Solution Manual for Thomas Calculus, 13th Edition

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SOLUTIONSMANUALMULTIVARIABLEELKABLOCKFRANKPURCELLTHOMAS’CALCULUSTHIRTEENTHEDITIONANDTHOMAS’CALCULUSEARLYTRANSCENDENTALSTHIRTEENTHEDITIONBased on the original work byGeorge B. Thomas, JrMassachusetts Institute of Technologyas revised byMaurice D. WeirNaval Postgraduate SchoolJoel HassUniversity of California, Daviswith the assistance ofChristopher HeilGeorgia Institute of Technology

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Learning ObjectivesL-1LEARNING OBJECTIVESCHAPTER 10. Infinite Sequences and SeriesSection 1. Sequences1. Find terms of a sequence.2. Find the formula for the nth term of a sequence.3. Determine if a sequence is monotonic and bounded.4. Determine if a sequence converges or diverges.5. Find the limit of a sequence, if one exists.6. Find the limit of a recursively defined sequence.7. Solve theory and application problems involving sequences.Section 2. Infinite Series1. Find the formula for the nth partial sum of a series.2. Find the sum of a series, if it converges.3. Express repeating decimals as the ratio of two integers.4. Use thenth-term test for divergence.5. Find the sum of a geometric series and the values for which it converges.6. Solve theory and application problems involving series.Section 3. The Integral Test1. Use the integral test to determine if a series converges or diverges.2. Estimate bounds for the remainder when using the integral test.3. Use the integral test to solve theory and application problems involving series.Section 4. Comparison Tests1. Use the comparison test to determine if a series converges or diverges.2. Use the limit comparison test to determine if a series converges or diverges.3. Use cmparison tests to solve theory and application problems involving series.Section 5. Absolute Convergence; The Ratio and Root Tests1. Use the Ratio Test to determine whether a series converges absolutely or diverges.2. Use the Root Test to determine whether a series converges absolutely or diverges.3. Solve theory problems involving the Root and Ratio Tests.Section 6. Alternating Series, Absolute and Conditional Convergence1. Determine if a series converges absolutely, converges conditionally, or diverges.2. Estimate the error in approximating the sum of an alternating series.3. Determine the number of terms needed to estimate the sum of an alternating series.4. Approximate the sum of an alternating series given a specific magnitude of error.5. Solve theory and application problems involving alternating series.Section 7. Power Series1. Find the radius and interval of convergence of a power series.2. Determine whether a power series diverges, converges conditionally, or converges absolutelyat the endpoints of the interval of convergence.

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L-2Learning Objectives3. Use algebraic operations, term-by-term differentiation, and term-by-term integration to findthe sum of a power series.4. Solve theory and application problems involving power series.Section 8. Taylor and Maclaurin Series1. Find thenth Taylor polynomial for a functionfat a pointx=a.2. Find the Taylor series for a functionfat a pointx=a.3. Find the Maclaurin series for a functionf.4. Find the values ofxfor which a Taylor or Maclaurin series converges absolutely.5. Solve theory problems involving Taylor or Maclaurin series.Section 9. Convergence of Taylor Series1. Use substitution and power series operations to find a Taylor series.2. Show that a Taylor series converges at a given point by estimating the remainder term.3. Estimate the error whenf(x) is approximated by thenth Taylor polynomialPn(x).4. Determine how largenmust be in order that the Taylor polynomialPn(x) approximatef(x) towithin a given accuracy.5. Solve theory and application problems involving Taylor series.Section 10. The Binomial Series and Applications of Taylor Series1. Find terms of a binomial series.2. Find a binomial series.3. Use series to estimate the value of an integral within a specific error.4. Find a polynomial that will approximate a function given by an integral to a given accuracy.5. Use series to evaluate limits that involve indeterminate forms.6. Use algebraic operations and common Taylor series to find the sum of a given series.7. Solve theory and application problems involving Taylor series.8. Use Euler's identity.CHAPTER 11. Parametric Equations and Polar CoordinatesSection 1. Parametrizations of Plane Curves1. Graph a curve given by a parametric equation.2. Find and graph a Cartesian equation corresponding to a given parametric equation.3. Find parametric equations that define a curve or the motion of a particle.4. Graph parametric curves using a software package.Section 2. Calculus with Parametric Curves1. Given a parametric equation, find the parametric formulas fordy/dxandd2y/dx2.2. Find the tangent to a curve given by a parametric equation.3. Find the area enclosed by a parametrically defined curve.4. Find the length of a parametrically defined curve.5. Find the area of a surface of revolution corresponding to a parametrized curve.6. Find the coordinates of the centroid of a region defined by a parametrized curve.7. Solve theory and application problems involving parametric curves.Section 3. Polar Coordinates1. Find all of the polar coordinates of a given point.

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Learning ObjectivesL-32. Write Cartesian coordinates for given polar coordinates.3. Write polar coordinates for given Cartesian coordinates.4. Graph sets of points whose polar coordinates satisfy a given equation or inequality.5. Convert polar equations to Cartesian equations.6. Convert Cartesian equations to polar equations.Section 4. Graphing in Polar Coordinates1. Identify the symmetries of a curve and sketch its graph.2. Find the slope of a curve given in polar coordinates at a given point.3. Graph curves given in polar coordinates.Section 5. Areas and Lengths in Polar Coordinates1. Find the area of a region enclosed by a curve given in polar coordinates.2. Find the length of a curve given in polar coordinates.Section 6. Conic Sections1. Sketch conic section and find quantities related to the conic section, such as vertices, foci,directrix, or asymptotes.2. Find the standard form of a conic equation.3. Solve problems involving shifted conic sections.4. Solve theory and application problems related to conic sections.Section 7. Conics in Polar Coordinates1. Find the eccentricity, foci, and directrix of a conic section.2. Find a standard-form equation in Cartesian coordinates.3. Find the polar equation for a conic section.4. Graph a conic section.CHAPTER 12. Vectors and the Geometry of SpaceSection 1. Three-Dimensional Coordinate Systems1. Describe the set whose coordinates satisfy the given information.2. Find the distance between points.3. Find the center and radius of a sphere.4. Write an equation for a sphere.5. Solve theory and application problems related to points in space.Section 2. Vectors1. Find the component form of a vector.2. Sketch vectors.3. Find sums and scalar multiples of vectors.4. Find the length and direction of a vector.5. Find the midpoint of a line segment.6. Solve theory and application problems involving vectors.Section 3. The Dot Product1. Find the dot product of two vectors.2. Find the angle between two vectors.

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L-4Learning Objectives3. Determine if vectors are orthogonal.4. Find the projection of one vector onto another.5. Solve theory and application problems involving dot products and orthogonal vectors.Section 4. The Cross Product1. Calculate the cross product of two vectors inR3.2. Find the length and direction of a cross product of two vectors.3. Find the area of a triangle or parallelogram in space.4. Compute a triple scalar product of three vectors.5. Find the volume of a parallelepiped.6. Solve theory and application problems related to cross products.Section 5. Lines and Planes in Space1. Find parametrizations for lines and line segments in space.2. Find the equation of a plane.3. Find the distance from a point to a line or a plane.4. Find the line of intersection of two planes and the angle between them.5. Find the point at which a line meets a plane.6. Solve theory and application problems involving lines and planes.Section 6. Cylinders and Quadric Surfaces1. Sketch cylinders and quadric surfaces.2. Solve theory and application problems related to cylinders and quadric surfaces.CHAPTER 13. Vector-Valued Functions and Motion in SpaceSection 1. Curves in Space and Their Tangents1. Find a particle's velocity and acceleration vectors.2. Find the angle between the velocity and acceleration vectors.3. Find parametric equations for the line tangent to a curve.4. Solve theory and application problems involving motion along a curve.Section 2. Integrals of Vector Functions; Projectile Motion1. Integrate vector-valued functions.2. Solve initial value problems.3. Solve applications involing projectile motion.4. Solve theory problems related to integration of vector functions.Section 3. Arc Length in Space1. Find the arc length of a curve.2. Find the unit tangent vector to a curve.3. Solve theory and application problems involving arc length.Section 4. Curvature and Normal Vectors of a Curve1. Find the unit tangent vectorT, the curvature kappa, and the principal unit norm vectorNfor aplane curve.2. Find the unit tangent vectorT, the curvature kappa, and the principal unit norm vectorNfor aspace curve.

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Learning ObjectivesL-53. Solve theory problems involving curvature.Section 5. Tangential and Normal Components of Acceleration1. Find tangential and normal components of acceleration.2. Find the torsion function of a smooth curve.3. Find the TNB frame for a curve.4. Solve theory and application problems involving acceleration.Section 6. Velocity and Acceleration in Polar Coordinates1. Find velocity and acceleration in polar coordinates.2. Solve problems related to Kepler's Laws.CHAPTER 14. Partial DerivativesSection 1. Functions of Several Variables1. Evaluate a function of several variables at specified points.2. Find the domain and range a function of two variables.3. Sketch level curves of a function of two variables, or match level curves with a surface.4. Sketch functions of two variables.5. Sketch level surfaces for a function of three variables.6. Find an equation for a level curve or level surface that passes through a given point.Section 2. Limits and Continuity in Higher Dimensions1. Determine if the limit of a function of several variables exists, and find the limit if it doesexist.2. Determine points of continuity for functions of several variables.3. Use the two-path test to prove the nonexistence of a limit.4. Use the sandwich theorem to find limits.5. Use polar coordinates to find limits.6. Use the epsilon-delta definition of a limit.Section 3. Partial Derivatives1. Calculate first-order partial derivatives.2. Calculate second-order partial derivatives.3. Use the limit definition to compute a partial derivative.4. Use implicit differentiation to find a partial derivative.5. Solve theory and application problems involving partial derivatives or partial differentialequations.Section 4. The Chain Rule1. Use the chain rule with one independent variable.2. Use the chain rule with multiple independent variables.3. Use a branch diagram to write a chain rule formula for a derivative.4. Use implicit differentiation.5. Find partial derivatives at specified points.6. Apply the multi-dimensional chain rule to solve applications.

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L-6Learning ObjectivesSection 5. Directional Derivatives and Gradient Vectors1. Calculate the gradient of a function at a given point.2. Find directional derivatives.3. Find the equation for the tangent line to a level curve and illustrate with a sketch.4. Apply knowledge of gradients and directional derivatives to solve applications.Section 6. Tangent Planes and Differentials1. Find equations for tangent planes and normal lines to a surface.2. Find parametric equations for the line tangent to a curve at a given point.3. Estimate the change in a function of two or three variables.4. Find the linearization of a function of two or three variables.5. Find an upper bound for the error in the linearization.6. Estimate error and sensitivity to change.7. Solve theory and application problems related to tangent planes and differentials.Section 7. Extreme Values and Saddle Points1. Use the first derivative test to find local extrema of a function of two variables.2. Use the second derivative test to find local extrema and saddle points of functions of twovariables.3. Find absolute extrema of a function of two variables.4. Find extreme values on parameterized curves.5. Solve theory and application problems involving extreme values and saddle points.Section 8. Lagrange Multipliers1. Solve applications involving two independent variables with one constraint.2. Solve applications involving three independent variables with one constraint.3. Solve applications involving three independent variables with two constraints.4. Solve theoretical problem involving Lagrange multipliers.Section 9. Taylor's Formula for Two Variables1. Find quadratic and cubic approximations to a function of two variables.Section 10. Partial Derivatives with Constrained Variables1. Find partial derivatives of functions of constrained variables.CHAPTER 15. Multiple IntegralsSection 1. Double and Iterated Integrals over Rectangles1. Evaluate iterated integrals.2. Evaluate double integrals over rectangles.3. Find the volume beneath a surface.Section 2. Double Integrals over General Regions1. Sketch the region of integration.2. Find limits of integration that define a region, and write an iterated integral that gives the areaof a region.3. Evaluate integrals over a region.4. Write an equivalent double integral with the order of integration reversed.

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Learning ObjectivesL-75. Evaluate an integral by reversing the order of integration.6. Find the volume beneath a surface.7. Evaluate an integral over an unbounded region.8. Approximate an integral with a finite sum.9. Solve theoretical and applied problems related to double integrals.Section 3. Area by Double Integration1. Express the area of a region as a double integral and evaluate the integral.2. Sketch the region indicated by the double integral, find the equations of the bounding curves,and evaluate the integral.3. Find the average value of a function over a region.4. Solve theory and application problems related to double integrals.Section 4. Double Integrals in Polar Form1. Describe a region in polar coordinates.2. Change a Cartesian integral to polar form and evaluate.3. Change a polar integral into Cartesian form and evaluate.4. Find the area of a region using a polar double integral.5. Find the average value of a function using a polar integral.6. Solve theory and application problems involving polar integrals.Section 5. Triple Integrals in Rectangular Coordinates1. Evaluate triple integrals.2. Write triple integrals in multiple orders of integration and evaluate.3. Find volumes by using triple integrals.4. Find the average value of a function of three variables.5. Integrate by changing the order of integration.6. Solve theory and application problems involving triple integrals.Section 6. Moments and Centers of Mass1. Find the mass, first moments, center of mass, and moments of intertia for plates of constant orvarying density.2. Find the mass, first moments, center of mass, and moments of intertia for solids of constant orvarying density.3. Solve theory and application problems involving moments and centers of mass.Section 7. Triple Integrals in Cylindrical and Spherical Coordinates1. Evaluate integrals in cylindrical or spherical coordinates.2. Change the order of integration in cylindrical or spherical coordinates.3. Find iterated integrals in cylindrical or spherical coordinates.4. Find the volume of a solid using triple integrals.5. Find the average value of a function over a solid.6. Find the mass, center of mass, or moments of a solid.7. Solve theory and application problems involving triple integrals.Section 8. Substitutions in Multiple Integrals1. Calculate the Jacobian of a transformation and sketch the transformed region.

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L-8Learning Objectives2. Use transformations to evaluate double integrals.3. Use transformations to evaluate triple integrals.7. Solve theory and application problems involving substitutions in multiple integrals.CHAPTER 16. Integration in Vector FieldsSection 1. Line Integrals1. Graph vector equations.2. Evaluate a line integral by finding a smooth parametrization of a curve.3. Find masses and moments for coil springs, wires, and thin rods.Section 2. Vector Fields and Line Integrals: Work, Circulation, and Flux1. Find the gradient field of a function.2. Find a line integral of a vector field over a given curve.3. Find the work done by a force field moving an object over a curve in space.4. Find the flow or circulation around a curve in a velocity field.5. Find the flux across a simple closed plane curve.6. Find a vector field that has given properties.Section 3. Path Independence, Conservative Fields, and Potential Functions1. Determine if a field is conservative.2. Find a potential function for a given field.3. Determine if a differential form is exact.4. Use potential functions to evaluate line integrals.5. Solve theory and application problems related to conservative fields.Section 4. Green's Theorem in the Plane1. Verify that Green's theorem holds for a given field over a given region.2. Find the counterclockwise circulation and outward flux for a given field over a given curve.3. Find the work done by a field in moving a particle along a curve.4. Using Green's Theorem to evaluate line integrals in a plane.5. Calculate areas by using Green's theorem.6. Solve applied problems by using Green's theorem.Section 5. Surfaces and Area1. Find a parametrization of a surface.2. Find the area of a surface.3. Find a tangent plane to a parametrized surface.4. Solve applied problems related to surfaces and area.Section 6. Surface Integrals1. Find the surface integral of a scalar function over a given surface.2. Find the surface integral of a vector field over a given surface.3. Find the flux of a vector field across a given surface.4. Find masses and moments of thin shells.Section 7. Stokes' Theorem1. Find the curl of a vector field.

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Learning ObjectivesL-92. Use Stokes' theorem to find a circulation of a field around a given curve.3. Find the integral of a curl vector field.4. Use Stokes' theorem to calculate the flux of the curl of a field across a given surface.5. Solve applied problems by using Stokes' theorem.Section 8. The Divergence Theorem and a Unified Theory1. Find the divergence of a field.2. Use the divergence theorem to calculate outward flux across the boundary of a given region.3. Solve theory and aplication problems related to divergence.CHAPTER 17. Second-Order Differential EquationsSection 1. Second-Order Linear Equations1. Find the general solution of a second-order linear differential equation.2. Solve initial value problems involving second-order linear differential equations.Section 2. Nonhomogeneous Linear Equations1. Solve differential equations by the method of undetermined coefficients.2. Solve differential equations by the method of variation of parameters.3. Solve initial value problems by using the methods of this section.Section 3. Applications1. Solve applications involving differential equations.Section 4. Euler Equations1. Find the general solution to an Euler equation.2. Solve initial value problems related to Euler equations.Section 5. Power-Series Solutions1. Use power series to find the general solution of a differential equation.

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TABLE OF CONTENTS10Infinite Sequences and Series 70110.1 Sequences 70110.2 Infinite Series 71210.3 The Integral Test 72010.4 Comparison Tests 72810.5 Absolute Convergence; The Ratio and Root Tests 73810.6 Alternating Series and Conditional Convergence 74410.7 Power Series 75210.8 Taylor and Maclaurin Series 76410.9 Convergence of Taylor Series 76910.10 The Binomial Series and Applications of Taylor Series 777Practice Exercises 786Additional and Advanced Exercises 79511Parametric Equations and Polar Coordinates 80111.1 Parametrizations of Plane Curves 80111.2 Calculus with Parametric Curves 80911.3 Polar Coordinates 81911.4 Graphing Polar Coordinate Equations 82511.5 Areas and Lengths in Polar Coordinates 83211.6 Conic Sections 83811.7 Conics in Polar Coordinates 849Practice Exercises 860Additional and Advanced Exercises 87112Vectors and the Geometry of Space 87712.1 Three-Dimensional Coordinate Systems 87712.2 Vectors 88112.3 The Dot Product 88612.4 The Cross Product 89212.5 Lines and Planes in Space 89812.6 Cylinders and Quadric Surfaces 906Practice Exercises 912Additional and Advanced Exercises 92013Vector-Valued Functions and Motion in Space 92713.1 Curves in Space and Their Tangents 92713.2 Integrals of Vector Functions; Projectile Motion 93313.3 Arc Length in Space 94113.4 Curvature and Normal Vectors of a Curve 94513.5 Tangential and Normal Components of Acceleration 952

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13.6 Velocity and Acceleration in Polar Coordinates 959Practice Exercises 962Additional and Advanced Exercises 96914Partial Derivatives 97314.1 Functions of Several Variables 97314.2 Limits and Continuity in Higher Dimensions 98414.3 Partial Derivatives 99014.4 The Chain Rule 99914.5 Directional Derivatives and Gradient Vectors 100814.6 Tangent Planes and Differentials 101414.7 Extreme Values and Saddle Points 102414.8 Lagrange Multipliers 104014.9 Taylor's Formula for Two Variables 105214.10 Partial Derivatives with Constrained Variables 1055Practice Exercises 1059Additional and Advanced Exercises 107615Multiple Integrals 108315.1 Double and Iterated Integrals over Rectangles 108315.2 Double Integrals over General Regions 108615.3 Area by Double Integration 110015.4 Double Integrals in Polar Form 110515.5 Triple Integrals in Rectangular Coordinates 111215.6 Moments and Centers of Mass 111815.7 Triple Integrals in Cylindrical and Spherical Coordinates 112415.8 Substitutions in Multiple Integrals 1134Practice Exercises 1142Additional and Advanced Exercises 114916Integrals and Vector Fields 115516.1 Line Integrals 115516.2 Vector Fields and Line Integrals: Work, Circulation, and Flux 116116.3 Path Independence, Conservative Fields, and Potential Functions 117216.4 Green's Theorem in the Plane 117816.5 Surfaces and Area 118516.6 Surface Integrals 119616.7 Stokes' Theorem 120616.8 The Divergence Theorem and a Unified Theory 1213Practice Exercises 1219Additional and Advanced Exercises 1230

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701CHAPTER 10INFINITE SEQUENCES AND SERIES10.1SEQUENCES1.21 1110,a21 21242,a 21 32393,a 231 44164a 2.111!1,a1122!2,a1133!6,a1144!24a3.2( 1)12 11,a3( 1)124 13,a 4( 1)136 15,a5( 1)148 17a 4.112( 1)1,a 222( 1)3,a 332( 1)1,a 442( 1)3a 5.221122,a2321222,a3421322,a4521422a6.2 11122,a22321242,a33721382,a4415214162a7.11,a312221,a23713242,a371514482,a41531158162,a63632,a127764,a2558128,a5119256,a102310512a8.11,a122,a121336,a1614424,a124155120,a16720,a175040,a1840,320,a19362,880,a1103,628,800a9.12,a2( 1) (2)221,a3( 1) (1)1322,a 412( 1)1424,a 514( 1)1528,a1616,a1732,a 1864,a 19128,a110256a10.12,a 1 ( 2)221,a  2 ( 1)2333,a  2331442,a  1242555,a  163,a 277,a 184,a 299,a 1105a11.11,a21,a3112,a4213,a5325,a68,a713,a821,a934,a1055a12.12,a21,a 132,a 121412,a121251,a 62,a72,a81,a192,a 1102a13.1( 1),nna 1, 2,n14.( 1) ,nna 1, 2,n15.12( 1),nnan 1, 2,n16.12( 1),nnna1, 2,n

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702Chapter 10 Infinite Sequences and Series17.123(2),nnna1, 2,n18.25(1),nnn na1, 2,n19.21,nan1, 2,n20.4,nan1, 2,n21.43,nan1, 2,n22.42,nan1, 2,n23.32!,nnna1, 2,n24.315,nnna1, 2,n25.11 ( 1)2,nna 1, 2,n26.1122( 1)22,nnnna  1, 2,n27.lim2(0.1)2nnconverges(Theorem 5, #4)28.( 1)( 1)limlim11nnnnnnn converges29.1121 221 222limlimlim1nnnnnnn converges30.112211 33limlimnnnnnnn  diverges31.14443851 581limlim5nnnnnnn converges32.2331(3)(2)256limlimlim0nnnnnnnnnnconverges33.2(1)(1)2111limlimlim (1)nnnnnnnnnn  diverges34.132270217044limlimnnnnnnn  diverges35.lim1( 1)nn does not existdiverges36.1lim ( 1)1nnndoes not existdiverges37. 1111112222lim1lim1nnnnnnnconverges38. 1122lim236nnnconverges39.1( 1)21lim0nnnconverges

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Section 10.1 Sequences70340.( 1)122limlim0nnnnnconverges41.1222111limlimlim2nnnnnnnnconverges42.1019(0.9)limlimnnnn  diverges43.11222lim sinsinlimsin1nnnnconverges44.limcos ()lim ()( 1)nnnnnndoes not existdiverges45.sinlim0nnnbecausesin11nnnnconverges by the Sandwich Theorem for sequences46.2sin2lim0nnnbecause2sin1220nnnconverges by the Sandwich Theorem for sequences47.122ln 2limlim0nnnnnconvergesˆ(using l'Hopital's rule)48.23323 ln 33 (ln 3)3 (ln 3)3663limlimlimlimnnnnnnnnnnn  divergesˆ(using l'Hopital's rule)49.211112ln (1)211limlimlimlim0nnnnnnnnnnnnconverges50.122lnln 2limlim1nnnnnnconverges51.1lim 81nnconverges(Theorem 5, #3)52.1lim (0.03)1nnconverges(Theorem 5, #3)53.77lim1nnneconverges(Theorem 5, #5)54.( 1)11lim1lim1nnnnnneconverges(Theorem 5, #5)55.11lim10lim 101 11nnnnnnnconverges(Theorem 5, #3 and #2)

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