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All Textbook Solutions for Multivariable Calculus

21E 22E 23E 24E 25E 26E 27E Evaluate the indefinite integral as a power series. What is the radius of convergence? 28. tan1xxdx Use a power series to approximate the definite integral to six decimal places. 29. 00.3x1+x3dx 30E 31E 32E 33E 34E (a) Show that J0 (the Bessel function of order 0 given in Example 4) satisfies the differential equation x2J0(x)+xJ0(x)+x2J0(x)=0 (b) Evaluate 01J0(x)dx correct to three decimal places.36E 37E 38E Let f(x)=n=1xnn2 Find the intervals of convergence for f, f, and f.(a) Starting with the geometric series n=0xn, find the sum of the series n=1nxn1x1 (b) Find the sum of each of the following series. (i) n=1nxn,x1 (ii) n=1n2n (c) Find the sum of each of the following series. (i) n=2n(n1)xn,x1 (ii) n=2n2n2n (iii) n=1n22n 41E 42E If f(x)=n=0bn(x5)n for all x, write a formula for b8.2E If f(n)(0) = (n + 1)! for n = 0, 1, 2, , find the Maclaurin series for f and its radius of convergence.Find the Taylor series for f centered at 4 if f(n)(4)=(1)nn!3n(n+1) What is the radius of convergence of the Taylor series?Use the definition of a Taylor series to find the first four nonzero terms of the series for f(x) centered at the given value of a. 5. f(x) = xex, a = 0 6E Use the definition of a Taylor series to find the first four nonzero terms of the series for f(x) centered at the given value of a. 7. f(x)=x3, a = 8 8E 9E 10E Find the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that Rn(x) 0.] Also find the associated radius of convergence. 11. f(x) (1 x)2 Find the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that Rn(x) 0.] Also find the associated radius of convergence. 12. f(x) = ln(1 + x)Find the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that Rn(x) 0.] Also find the associated radius of convergence. 13. f(x) = cos x 14E 15E Find the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that Rn(x) 0.] Also find the associated radius of convergence. 16. f(x) = x cos x 17E Find the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that Rn(x) 0.] Also find the associated radius of convergence. 18. f(x) = cosh x 19E 20E Find the Taylor series for f(x) centered at the given value of a. [Assume that f has a power series expansion. Do not show that Rn(x) 0.] Also find the associated radius of convergence. 21. f(x) = ln x, a = 2 22E 23E Find the Taylor series for f(x) centered at the given value of a. [Assume that f has a power series expansion. Do not show that Rn(x) 0.] Also find the associated radius of convergence. 24. f(x) = cos x, a = /2 25E 26E 27E 28E 29E 30E 31E 32E 33E Use the binomial series to expand the function as a power series. State the radius of convergence. 34. (1 x)3/4 Use a Maclaurin series in Table 1 to obtain the Maclaurin series for the given function. 35. f(x) = arctan (x2)36E 37E 38E 39E 40E 41E 42E Use a Maclaurin series in Table 1 to obtain the Maclaurin series for the given function. 43. f(x) = sin2 x [Hint: Use sin2x=12(1cos2x).]44E 45E 46E 47E 48E Use the Maclaurin series for cos x to compute cos 5 correct to five decimal places.Use the Maclaurin series for ex to calculate 1/e10 correct to five decimal places.51E 52E 53E 54E 55E 56E Use series to approximate the definite integral to within the indicated accuracy. 57. 01/2x3arctanxdx (four decimal places)Use series to approximate the definite integral to within the indicated accuracy. 58. 01sin(x4)dx (four decimal places)59E 60E 61E 62E 63E 64E 65E 66E 67E 68E 69E Use multiplication or division of power series to find the first three nonzero terms in the Maclaurin series for each function. 70. y = ex ln(1 + x)71E Use multiplication or division of power series to find the first three nonzero terms in the Maclaurin series for each function. 72. y = ex sin2 x 73E 74E 75E 76E 77E 78E 79E 80E 81E 82E Prove Taylors Inequality for n = 2, that is, prove that if |f(x)| M for |x a| d, then R2(x)M6xa3forxad (a) Show that the function defined by f(x)={e1/x2ifx00ifx=0 is not equal to its Maclaurin series. (b) Graph the function in part (a) and comment on its behavior near the origin.85E 86E (a) Find the Taylor polynomials up to degree 5 for f(x) = sin x centered at a = 0. Graph f and these polynomials on a common screen. (b) Evaluate f and these polynomials at x = /4, /2, and . (c) Comment on how the Taylor polynomials converge to f(x).(a) Find the Taylor polynomials up to degree 3 for f(x) = tan x centered a = 0. Graph f and these polynomials on a common screen. (b) Evaluate f and these polynomials at x = /6, /4, and /3. (c) Comment on how the Taylor polynomials converge to f(x).3E 4E Find the Taylor polynomial T3(x) for the function f centered at the number a. Graph f and T3 on the same screen. 5. f(x) = cos x, a = /2 6E Find the Taylor polynomial T3(x) for the function f centered at the number a. Graph f and T3 on the same screen. 7. f(x) = ln x, a = 1 8E Find the Taylor polynomial T3(x) for the function f centered at the number a. Graph f and T3 on the same screen. 9. f(x) = xe2x, a = 0 10E 13E 14E (a) Approximate f by a Taylor polynomial with degree n at the number a. (b) Use Taylors Inequality to estimate the accuracy of the approximation f(x) Tn(x) when x lies in the given interval. (c) Check your result in part (b) by graphing | Rn(x) |. 15. f(x) = x2/3, a = 1, n = 3, 0.8 x 1.2 16E (a) Approximate f by a Taylor polynomial with degree n at the number a. (b) Use Taylors Inequality to estimate the accuracy of the approximation f(x) Tn(x) when x lies in the given interval. (c) Check your result in part (b) by graphing | Rn(x) |. 17. f(x) = sec x, a = 0, n = 2, 0.2 x 0.2 18E 19E (a) Approximate f by a Taylor polynomial with degree n at the number a. (b) Use Taylors Inequality to estimate the accuracy of the approximation f(x) Tn(x) when x lies in the given interval. (c) Check your result in part (b) by graphing | Rn(x) |. 20. f(x) = x ln x, a = 1, n = 3, 0.5 x 1.5 21E 22E Use the information from Exercise 5 to estimate cos 80 correct to five decimal places. 310 Find the Taylor polynomial T3(x) for the function f centered at the number a. Graph f and T3 on the same screen. 5. f(x) = cos x, a = /2 Use the information from Exercise 16 to estimate sin 38 correct to five decimal places. 1322 (a) Approximate f by a Taylor polynomial with degree n at the number a. (b) Use Taylors Inequality to estimate the accuracy of the approximation f(x) Tn(x) when x lies in the given interval. (c) Check your result in part (b) by graphing | Rn(x) |. 16. f(x) = sin x, a = /6, n = 4, 0 x /3 Use Taylors Inequality to determine the number of terms of the Maclaurin series for ex that should be used to estimate e0.1 to within 0.00001.26E Use the Alternating Series Estimation Theorem or Taylors inequality to estimate the range of values of x for which the given approximation is accurate to within the stated error. Check your answer graphically. 27. sinxxx36(error0.01)Use the Alternating Series Estimation Theorem or Taylors inequality to estimate the range of values of x for which the given approximation is accurate to within the stated error. Check your answer graphically. 28. cosx1x22+x424(error0.005)Use the Alternating Series Estimation Theorem or Taylors inequality to estimate the range of values of x for which the given approximation is accurate to within the stated error. Check your answer graphically. 29. arctanxxx33+x55(error0.05)Suppose you know that f(n)(4)=(1)nn!3n(n+1) and the Taylor series of f centered at 4 converges to f(x) for all x in the interval of convergence. Show that the fifth-degree Taylor polynomial approximates f(5) with error less than 0.0002.31E The resistivity of a conducting wire is the reciprocal of the conductivity and is measured in units of ohm-meters (-m). The resistivity of a given metal depends on the temperature according to the equation (t) = 20 e(t20) where t is the temperature in C. There are tables that list the values of (called the temperature coefficient) and 20 (the resistivity at 20C) for various metals. Except at very low temperatures, the resistivity varies almost linearly with temperature and so it is common to approximate the expression for (t) by its first- or second-degree Taylor polynomial at t = 20. (a) Find expressions for these linear and quadratic approximations. (b) For copper, the tables give = 0.0039/C and 20 = 1.7 108 -m. Graph the resistivity of copper and the linear and quadratic approximations for 250C t 1000C. (c) For what values of t does the linear approximation agree with the exponential expression to within one percent?An electric dipole consists of two electric charges of equal magnitude and opposite sign. If the charges are q and q and are located at a distance d from each other, then the electric field E at the point P in the figure is E=qD2q(D+d)2 By expanding this expression for E as a series in powers of d/D, show that E is approximately proportional to 1/D3 when P is far away from the dipole.34E If a water wave with length L moves with velocity v across a body of water with depth d, as in the figure on page 822, then v2=gL2tanh2dL (a) If the water is deep, show that vgL/(2). (b) If the water is shallow, use the Maclaurin series for tanh to show that vgd. (Thus in shallow water the velocity of a wave tends to be independent of the length of the wave.) (c) Use the Alternating Series Estimation Theorem to show that if L 10d, then the estimate v2 gd is accurate to within 0.014gL.36E 37E 38E 39E (a) What is a convergent sequence? (b) What is a convergent series? (c) What does limn an = 3 mean? (d) What does n=1an=3 mean?2RCC 3RCC 4RCC 5RCC 6RCC 7RCC 8RCC 9RCC 10RCC 11RCC 12RCC 1RQ 2RQ 3RQ 4RQ Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. 5. If cn6n is convergent, then cn(6)n is convergent.Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. 6. If cnxn diverges when x = 6, then it diverges when x = 10.7RQ Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. 8. The Ratio Test can be used to determine whether 1/n! converges.9RQ 10RQ 11RQ 12RQ Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. 13. If f(x)=2xx2+13x3 converges for all x, then f(0) = 2.14RQ 15RQ Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. 16. If {an} is decreasing and an 0 for all n, then {an} is convergent.17RQ 18RQ Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. 19. 0.99999 = 1 20RQ 21RQ 22RQ 1RE 2RE 3RE 4RE 5RE 6RE 7RE 8RE 9RE 10RE 11RE 12RE 13RE 14RE 15RE 16RE 17RE 18RE 19RE 20RE 21RE 22RE 23RE 24RE 25RE 26RE 27RE 28RE 29RE 30RE 31RE 32RE 33RE 34RE 35RE 36RE 37RE 38RE 39RE 40RE 41RE 42RE 43RE 44RE Find the Taylor series of f(x) = sin x at a = /6.46RE 47RE 48RE 49RE 50RE 51RE 52RE 53RE 54RE 55RE 56RE 57RE 58RE 59RE 60RE 61RE 62RE If f(x) = sin(x3), find f(15)(0).A function f is defined by f(x)=limnx2n1x2n+1 Where is f continuous?3P Let {Pn} be a sequence of points determined as in the figure. Thus |AP1| = 1, |PnPn+1| = 2n1, and angle APnPn+1 is a right angle. Find limn PnAPn+1. FIGURE FOR PROBLEM 4 To construct the snowflake curve, start with an equilateral triangle with sides of length 1. Step 1 in the construction is to divide each side into three equal parts, construct an equilateral triangle on the middle part, and then delete the middle part (see the figure). Step 2 is to repeat step 1 for each side of the resulting polygon. This process is repeated at each succeeding step. The snowflake curve is the curve that results from repeating this process indefinitely. (a) Let sn, ln, and pn represent the number of sides, the length of a side, and the total length of the nth approximating curve (the curve obtained alter step n of the construction), respectively. Find formulas for sn, ln, and pn. (b) Show that pn as n . (c) Sum an infinite series to find the area enclosed by the snowflake curve. Note: Parts (b) and (c) show that the snowflake curve is infinitely long but encloses only a finite area. FIGURE FOR PROBLEM 5 Find the sum of the series 1+12+13+14+16+18+19+112+ where the terms are the reciprocals of the positive integers whose only prime factors are 2s and 3s.7P 8P 9P 10P Find the interval of convergence of n=1n3xn and find its sum.Suppose you have a large supply of books, all the same size, and you stack them at the edge of a table, with each book extending farther beyond the edge of the table than the one beneath it. Show that it is possible to do this so that the top book extends entirely beyond the table. In fact, show that the top book can extend any distance at all beyond the edge of the table if the stack is high enough. Use the following method of stacking: The top book extends half its length beyond the second book. The second book extends a quarter of its length beyond the third. The third extends one-sixth of its length beyond the fourth, and so on. (Try it yourself with a deck of cards.) Consider centers of mass. FIGURE FOR PROBLEM 12 13P If p 1. evaluate the expression 1+12P+13P+14P+112P+13P14P+Suppose that circles of equal diameter are packed tightly in n rows inside an equilateral triangle. (The figure illustrates the case n = 4.) If A is the area of the triangle and An is the total area occupied by the n rows of circles, show that limnAnA=23 FIGURE FOR PROBLEM 15 16P If the curve y = ex/10 sin x, x 0, is rotated about the x-axis, the resulting solid looks like an infinite decreasing string of beads. (a) Find the exact volume of the nth bead. (Use either a table of integrals or a computer algebra system.) (b) Find the total volume of the beads.Starting with the vertices P1(0, 1), P2(1, 1), P3(1, 0), P4(0, 0) of a square, we construct further points as shown in the figure: P5 is the midpoint of P1P2, P6 is the midpoint of P2P3, P7 is the midpoint of P3P4, and so on. The polygonal spiral path P1P2P3P4P5P6P7 approaches a point P inside the square. (a) If the coordinates of Pn are (xn, yn), show that 12xn+xn+1+xn+2+xn+3=2 and find a similar equation for the y-coordinates. (b) Find the coordinates of P. FIGURE FOR PROBLEM 18 19P 20P 21P Right-angled triangles are constructed as in the figure. Each triangle has height 1 and its base is the hypotenuse of the preceding triangle. Show that this sequence of triangles makes indefinitely many turns around P by showing that n is a divergent series. FIGURE FOR PROBLEM 22 23P (a) Show that the Maclaurin series of the function f(x)=x1xx2isn=1fnxn (b) where fn is the nth Fibonacci number, that is, f1 = 1, f2 = 1, and fn = fn1 + fn2 for n 3. (Hint: Write x/(1xx2)=c0+c1x+c2x2+ and multiply both sides of this equation by 1 x x2.] (c) By writing f(x) as a sum of partial fractions and thereby obtaining the Maclaurin series in a different way, find an explicit formula for the nth Fibonacci number.Let u=1+x33!+x66!+x99!+v=x+x44!+x77!+x1010!+w=x22!+x55!+x88!+ Show that u3 + v3 + w3 3uvw = 1.Prove that if n 1, the nth partial sum of the harmonic series is not an integer. Hint: Let 2k be the largest power of 2 that is less than or equal to n and let M be the product of all odd integers that are less than or equal to n. Suppose that sn = m, an integer. Then M2ksn = M2km. The right side of this equation is even. Prove that the left side is odd by showing that each of its terms is an even integer, except for the last one.Suppose you start at the origin, move along the x-axis a distance of 4 units in the positive direction, and then move downward a distance of 3 units. What are the coordinates of your position?Sketch the points (1, 5, 3), (0, 2, 3), (3, 0, 2), and (2, 2, 1) on a single set of coordinate axes.Which of the points A(4, 0, 1), B(3, 1, 5), and C(2, 4, 6) is closest to the yz-plane? Which point lies in the xz-plane?4E What does the equation x = 4 represent in 2? What does it represent in 3? Illustrate with sketches.What does the equation y = 3 represent in 3? What does z = 5 represent? What does the pair of equations y = 3, z = 5 represent? In other words, describe the set of points (x, y, z) such that y = 3 and z = 5. Illustrate with a sketch.Describe and sketch the surface in 3 represented by the equation x + y = 2.Describe and sketch the surface in 3 represented by the equation x2 + z2 = 9.Find the lengths of the sides of the triangle PQR. Is it a right triangle? Is it an isosceles triangle? 9. P(3. 2, 3), Q(7.0. 1), R(1, 2, 1)Find the lengths of the sides of the triangle PQR. Is it a right triangle? Is it an isosceles triangle? 10. P(2, 1, 0), Q(4, 1, 1), R(4, 5, 4)Determine whether the points lie on a straight line. (a) A(2, 4, 2), B(3, 7, 2), C(1, 3, 3) (b) D(0,5, 5), E(1, 2, 4), F(3, 4, 2)Find the distance from (4, 2, 6) to each of the following. (a) The xy-plane (b) The yz-plane (c) The xz-plane (d) The x-axis (e) The y-axis (f) The z-axis Find an equation of the sphere with center (3, 2, 5) and radius 4. What is the intersection of this sphere with the yz-plane?Find an equation of the sphere with center (2, 6, 4) and radius 5. Describe its intersection with each of the coordinate planes.Find an equation of the sphere that passes through the point (4, 3, 1) and has center (3, 8, 1).Find an equation of the sphere that passes through the origin and whose center is (1, 2, 3).Show that the equation represents a sphere, and find its center and radius. 17. x2 + y2 + z2 2x - 4y + 8z = 15 18E 19E 20E (a) Prove that the midpoint of the line segment from P1(x1, y1, z1) to P2(x2, y2, z2) is (x1+x22,y1+y22,z1+z22) (b) Find the lengths of the medians of the triangle with vertices A(1, 2, 3), B(2, 0, 5), and C(4, 1, 5).(A median of a triangle is a line segment that joins a vertexto the midpoint of the opposite side.)Find an equation of a sphere if one of its diameters has end points (5, 4, 3) and (1, 6, 9).Find equations of the spheres with center (2, 3, 6) that touch (a) the xy-plane, (b) the yz-plane, (c) the xz-plane.24E 25E 26E Describe in words the region of 3 represented by the equation(s) or inequality. 27. y 8 Describe in words the region of 3 represented by the equation(s) or inequality. 28. z 1 Describe in words the region of 3 represented by the equation(s) or inequality. 29. 0 z 6 30E Describe in words the region of 3 represented by the equation(s) or inequality. 31. x2 + y2 = 4, z = 1 32E Describe in words the region of 3 represented by the equation(s) or inequality. 33. x2 + y2 + z2 = 4

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