Question: Find a power series representation for the

> Find a power series representation for f, and graph f and several partial sums sn(x) on the same screen. What happens as n increases? f(x) = In(1 + x*)

> Use a computer algebra system to evaluate the integral. Compare the answer with the result of using tables. If the answers are not the same, show that they are equivalent. - x² dx

> Use a computer algebra system to evaluate the integral. Compare the answer with the result of using tables. If the answers are not the same, show that they are equivalent. | cos*x dx

> Use a computer algebra system to evaluate the integral. Compare the answer with the result of using tables. If the answers are not the same, show that they are equivalent. dx e*(3e* + 2)

> Use a computer algebra system to evaluate the integral. Compare the answer with the result of using tables. If the answers are not the same, show that they are equivalent. fxVF + 4 dx

> Use a computer algebra system to evaluate the integral. Compare the answer with the result of using tables. If the answers are not the same, show that they are equivalent. Sese'x dx

> Use a computer algebra system to evaluate the integral. Compare the answer with the result of using tables. If the answers are not the same, show that they are equivalent. | sec'x dx

> Verify Formula 53 in the Table of Integrals (a) by differentiation and (b) by using the substitution t = a + bu.

> Find the volume of the solid obtained when the region under the curve y = arcsin x, x > 0, is rotated about the y-axis.

> Find a power series representation for f, and graph f and several partial sums sn(x) on the same screen. What happens as n increases? 2 S(x) – %3D x? + 1

> Use the Table of Integrals on Reference Pages 6–10 to evaluate the integral. sec20 tan´0 do 9 – tan?0

> Use the Table of Integrals on Reference Pages 6–10 to evaluate the integral. x*dx r10 – 2 - 2

> Use the Table of Integrals on Reference Pages 6–10 to evaluate the integral. |e' sin(at – 3) dt

> Use the Table of Integrals on Reference Pages 6–10 to evaluate the integral. 2x – 1 dx

> Use the Table of Integrals on Reference Pages 6–10 to evaluate the integral. dx 2x

> Use the Table of Integrals on Reference Pages 6–10 to evaluate the integral. cos '(x-2)

> Use the Table of Integrals on Reference Pages 6–10 to evaluate the integral. x*e* dx

> Use the Table of Integrals on Reference Pages 6–10 to evaluate the integral. 4 + (In x)² – dx

> Use the Table of Integrals on Reference Pages 6–10 to evaluate the integral. Sx'arcsin(x²) dx

> Use the Table of Integrals on Reference Pages 6–10 to evaluate the integral. ( sec'x dx

> Find a power series representation for the function and determine the radius of convergence. x² + x f(x)- (1 – x)'

> Use the Table of Integrals on Reference Pages 6–10 to evaluate the integral. x'/4x² – xª dx Jo

> Use the Table of Integrals on Reference Pages 6–10 to evaluate the integral. · dx 3 - e2x

> Use the Table of Integrals on Reference Pages 6–10 to evaluate the integral. sin 20 do /5 – sin 0

> Use the Table of Integrals on Reference Pages 6–10 to evaluate the integral. S sin'x cos x In(sin x) dx

> Use the Table of Integrals on Reference Pages 6–10 to evaluate the integral. dx 2x – 3x?

> Use the Table of Integrals on Reference Pages 6–10 to evaluate the integral. S V6 + 4y – 4y² dy

> Use the Table of Integrals on Reference Pages 6–10 to evaluate the integral. dt 21

> Use the Table of Integrals on Reference Pages 6–10 to evaluate the integral. coth(1/y) dy y?

> Use the Table of Integrals on Reference Pages 6–10 to evaluate the integral. x'sinx dx

> Use the Table of Integrals on Reference Pages 6–10 to evaluate the integral. arctan Vr dx

> Find a power series representation for the function and determine the radius of convergence. 1 + x (1 – x)? S(x -

> Use the Table of Integrals on Reference Pages 6–10 to evaluate the integral. fx/2 + x* dx

> Use the Table of Integrals on Reference Pages 6–10 to evaluate the integral. ( cos"0 do

> Use the Table of Integrals on Reference Pages 6–10 to evaluate the integral. /2y² – 3 dy ,2

> Use the Table of Integrals on Reference Pages 6–10 to evaluate the integral. 9x² x? /9х2 + 4 dx

> Use the Table of Integrals on Reference Pages 6–10 to evaluate the integral. w/8 arctan 2x dx

> Use the indicated entry in the Table of Integrals on the Reference Pages to evaluate the integral. tan(Tx/6) dx; entry 69

> Use the indicated entry in the Table of Integrals on the Reference Pages to evaluate the integral. I V - x dx; entry 113

> Use the indicated entry in the Table of Integrals on the Reference Pages to evaluate the integral. w/2 cos 5x cos 2x dx; entry 80

> Show that 1 3 Tn + 2 3 Mn = S2n

> Show that 1 2 (Tn + Mn) = T2n

> Find a power series representation for the function and determine the radius of convergence. 3 f(x) = 2 - x

> Sketch the graph of a continuous function on [0, 2] for which the right endpoint approximation with n = 2 is more accurate than Simpson’s Rule.

> Sketch the graph of a continuous function on [0, 2] for which the Trapezoidal Rule with n = 2 is more accurate than the Midpoint Rule.

> The region bounded by the curve y = 1/(1+e-x) the x- and y-axes, and the line x = 10 is rotated about the x-axis. Use Simpson’s Rule with n = 10 to estimate the volume of the resulting solid.

> The table shows values of a force function f(x), where x is measured in meters and f(x) in newtons. Use Simpson’s Rule to estimate the work done by the force in moving an object a distance of 18 m. 6 9 3 12 15 18 f(x) 9.8 9.1 8.5 8

> Use Simpson’s Rule with n = 8 to estimate the volume of the solid obtained by rotating the region shown in the figure about (a) the x-axis and (b) the y-axis. y. 4 8 10 x 4. 2. 2.

> Shown is the graph of traffic on an Internet service provider’s T1 data line from midnight to 8:00 am. D is the data throughput, measured in megabits per second. Use Simpson’s Rule to estimate the total amount of data

> The table (supplied by San Diego Gas and Electric) gives the power consumption P in megawatts in San Diego County from midnight to 6:00 am on a day in December. Use Simpson’s Rule to estimate the energy used during that time period. (Us

> Water leaked from a tank at a rate of r(t) liters per hour, where the graph of r is as shown. Use Simpson’s Rule to estimate the total amount of water that leaked out during the first 6 hours. 4 2 0 2 4 6 1 (seconds)

> The graph of the acceleration a(t) of a car measured in ft/s2 is shown. Use Simpson’s Rule to estimate the increase in the velocity of the car during the 6-second time interval. 12 8 4 4 6 t(seconds) 2.

> A radar gun was used to record the speed of a runner during the first 5 seconds of a race (see the table). Use Simpson’s Rule to estimate the distance the runner covered during those 5 seconds. t (s) v (m/s) 1 (s) v (m/

> A graph of the temperature in Boston on August 11, 2013, is shown. Use Simpson’s Rule with n = 12 to estimate the average temperature on that day. TA (F) 80 70 60- noon 4 8 4.

> The widths (in meters) of a kidney-shaped swimming pool were measured at 2-meter intervals as indicated in the figure. Use Simpson’s Rule to estimate the area of the pool. 5.6 5.0 6.8 7.2 4.8 4.8 6.2

> Estimate the area under the graph in the figure by using (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson’s Rule, each with n = 6. y4 1 1 3 4 5 6 x 2.

> Find the approximations Tn, Mn, and Sn for n = 6 and 12. Then compute the corresponding errors ET, EM, and ES. (Round your answers to six decimal places. You may wish to use the sum command on a computer algebra system.) What observations can you make? I

> Find a power series representation for the function and determine the radius of convergence. f(x) = x² tan '(x')

> Find the approximations Tn, Mn, and Sn for n = 6 and 12. Then compute the corresponding errors ET, EM, and ES. (Round your answers to six decimal places. You may wish to use the sum command on a computer algebra system.) What observations can you make? I

> Find the approximations Ln, Rn, Tn, and Mn for n = 5, 10, and 20. Then compute the corresponding errors EL, ER, ET, and EM. (Round your answers to six decimal places. You may wish to use the sum command on a computer algebra system.) What observations ca

> Find the approximations Ln, Rn, Tn, and Mn for n = 5, 10, and 20. Then compute the corresponding errors EL, ER, ET, and EM. (Round your answers to six decimal places. You may wish to use the sum command on a computer algebra system.) What observations ca

> The trouble with the error estimates is that it is often very difficult to compute four derivatives and obtain a good upper bound K for / by hand. But computer algebra systems have no problem computing f (4) and graphing it, so we can easily find a value

> (a) Find the approximations T10 and M10 for / (b) Estimate the errors in the approximations of part (a). (c) How large do we have to choose n so that the approximations Tn and Mn to the integral in part (a) are accurate to within 0.0001?

> (a) Find the approximations T8 and M8 for the integral / (b) Estimate the errors in the approximations of part (a). (c) How large do we have to choose n so that the approximations Tn and Mn to the integral in part (a) are accurate to within 0.0001?

> Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson’s Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.) ( Va + x* dx, n n = 10 п

> Find a power series representation for the function and determine the radius of convergence. f(x) = In(5 – x)

> Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson’s Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.) L'In(1 'In(1 + e*) dx, n=8

> Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson’s Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.) sin t dt, n = 4

> Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson’s Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.) x? dx, n = 10 O 1 + x* 1+3

> Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson’s Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.) dt, n=10 J2 In t

> Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson’s Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.) Vy cos y dy, n= 8

> Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson’s Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.) S el 1/x dx, n п — 8 %3D

> Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson’s Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.) x'sin x dx, n= 8 Jo

> Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson’s Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.) V1 n= 4 Jo + cos x dx,

> Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson’s Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.) dx, п — 10 Jo 1 + x²

> Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson’s Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.) dx, п — 8 Jo 1 + x6

> (a) Use Equation 1 to find a power series representation for f(x) = ln(1 - x). What is the radius of convergence? (b) Use part (a) to find a power series for f(x) = x ln(1 – x). (c) By putting / in your result from part (a), express ln 2 as the sum of an

> (a) Approximate f by a Taylor polynomial with degree n at the number a. (b) Use Taylor’s Inequality to estimate the accuracy of the approximation / when x lies in the given interval. (c) Check your result in part (b) by graphing / S

> Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson’s Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.) x - I dх, п — – 1 dx,

> Use (a) the Midpoint Rule and (b) Simpson’s Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.) Compare your results to the actual value to determine the error in each approx

> Use (a) the Midpoint Rule and (b) Simpson’s Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.) Compare your results to the actual value to determine the error in each approx

> Draw the graph of f(x) = sin( 1 2 ) x2 in the viewing rectangle / (a) Use the graph to decide whether L2, R2, M2, and T2 underestimate or overestimate I. (b) For any value of n, list the numbers Ln, Rn, Mn, Tn, and I in increasing order. (c) Compute L5,

> The left, right, Trapezoidal, and Midpoint Rule approximations were used to estimate / where f is the function whose graph is shown. The estimates were 0.7811, 0.8675, 0.8632, and 0.9540, and the same number of subintervals were used in each case. (a) Wh

> Show that if a > -1 and b > a + 1, then the following integral is convergent. x" 1+x*

> Find the value of the constant C for which the integral converges. Evaluate the integral for this value of C. dx ² + 13x + 1 .2

> (a) Use differentiation to find a power series representation for What is the radius of convergence? (b) Use part (a) to find a power series for (c) Use part (b) to find a power series for 1 S(x) (1 + x)* 1 S(x) (1 + x)' x? S(x)- (1 + x)'

> Find the value of the constant C for which the integral converges. Evaluate the integral for this value of C. 1 C dx x + 2 x² + 4

> Show that 0 ≤ f(t) ≤ Meat for t ≥ 0, where M and a are constants, then the Laplace transform F(s) exists for s > a.

> If f(t) is continuous for t > 0, the Laplace transform of f is the function F defined by and the domain of F is the set consisting of all numbers s for which the integral converges. Find the Laplace transforms of the following functions. F(s) = [

> Determine how large the number a has to be so that 1 -dx < 0.001 x? + 1

> Dialysis treatment removes urea and other waste products from a patient&acirc;&#128;&#153;s blood by diverting some of the blood flow externally through a machine called a dialyzer. The rate at which urea is removed from the blood (in mg/min) is often we

> Express the function as the sum of a power series by first using partial fractions. Find the interval of convergence. 2x + 3 f(x) = x? + 3x + 2

> In a study of the spread of illicit drug use from an enthusiastic user to a population of N users, the authors model the number of expected new users by the equation where c, k and are positive constants. Evaluate this integral to express in terms of c

> As we saw in Section 3.8, a radioactive substance decays exponentially: The mass at time t is / where m(0) is the initial mass and k is a negative constant. The mean life M of an atom in the substance is For the radioactive carbon isotope, 14C, used in r

> A manufacturer of lightbulbs wants to produce bulbs that last about 700 hours but, of course, some bulbs burn out faster than others. Let F(t) be the fraction of the company’s bulbs that burn out before t hours, so F(t) always lies between 0 and 1. (a) M

> Astronomers use a technique called stellar stereography to determine the density of stars in a star cluster from the observed (two-dimensional) density that can be analyzed from a photograph. Suppose that in a spherical cluster of radius R the density of

> Find the escape velocity v0 that is needed to propel a rocket of mass m out of the gravitational field of a planet with mass M and radius R. Use Newton’s Law of Gravitation and the fact that the initial kinetic energy of / supplies the needed work.

> Use the information and data in Exercise 6.4.33 to find the work required to propel a 1000-kg space vehicle out of the earth&acirc;&#128;&#153;s gravitational field. Exercise 6.4.33: (a) Newton&acirc;&#128;&#153;s Law of Gravitation states that two bod

> The average speed of molecules in an ideal gas is where M is the molecular weight of the gas, R is the gas constant, T is the gas temperature, and v is the molecular speed. Show that 3/2 v’e Mv³/(2RT") 4 M dv IT 2RT 8RT v = V TM

> Find the values of p for which the integral converges and evaluate the integral for those values of p. (x' In x dx