s rule. Then, find the exact value by integration. Express your answers to five decimal places. ∫0 2 2x ex2 dx; n = 4 18.
> Approximate the following integrals by the midpoint rule, the trapezoidal rule, and Simpson’s rule. Then, find the exact value by integration. Express your answers to five decimal places. ∫10 20 ln x / x dx; n = 5
> Approximate the following integrals by the midpoint rule, the trapezoidal rule, and Simpson’s rule. Then, find the exact value by integration. Express your answers to five decimal places. ∫1 4 (2x - 3)3 dx; n = 3
> Approximate the following integrals by the trapezoidal rule; then, find the exact value by integration. Express your answers to five decimal places. ∫- 1 e2x dx; n = 2, 4
> Approximate the following integrals by the trapezoidal rule; then, find the exact value by integration. Express your answers to five decimal places. ∫1 5 1/x2 dx; n = 3
> Determine the integrals by making appropriate substitutions. ∫ ln(2x)/x dx
> Approximate the following integrals by the trapezoidal rule; then, find the exact value by integration. Express your answers to five decimal places. ∫4 9 1 / (x – 3) dx; n = 5
