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mathematics
calculus early transcendentals
Calculus Early Transcendentals 2nd edition William L. Briggs, Lyle Cochran, Bernard Gillett - Solutions
Evaluate the following integrals or state that they diverge. dx -3 (2x + 6)²/3
Evaluate the following integrals or state that they diverge. Г dx Vx – 1 /х —
Evaluate the following integrals or state that they diverge. TT/2 tan 0 do
Evaluate the following integrals or state that they diverge. dx Vx
Find the volume of the described solid of revolution or state that it does not exist.The region bounded by and the x-axis on the interval [0, ∞] is revolved about the x-axis. Vx f(x) Vx? + 1
Find the volume of the described solid of revolution or state that it does not exist.The region bounded by and the x-axis on the interval [2, ∞) is revolved about the x-axis. f(x) Vx In x
Find the volume of the described solid of revolution or state that it does not exist.The region bounded by f(x) = (x + 1)-3 and the x-axis on the interval [0, ∞] is revolved about the y-axis.
Find the volume of the described solid of revolution or state that it does not exist.The region bounded by and the x-axis on the interval [1, ∞] is revolved about the x-axis. f(x)
Find the volume of the described solid of revolution or state that it does not exist.The region bounded by f(x) = (x2 + 1)-1/2 and the x-axis on the interval [2, ∞] is revolved about the x-axis.
Find the volume of the described solid of revolution or state that it does not exist.The region bounded by f(x) = x-2 and the x-axis on the in terval [1, ∞] is revolved about the x-axis.
Evaluate the following integrals or state that they diverge. dy y In y
Evaluate the following integrals or state that they diverge. tan-1 ds s- + 1
Evaluate the following integrals or state that they diverge. dx (x + 2)² 2 8.
Evaluate the following integrals or state that they diverge. TT dz sin z? Z. 8.
Evaluate the following integrals or state that they diverge. .2 3x² + 1 dx x* + x .3 8.
Evaluate the following integrals or state that they diverge. dx x²(x + 1)
Evaluate the following integrals or state that they diverge. 00 dv v(v + 1)
Evaluate the following integrals or state that they diverge. Ve dx, a real -00
Evaluate the following integrals or state that they diverge. el du + 1 ,2u
Evaluate the following integrals or state that they diverge. 00 dx x2 + 2x + 5
Evaluate the following integrals or state that they diverge. cos (7/x) dx x2
Evaluate the following integrals or state that they diverge. cos x dx CO
Evaluate the following integrals or state that they diverge. Г dx хе -00
Evaluate the following integrals or state that they diverge. dp Vp? + 1 8.
Evaluate the following integrals or state that they diverge. dx e2 x Inº x 8.
Evaluate the following integrals or state that they diverge. —ах dх, а > 0
Evaluate the following integrals or state that they diverge. sec? x? dx 4/7 TT
Evaluate the following integrals or state that they diverge. —2х dx
Evaluate the following integrals or state that they diverge. dx V2 - x -0-
Evaluate the following integrals or state that they diverge. dx Vx X.
Evaluate the following integrals or state that they diverge. 2* dx X-
Evaluate the following integrals or state that they diverge. e* dx
Evaluate the following integrals or state that they diverge. dx 3 (х + 1)?
Evaluate the following integrals or state that they diverge. -2 dx
For what values of p doesconverge? dx 00 х
Explain how to evaluate --1/2
Explain how to evaluate L(x) dx. a
What are the two general ways in which an improper integral may occur?
Another Simpson’s Rule formula isUse this rule to estimate using n = 10 subintervals. 2M(n) + T(n) for n 2 1 S(2n) 3. Si 1/x dx
Using the notation of the text, prove that 4T(2n) – T(n) S(2n) for n > 1. 3
Suppose f is positive and its first two derivatives are continuous on [a, b]. If f" is positive on [a, b], then is a Trapezoid Rule estimate of an underestimate or overestimate of the integral? Justify your answer using Theorem 7.2 and an illustration. ГЛО f(x) dx a
Given a Midpoint Rule approximation M(n) and a Trapezoid Rule approximation T(n) for a continuous function on [a, b] with n subintervals, show that T(2n) = (T(n) + M(n))/2.
a. Use Simpson’s Rule to approximateusing two subintervals (n = 2); compare the approximation to the value of the integral.b. Use Simpson’s Rule to approximateusing four subintervals 1n = 42; compare the approximation to the value of the integral.c. Use the error bound associated with
Prove that the Trapezoid Rule is exact (no error) when approximating the definite integral of a linear function.
Refer to Theorem 7.2 and let f(x) = sin ex. a. Find a Trapezoid Rule approximation to using = 40 subintervals.b. Calculate f"(x).c. Explain why |f"(x)| < 6 on [0, 1], given that e < 3. Graph f".d. Find an upper bound on the absolute error in the estimate found in part (a) using
Refer to Theorem 7.2 and let f(x) = ex2. a. Find a Trapezoid Rule approximation tousing n = 50 subintervals.b. Calculate f"(x).c. Explain why |f"(x)| < 18 on [0, 1], given that e < 3.d. Use Theorem 7.2 to find an upper bound on the absolute error in the estimate found in part (a). Soet dx
The figure shows the rate at which U.S. oil was produced and imported between 1920 and 2005 in units of millions of barrels per day. The total amount of oil produced or imported is given by the area of the region under the corresponding curve. Be careful with units because both days and years are
A recent study revealed that the lengths of U.S. movies are normally distributed with a mean of 110 minutes and a standard deviation of 22 minutes. This means that the fraction of movies with lengths between a and b minutes (with a < b) is given by the integralWhat percentage of U.S. movies are
The heights of U.S. men are normally distributed with a mean of 69 inches and a standard deviation of 3 inches. This means that the fraction of men with a height between a and b (with a < b) inches is given by the integralWhat percentage of American men are between 66 and 72 inches tall? Use the
The theory of diffraction produces the sine integral function Use the Midpoint Rule to approximate Si (1) and Si (10). (Recall that Experiment with the number of subintervals until you obtain approximations that have an error less than 10-3. A rule of thumb is that if two successive approximations
The length of an ellipse with axes of length 2a and 2b isUse numerical integration and experiment with different values of n to approximate the length of an ellipse with a = 4 and b = 8. 2T Va² cos² t + b² sin² t dt.
A standard pendulum of length L swinging under only the influence of gravity (no resistance) has a period ofwhere ω2 = g/L, k2 = sin2 (θ0/2), g ≈ 9.8 m/s2 is the acceleration due to gravity, and θ0 is the initial angle from which the pendulum is released (in radians). Use numerical integration
Approximate the following integrals using Simpson’s Rule. Experiment with values of n to ensure that the error is less than 10-3. т sin 6x cos 3x dx
Approximate the following integrals using Simpson’s Rule. Experiment with values of n to ensure that the error is less than 10-3. 2 + V3 In (2 + cos x) dx = 1 In 2
Approximate the following integrals using Simpson’s Rule. Experiment with values of n to ensure that the error is less than 10-3. 2т 4 cos x -dx 5 - 4 cos X 3
Approximate the following integrals using Simpson’s Rule. Experiment with values of n to ensure that the error is less than 10-3. dx 5т (5 + 3 sin x)² 32 ||
Compare the errors in the Midpoint and Trapezoid Rules with n = 4, 8, 16, and 32 subintervals when they are applied to the following integrals (with their exact values given). TT 9. In (5 + 3cosx) dr =D ㅠ In
Compare the errors in the Midpoint and Trapezoid Rules with n = 4, 8, 16, and 32 subintervals when they are applied to the following integrals (with their exact values given).
Compare the errors in the Midpoint and Trapezoid Rules with n = 4, 8, 16, and 32 subintervals when they are applied to the following integrals (with their exact values given). T/2 128 cos' x dx = 315
Compare the errors in the Midpoint and Trapezoid Rules with n = 4, 8, 16, and 32 subintervals when they are applied to the following integrals (with their exact values given). п /2 sin° x dx 5п 32
Determine whether the following statements are true and give an explanation or counterexample.a. The Trapezoid Rule is exact when used to approximate the definite integral of a linear function. b. If the number of subintervals used in the Midpoint Rule is increased by a factor of 3, the error
Apply Simpson’s Rule to the following integrals. It is easiest to obtain the Simpson’s Rule approximations from the Trapezoid Rule approximations. Make a table similar to Table 7.8 showing the approximations and errors for n = 4, 8, 16, and 32. The exact values of the integrals are given for
Apply Simpson’s Rule to the following integrals. It is easiest to obtain the Simpson’s Rule approximations from the Trapezoid Rule approximations. Make a table similar to Table 7.8 showing the approximations and errors for n = 4, 8, 16, and 32. The exact values of the integrals are given for
Apply Simpson’s Rule to the following integrals. It is easiest to obtain the Simpson’s Rule approximations from the Trapezoid Rule approximations. Make a table similar to Table 7.8 showing the approximations and errors for n = 4, 8, 16, and 32. The exact values of the integrals are given for
Apply Simpson’s Rule to the following integrals. It is easiest to obtain the Simpson’s Rule approximations from the Trapezoid Rule approximations. Make a table similar to Table 7.8 showing the approximations and errors for n = 4, 8, 16, and 32. The exact values of the integrals are given for
Consider the following integrals and the given values of n.a. Find the Trapezoid Rule approximations to the integral using n and 2n subintervals.b. Find the Simpson’s Rule approximation to the integral using 2n subintervals. It is easiest to obtain Simpson’s Rule approximations from the
Consider the following integrals and the given values of n.a. Find the Trapezoid Rule approximations to the integral using n and 2n subintervals.b. Find the Simpson’s Rule approximation to the integral using 2n subintervals. It is easiest to obtain Simpson’s Rule approximations from the
Consider the following integrals and the given values of n.a. Find the Trapezoid Rule approximations to the integral using n and 2n subintervals.b. Find the Simpson’s Rule approximation to the integral using 2n subintervals. It is easiest to obtain Simpson’s Rule approximations from the
Consider the following integrals and the given values of n.a. Find the Trapezoid Rule approximations to the integral using n and 2n subintervals.b. Find the Simpson’s Rule approximation to the integral using 2n subintervals. It is easiest to obtain Simpson’s Rule approximations from the
Use the indicated methods to solve the following problems with nonuniform grids.A piece of wood paneling must be cut in the shape shown in the figure. The coordinates of several points on its curved surface are also shown (with units of inches).a. Estimate the surface area of the paneling using the
Use the indicated methods to solve the following problems with nonuniform grids.A hot-air balloon is launched from an elevation of 5400 ft above sea level. As it rises, its vertical velocity is computed using a device (called a variometer) that measures the change in atmospheric pressure. The
Use the indicated methods to solve the following problems with nonuniform grids.The function f is twice differentiable on (-∞, ∞). Values of f at various points on [0, 20] are given in the table.a. Approximatein three ways: using a left Riemann sum, a right Riemann sum, and the Trapezoid
Use the indicated methods to solve the following problems with nonuniform grids.A curling iron is plugged into an outlet at time t = 0. Its temperature T in degrees Fahrenheit, assumed to be a continuous function that is strictly increasing and concave down on 0 ≤ t ≤ 120, is given at various
Hourly temperature data for Boulder, Colorado, San Francisco, California, Nantucket, Massachusetts, and Duluth, Minnesota, over a 12 hr period on the same day of January are shown in the figure. Assume that these data are taken from a continuous temperature function T(t). The average temperature
Hourly temperature data for Boulder, Colorado, San Francisco, California, Nantucket, Massachusetts, and Duluth, Minnesota, over a 12 hr period on the same day of January are shown in the figure. Assume that these data are taken from a continuous temperature function T(t). The average temperature
Hourly temperature data for Boulder, Colorado, San Francisco, California, Nantucket, Massachusetts, and Duluth, Minnesota, over a 12 hr period on the same day of January are shown in the figure. Assume that these data are taken from a continuous temperature function T(t). The average temperature
Hourly temperature data for Boulder, Colorado, San Francisco, California, Nantucket, Massachusetts, and Duluth, Minnesota, over a 12 hr period on the same day of January are shown in the figure. Assume that these data are taken from a continuous temperature function T(t). The average temperature
Apply the Midpoint and Trapezoid Rules to the following integrals. Make a table similar to Table 7.5 showing the approximations and errors for n = 4, 8, 16, and 32. The exact values of the integrals are given for computing the error. -2х dx e`16 -16
Apply the Midpoint and Trapezoid Rules to the following integrals. Make a table similar to Table 7.5 showing the approximations and errors for n = 4, 8, 16, and 32. The exact values of the integrals are given for computing the error. sin x cos 3x dx = 0
Apply the Midpoint and Trapezoid Rules to the following integrals. Make a table similar to Table 7.5 showing the approximations and errors for n = 4, 8, 16, and 32. The exact values of the integrals are given for computing the error. In x dx = 1
Apply the Midpoint and Trapezoid Rules to the following integrals. Make a table similar to Table 7.5 showing the approximations and errors for n = 4, 8, 16, and 32. The exact values of the integrals are given for computing the error. T/4 3 sin 2x dx = ||
Apply the Midpoint and Trapezoid Rules to the following integrals. Make a table similar to Table 7.5 showing the approximations and errors for n = 4, 8, 16, and 32. The exact values of the integrals are given for computing the error. Ге 3 х) dx х = 4 16 -2
Apply the Midpoint and Trapezoid Rules to the following integrals. Make a table similar to Table 7.5 showing the approximations and errors for n = 4, 8, 16, and 32. The exact values of the integrals are given for computing the error. .5 2х) dx %3D 100 (Зr?
Find the Midpoint and Trapezoid Rule approximations tousing n = 50 subintervals. Compute the relative error of each approximation. Soe* dx 0.
Find the Midpoint and Trapezoid Rule approximations to using n = 25 subintervals. Compute the relative error of each approximation. dx Lo sin T
Find the indicated Trapezoid Rule approximations to the following integrals.using n = 8 subintervals X- xP x_a Jo
Find the indicated Trapezoid Rule approximations to the following integrals.using n = 6 subintervals sin Tx dx Jo
Find the indicated Trapezoid Rule approximations to the following integrals.using n = 2, 4, and 8 subintervals Г- х3 dx
Find the indicated Trapezoid Rule approximations to the following integrals.using n = 2, 4, and 8 subintervals 10 2x² dx
Find the indicated Midpoint Rule approximations to the following integrals.using n = 8 subintervals e dx —х
Find the indicated Midpoint Rule approximations to the following integrals.using n = 6 subintervals sin Tx dx Jo
Find the indicated Midpoint Rule approximations to the following integrals.using n = 1, 2, and 4 subintervals dx
Find the indicated Midpoint Rule approximations to the following integrals.using n = 1, 2, and 4 subintervals 10 2x² dx
Compute the absolute and relative errors in using c to approximate x.x = e; c = 2.718
Compute the absolute and relative errors in using c to approximate x.x = e; c = 2.72
Compute the absolute and relative errors in using c to approximate x.x = √2; c = 1.414
Compute the absolute and relative errors in using c to approximate x.x = π; c = 3.14
State how to compute the Simpson’s Rule approximation S(2n) if the Trapezoid Rule approximations T(2n) and T(n) are known.
If the Trapezoid Rule is used on the interval [-1, 9] with n = 5 subintervals, at what x-coordinates is the integrand evaluated?
If the Midpoint Rule is used on the interval [-1, 11] with n = 3 subintervals, at what x-coordinates is the integrand evaluated?
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