CALCULUS Early TranscEndEnTals EighTh EdiTion JamEs sTEwarT M C Master University and University of toronto Australia • Brazil • Mexico • Singapore • United Kingdom • United States Copyright 2016 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it This is an electronic version of the print textbook Due to electronic rights restrictions, some third party content may be suppressed Editorial review has deemed that any suppressed content does not materially affect the overall learning experience The publisher reserves the right to remove content from this title at any time if subsequent rights restrictions 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Contents PrEfacE xi To ThE sTudEnT xxiii calculaTors, comPuTErs, and oThEr graPhing dEvicEs diagnosTic TEsTs xxiv xxvi a Preview of calculus 1 1.1 1.2 1.3 1.4 1.5 Four Ways to Represent a Function 10 Mathematical Models: A Catalog of Essential Functions 23 New Functions from Old Functions 36 Exponential Functions 45 Inverse Functions and Logarithms 55 Review 68 Principles of Problem Solving 71 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 The Tangent and Velocity Problems 78 The Limit of a Function 83 Calculating Limits Using the Limit Laws 95 The Precise Definition of a Limit 104 Continuity 114 Limits at Infinity; Horizontal Asymptotes 126 Derivatives and Rates of Change 140 Writing Project • Early Methods for Finding Tangents 152 The Derivative as a Function 152 Review 165 Problems Plus 169 iii Copyright 2016 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it iv Contents 3.1 Derivatives of Polynomials and Exponential Functions 172 Applied Project • Building a Better Roller Coaster 182 3.2 The Product and Quotient Rules 183 3.3 Derivatives of Trigonometric Functions 190 3.4 The Chain Rule 197 Applied Project • Where Should a Pilot Start Descent? 208 3.5 Implicit Differentiation 208 Laboratory Project • Families of Implicit Curves 217 3.6 Derivatives of Logarithmic Functions 218 3.7 Rates of Change in the Natural and Social Sciences 224 3.8 Exponential Growth and Decay 237 Applied Project • Controlling Red Blood Cell Loss During Surgery 244 3.9 Related Rates 245 3.10 Linear Approximations and Differentials 251 Laboratory Project • Taylor Polynomials 258 3.11 Hyperbolic Functions 259 Review 266 Problems Plus 270 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 Maximum and Minimum Values 276 Applied Project • The Calculus of Rainbows 285 The Mean Value Theorem 287 How Derivatives Affect the Shape of a Graph 293 Indeterminate Forms and l’Hospital’s Rule 304 Writing Project • The Origins of l’Hospital’s Rule 314 Summary of Curve Sketching 315 Graphing with Calculus and Calculators 323 Optimization Problems 330 Applied Project • The Shape of a Can 343 Applied Project • Planes and Birds: Minimizing Energy 344 Newton’s Method 345 Antiderivatives 350 Review 358 Problems Plus 363 Copyright 2016 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Contents 5.1 5.2 5.3 5.4 5.5 Areas and Distances 366 The Definite Integral 378 Discovery Project • Area Functions 391 The Fundamental Theorem of Calculus 392 Indefinite Integrals and the Net Change Theorem 402 Writing Project • Newton, Leibniz, and the Invention of Calculus 411 The Substitution Rule 412 Review 421 Problems Plus 425 6.1 6.2 6.3 6.4 6.5 Areas Between Curves 428 Applied Project • The Gini Index 436 Volumes 438 Volumes by Cylindrical Shells 449 Work 455 Average Value of a Function 461 Applied Project • Calculus and Baseball 464 Applied Project • Where to Sit at the Movies 465 Review 466 Problems Plus 468 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 Integration by Parts 472 Trigonometric Integrals 479 Trigonometric Substitution 486 Integration of Rational Functions by Partial Fractions 493 Strategy for Integration 503 Integration Using Tables and Computer Algebra Systems 508 Discovery Project • Patterns in Integrals 513 Approximate Integration 514 Improper Integrals 527 Review 537 Problems Plus 540 Copyright 2016 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it v vi Contents 8.1 8.2 8.3 8.4 8.5 Arc Length 544 Discovery Project • Arc Length Contest 550 Area of a Surface of Revolution 551 Discovery Project • Rotating on a Slant 557 Applications to Physics and Engineering 558 Discovery Project • Complementary Coffee Cups 568 Applications to Economics and Biology 569 Probability 573 Review 581 Problems Plus 583 9.1 9.2 9.3 9.4 9.5 9.6 Modeling with Differential Equations 586 Direction Fields and Euler’s Method 591 Separable Equations 599 Applied Project • How Fast Does a Tank Drain? 608 Applied Project • Which Is Faster, Going Up or Coming Down? 609 Models for Population Growth 610 Linear Equations 620 Predator-Prey Systems 627 Review 634 Problems Plus 637 10 10.1 10.2 10.3 10.4 Curves Defined by Parametric Equations 640 Laboratory Project • Running Circles Around Circles 648 Calculus with Parametric Curves 649 Laboratory Project • Bézier Curves 657 Polar Coordinates 658 Laboratory Project • Families of Polar Curves 668 Areas and Lengths in Polar Coordinates 669 Copyright 2016 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Contents 10.5 10.6 Conic Sections 674 Conic Sections in Polar Coordinates 682 Review 689 Problems Plus 692 11 11.1 Sequences 694 Laboratory Project • Logistic Sequences 707 11.2 Series 707 11.3 The Integral Test and Estimates of Sums 719 11.4 The Comparison Tests 727 11.5 Alternating Series 732 11.6 Absolute Convergence and the Ratio and Root Tests 737 11.7 Strategy for Testing Series 744 11.8 Power Series 746 11.9 Representations of Functions as Power Series 752 11.10 Taylor and Maclaurin Series 759 Laboratory Project • An Elusive Limit 773 Writing Project • How Newton Discovered the Binomial Series 773 11.11 Applications of Taylor Polynomials 774 Applied Project • Radiation from the Stars 783 Review 784 Problems Plus 787 12 12.1 12.2 12.3 12.4 12.5 12.6 Three-Dimensional Coordinate Systems 792 Vectors 798 The Dot Product 807 The Cross Product 814 Discovery Project • The Geometry of a Tetrahedron 823 Equations of Lines and Planes 823 Laboratory Project • Putting 3D in Perspective 833 Cylinders and Quadric Surfaces 834 Review 841 Problems Plus 844 Copyright 2016 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it vii viii Contents 13 13.1 13.2 13.3 13.4 Vector Functions and Space Curves 848 Derivatives and Integrals of Vector Functions 855 Arc Length and Curvature 861 Motion in Space: Velocity and Acceleration 870 Applied Project • Kepler’s Laws 880 Review 881 Problems Plus 884 14 14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8 Functions of Several Variables 888 Limits and Continuity 903 Partial Derivatives 911 Tangent Planes and Linear Approximations 927 Applied Project • The Speedo LZR Racer 936 The Chain Rule 937 Directional Derivatives and the Gradient Vector 946 Maximum and Minimum Values 959 Applied Project • Designing a Dumpster 970 Discovery Project • Quadratic Approximations and Critical Points 970 Lagrange Multipliers 971 Applied Project • Rocket Science 979 Applied Project • Hydro-Turbine Optimization 980 Review 981 Problems Plus 985 15 15.1 15.2 15.3 15.4 15.5 Double Integrals over Rectangles 988 Double Integrals over General Regions 1001 Double Integrals in Polar Coordinates 1010 Applications of Double Integrals 1016 Surface Area 1026 Copyright 2016 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it chaPTEr 14 ConCept CheCk answers Cut here and keep for reference (a) What is a function of two variables? A function f of two variables is a rule that assigns to each ordered pair sx, yd of real numbers in its domain a unique real number denoted by f sx, yd (b) Describe three methods for visualizing a function of two variables One way to visualize a function of two variables is by graphing it, resulting in the surface z − f sx, yd Another method is a contour map, consisting of level curves f sx, yd − k (k a constant), which are horizontal traces of the graph of the function projected onto the xy-plane Also, we can use an arrow diagram such as the one below z y f (x, y) (x, y) 0 x (a, b) D f (a, b) What is a function of three variables? How can you visualize such a function? A function f of three variables is a rule that assigns to each ordered triple sx, y, zd in its domain a unique real number f sx, y, zd We can visualize a function of three variables by examining its level surfaces f sx, y, zd − k, where k is a constant What does lim sx, yd l sa, bd f sx, yd − L mean? How can you show that such a limit does not exist? lim sx, yd l sa, bd f sx, yd − L means that the values of f sx, yd approach the number L as the point sx, yd approaches the point sa, bd along any path that is within the domain of f We can show that a limit at a point does not exist by inding two different paths approaching the point along which f sx, yd has different limits (a) What does it mean to say that f is continuous at sa, bd? A function f of two variables is continuous at sa, bd if lim sx, yd l sa, bd f sx, yd − f sa, bd (b) If f is continuous on R2, what can you say about its graph? If f is continuous on R2, its graph will appear as a surface without holes or breaks (a) Write expressions for the partial derivatives fx sa, bd and fy sa, bd as limits fx sa, bd − lim f sa h, bd f sa, bd h fy sa, bd − lim f sa, b hd f sa, bd h hl0 hl0 (b) How you interpret fx sa, bd and fy sa, bd geometrically? How you interpret them as rates of change? If f sa, bd − c, then the point Psa, b, cd lies on the surface S given by z − f sx, yd We can interpret fx sa, bd as the slope of the tangent line at P to the curve of intersection of the vertical plane y − b and S In other words, if we restrict ourselves to the path along S through P that is parallel to the xz-plane, then fx sa, bd is the slope at P looking in the positive x-direction Similarly, fy sa, bd is the slope of the tangent line at P to the curve of intersection of the vertical plane x − a and S If z − f sx, yd, then fx sx, yd can be interpreted as the rate of change of z with respect to x when y is ixed Thus fx sa, bd is the rate of change of z (with respect to x) when y is ixed at b and x is allowed to vary from a Similarly, fy sa, bd is the rate of change of z (with respect to y) when x is ixed at a and y is allowed to vary from b (c) If f sx, yd is given by a formula, how you calculate fx and fy ? To ind f x, regard y as a constant and differentiate f sx, yd with respect to x To ind f y , regard x as a constant and differentiate f sx, yd with respect to y What does Clairaut’s Theorem say? If f is a function of two variables that is deined on a disk D containing the point sa, bd and the functions f xy and f yx are both continuous on D, then Clairaut’s Theorem states that fx y sa, bd − fyx sa, bd How you ind a tangent plane to each of the following types of surfaces? (a) A graph of a function of two variables, z − f sx, yd If f has continuous partial derivatives, an equation of the tangent plane to the surface z − f sx, yd at the point sx 0, y 0, z0d is z z − f x sx 0, y 0dsx x 0d f y sx 0, y 0ds y y 0d (b) A level surface of a function of three variables, Fsx, y, zd − k The tangent plane to the level surface Fsx, y, zd − k at Psx 0, y 0, z 0d is the plane that passes through P and has normal vector =Fsx 0, y 0, z 0d: Fx sx 0, y 0, z0dsx x 0d Fy sx 0, y 0, z 0ds y y 0d Fz sx 0, y 0, z 0dsz z 0d − (continued) Copyright 2016 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it chaPTEr 14 ConCept CheCk answers (continued) Deine the linearization of f at sa, bd What is the corresponding linear approximation? What is the geometric interpretation of the linear approximation? The linearization of f at sa, bd is the linear function whose graph is the tangent plane to the surface z − f sx, yd at the point sa, b, f sa, bdd: 12 If z is deined implicitly as a function of x and y by an equation of the form Fsx, y, zd − 0, how you ind −zy−x and −zy−y? If F is differentiable and −Fy−z ± 0, then −F −z −x −2 −x −F −z Lsx, yd − f sa, bd f x sa, bdsx ad f y sa, bds y bd The linear approximation of f at sa, bd is f sx, yd < f sa, bd f xsa, bdsx ad f ysa, bds y bd Geometrically, the linear approximation says that function values f sx, yd can be approximated by values Lsx, yd from the tangent plane to the graph of f at sa, b, f sa, bdd when sx, yd is near sa, bd 13 (a) Write an expression as a limit for the directional derivative of f at sx , y0 d in the direction of a unit vector u − k a, b l How you interpret it as a rate? How you interpret it geometrically? The directional derivative of f at sx , y0 d in the direction of u is (a) What does it mean to say that f is differentiable at sa, bd? If z − f sx, yd, then f is differentiable at sa, bd if Dz can be expressed in the form Dz − f x sa, bd Dx f y sa, bd Dy «1 Dx « Dy where «1 and « l as sDx, Dyd l s0, 0d In other words, a differentiable function is one for which the linear approximation as stated above is a good approximation when sx, yd is near sa, bd (b) How you usually verify that f is differentiable? If the partial derivatives f x and f y exist near sa, bd and are continuous at sa, bd, then f is differentiable at sa, bd 10 If z − f sx, yd, what are the differentials dx, dy, and dz? The differentials dx and dy are independent variables that can be given any values If f is differentiable, the differential dz is then deined by dz − f x sx, yd dx f y sx, yd dy 11 State the Chain Rule for the case where z − f sx, yd and x and y are functions of one variable What if x and y are functions of two variables? Suppose that z − f sx, yd is a differentiable function of x and y, where x − tstd and y − hstd are both differentiable functions of t Then z is a differentiable function of t and Du f sx , y0 d − lim hl0 −z −z −x −z −y − −s −x −s −y −s −z −z −x −z −y − −t −x −t −y −t f sx ha, y0 hbd f sx , y0 d h if this limit exists We can interpret it as the rate of change of f (with respect to distance) at sx , y0 d in the direction of u Geometrically, if P is the point sx , y0, f sx 0, y0 dd on the graph of f and C is the curve of intersection of the graph of f with the vertical plane that passes through P in the direction of u, then Du f sx , y0 d is the slope of the tangent line to C at P (b) If f is differentiable, write an expression for Du f sx , y0 d in terms of fx and fy Du f sx , y0 d − fx sx , y0 d a fy sx , y0 d b 14 (a) Deine the gradient vector = f for a function f of two or three variables If f is a function of two variables, then = f sx, yd − k fx sx, yd, fy sx, yd l − −f −f i1 j −x −y For a function f of three variables, = f sx, y, zd − k fx sx, y, zd, fy sx, y, zd, fzsx, y, zd l dz −f dx −f dy − dt −x dt −y dt If z − f sx, yd is a differentiable function of x and y, where x − tss, td and y − hss, td are differentiable functions of s and t, then −F −z −y −2 −y −F −z − −f −f −f i1 j1 k −x −y −z (b) Express Du f in terms of = f Du f sx, yd − = f sx, yd ؒ u or Du f sx, y, zd − = f sx, y, zd ؒ u (continued) Copyright 2016 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it chaPTEr 14 ConCept CheCk answers (continued) Cut here and keep for reference (c) Explain the geometric signiicance of the gradient The gradient vector of f gives the direction of maximum rate of increase of f On the graph of z − f sx, yd, =f points in the direction of steepest ascent Also, the gradient vector is perpendicular to the level curves or level surfaces of a function 15 What the following statements mean? f has a local maximum at sa, bd if f sx, yd < f sa, bd when sx, yd is near sa, bd (b) f has an absolute maximum at sa, bd f has an absolute maximum at sa, bd if f sx, yd < f sa, bd for all points sx, yd in the domain of f (c) f has a local minimum at sa, bd f has a local minimum at sa, bd if f sx, yd > f sa, bd when sx, yd is near sa, bd (d) f has an absolute minimum at sa, bd (e) f has a saddle point at sa, bd f has a saddle point at sa, bd if f sa, bd is a local maximum in one direction but a local minimum in another 16 (a) If f has a local maximum at sa, bd, what can you say about its partial derivatives at sa, bd? If f has a local maximum at sa, bd and the irst-order partial derivatives of f exist there, then fx sa, bd − and fy sa, bd − (b) What is a critical point of f ? A critical point of f is a point sa, bd such that fx sa, bd − and fy sa, bd − or one of these partial derivatives does not exist 17 State the Second Derivatives Test Suppose the second partial derivatives of f are continuous on a disk with center sa, bd, and suppose that fx sa, bd − and fy sa, bd − [that is, sa, bd is a critical point of f ] Let ■ ■ A bounded set is one that is contained within some disk In other words, it is inite in extent If f is continuous on a closed, bounded set D in R 2, then f attains an absolute maximum value f sx 1, y1d and an absolute minimum value f sx , y2 d at some points sx 1, y1d and sx , y2 d in D (c) How you ind the values that the Extreme Value Theorem guarantees? ■ Find the values of f at the critical points of f in D ■ Find the extreme values of f on the boundary of D ■ f has an absolute minimum at sa, bd if f sx, yd > f sa, bd for all points sx, yd in the domain of f ■ A closed set in R is one that contains all its boundary points If one or more points on the boundary curve are omitted, the set is not closed (b) State the Extreme Value Theorem for functions of two variables (a) f has a local maximum at sa, bd D − Dsa, bd − fxx sa, bd fyy sa, bd f fx y sa, bdg 18 (a) What is a closed set in R 2? What is a bounded set? If D and fxx sa, bd 0, then f sa, bd is a local minimum The largest of the values from the above steps is the absolute maximum value; the smallest of these values is the absolute minimum value 19 Explain how the method of Lagrange multipliers works in inding the extreme values of f sx, y, zd subject to the constraint tsx, y, zd − k What if there is a second constraint hsx, y, zd − c? To ind the maximum and minimum values of f sx, y, zd subject to the constraint tsx, y, zd − k [assuming that these extreme values exist and =t ± on the surface tsx, y, zd − k], we irst ind all values of x, y, z, and where = f sx, y, zd − =tsx, y, zd and tsx, y, zd − k (Thus we are inding the points from the constraint where the gradient vectors =f and =t are parallel.) Evaluate f at all the resulting points s x, y, zd; the largest of these values is the maximum value of f, and the smallest is the minimum value of f If there is a second constraint hsx, y, zd − c, then we ind all values of x, y, z, , and such that = f sx, y, zd − =tsx, y, zd =hsx, y, zd Again we ind the extreme values of f by evaluating f at the resulting points s x, y, zd If D and fxx sa, bd , 0, then f sa, bd is a local maximum If D , 0, then f sa, bd is not a local maximum or minimum The point sa, bd is a saddle point of f Copyright 2016 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Copyright 2016 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Cut here and keep for reference chaPTEr 15 ConCept CheCk answers Suppose f is a continuous function deined on a rectangle R − fa, bg fc, d g (a) Write an expression for a double Riemann sum of f If f sx, yd > 0, what does the sum represent? A double Riemann sum of f is m n o o f sx ij*, yij*d DA i−1 j−1 (b) Write the deinition of yyR f sx, yd dA as a limit yy m n oo m, n l ` i−1 j−1 R yy f sx, yd dA − yy Fsx, yd dA D f sx ij*, yij*d DA (c) What is the geometric interpretation of yyR f sx, yd dA if f sx, yd > 0? What if f takes on both positive and negative values? If f sx, yd > 0, yyR f sx, yd dA represents the volume of the solid that lies above the rectangle R and below the surface z − f sx, yd If f takes on both positive and negative values, then yyR f sx, yd dA is V1 V2, where V1 is the volume above R and below the surface z − f sx, yd, and V2 is the volume below R and above the surface (d) How you evaluate yyR f sx, yd dA? (b) What is a type I region? How you evaluate yyD f sx, yd dA if D is a type I region? D − hsx, yd yy f sx, yd dA − y y a d c f sx, yd dy dx − y c y b a f sx, yd dx dy D − hsx, yd yy f sx, yd dA − y y d c D − yy f sx, yd dA yy tsx, yd dA D ■ yy c f sx, yd dA − c yy f sx, yd dA ■ D If f sx, yd > tsx, yd for all sx, yd in D, then yy f sx, yd dA > yy tsx, yd dA D ■ yy f sx, yd if sx, yd is in D if sx, yd is in R but not in D f sx, yd dA − ■ yy f sx, yd dA D1 D where AsRd is the area of R H D If D − D1 ø D2, where D1 and D2 don’t overlap except perhaps on their boundaries, then R Since D is bounded, it can be enclosed in a rectangular region R We deine a new function F with domain R by D where c is a constant yy f sx, yd dA (a) How you deine yyD f sx, yd dA if D is a bounded region that is not a rectangle? f sx, yd dx dy yy f f sx, yd tsx, ydg dA o o f sxi , yj d DA, where the sample AsRd h 2syd h1syd D points sxi , yjd are the centers of the subrectangles fave − h1s yd < x < h2s ydj (d) What properties double integrals have? i−1 j−1 Fsx, yd − | c < y < d, where h1 and h2 are continuous on fc, dg Then n (f ) Write an expression for the average value of f f sx, yd dy dx A region D is of type II if it lies between the graphs of two continuous functions of y, that is, D double Riemann sum t 2sxd t1sxd (c) What is a type II region? How you evaluate yyD f sx, yd dA if D is a type II region? (e) What does the Midpoint Rule for double integrals say? m b D R The Midpoint Rule for double integrals says that we approximate the double integral yyR f sx, yd dA by the t1sxd < y < t 2sxdj a yy f sx, yd dA − y y ■ d | a < x < b, where t1 and t are continuous on fa, bg Then We usually evaluate yyR f sx, yd dA as an iterated integral according to Fubini’s Theorem: b R A region D is of type I if it lies between the graphs of two continuous functions of x, that is, where DA is the area of each subrectangle and sx ij*, y ij*d is a sample point in each subrectangle If f sx, yd > 0, this sum represents an approximation to the volume of the solid that lies above the rectangle R and below the graph of f f sx, yd dA − lim Then we deine yy f sx, yd dA D2 yy dA − AsDd, the area of D D ■ If m < f sx, yd < M for all sx, yd in D, then mAsDd < yy f sx, yd dA < MAsDd D (continued) Copyright 2016 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it chaPTEr 15 ConCept CheCk answers (continued) How you change from rectangular coordinates to polar coordinates in a double integral? Why would you want to make the change? We may want to change from rectangular to polar coordinates in a double integral if the region D of integration is more easily described in polar coordinates: D − hsr, d | ␣ < < , h1sd < r < h2sdj To evaluate yyR f sx, yd dA, we replace x by r cos , y by r sin , and dA by r dr d (and use appropriate limits of integration): yy f sx, yd dA − y  ␣ y h2 sd h1sd f sr cos , r sin d r dr d (b) What properties does f possess? yy f sx, yd dA − f sx, yd > R2 (c) What are the expected values of X and Y ? The expected value of X is 1 − yy x f sx, yd dA R2 The expected value of Y is − yy y f sx, yd dA R2 Write an expression for the area of a surface with equation z − f sx, yd, sx, yd [ D D If a lamina occupies a plane region D and has density function sx, yd, write expressions for each of the following in terms of double integrals D (assuming that f x and f y are continuous) (a) Write the deinition of the triple integral of f over a rectangular box B m − yy sx, yd dA (a) The mass: AsSd − yy sf fxsx, ydg f fysx, ydg 1 dA D l (b) The moments about the axes: i jk My − yy xsx, yd dA D D (c) The center of mass: sx, y d, where x − n i jk i jk i−1 j−1 k−1 B Mx − yy ysx, yd dA m yyy f sx, y, zd dV − l, m,limn l ` o o o f sx* , y* , z* d DV where DV is the volume of each sub-box and sxi*jk, yi*jk, zi*jk d is a sample point in each sub-box (b) How you evaluate yyyB f sx, y, zd dV? My m y− and Mx m (d) The moments of inertia about the axes and the origin: We usually evaluate yyyB f sx, y, zd dV as an iterated integral according to Fubini’s Theorem for Triple Integrals: If f is continuous on B − fa, bg fc, d g fr, sg, then yyy f sx, y, zd dV − y y y s r d c b a f sx, y, zd dx dy dz B I x − yy y sx, yd dA Note that there are ive other orders of integration that we can use D I y − yy x sx, yd dA (c) How you deine yyyE f sx, y, zd dV if E is a bounded solid region that is not a box? D I − yy sx y d sx, yd dA D Let f be a joint density function of a pair of continuous random variables X and Y (a) Write a double integral for the probability that X lies between a and b and Y lies between c and d Psa < X < b, c < Y < d d − y b a y d c f sx, yd dy dx Since E is bounded, it can be enclosed in a box B as described in part (b) We deine a new function F with domain B by Fsx, y, zd − H f sx, y, zd if sx, y, zd is in E if sx, y, zd is in B but not in E Then we deine yyy f sx, y, zd dV − yyy Fsx, y, zd dV E B (continued) Copyright 2016 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Cut here and keep for reference chaPTEr 15 ConCept CheCk answers (continued) (d) What is a type solid region? How you evaluate yyyE f sx, y, zd dV if E is such a region? A region E is of type if it lies between the graphs of two continuous functions of x and y, that is, sx, y, zd E− | s x, yd [ D, u 1sx, yd < z < u 2sx, yd Suppose a solid object occupies the region E and has density function sx, y, zd Write expressions for each of the following (a) The mass: m − yyy sx, y, zd dV E where D is the projection of E onto the xy-plane Then F yyy f sx, y, zd dV − yy y D E u2sx, yd u1sx, yd G f sx, y, zd dz dA (b) The moments about the coordinate planes: Myz − yyy x sx, y, zd dV E (e) What is a type solid region? How you evaluate yyyE f sx, y, zd dV if E is such a region? Mxz − yyy y sx, y, zd dV E A type region is of the form E− sx, y, zd | s y, zd [ D, u1s y, zd < x < u 2s y, zd E where D is the projection of E onto the yz-plane Then F yyy f sx, y, zd dV − yy y D E u2sy, zd u1sy, zd G f sx, y, zd dx dA (f) What is a type solid region? How you evaluate yyyE f sx, y, zd dV if E is such a region? A type region is of the form E− sx, y, zd u1sx, zd < y < u 2sx, zd where D is the projection of E onto the xz-plane Then yyy f sx, y, zd dV − yy y E D (c) The coordinates of the center of mass: sx, y, z d, where x − Myz Mxz Mxy ,y− ,z− m m m (d) The moments of inertia about the axes: Ix − yyy s y z d sx, y, zd dV | sx, zd [ D, F Mx y − yyy z sx, y, zd dV u2sx, zd u1sx, zd G f sx, y, zd dy dA E Iy − yyy sx z d sx, y, zd dV E Iz − yyy sx y d sx, y, zd dV E (continued) Copyright 2016 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it chaPTEr 15 ConCept CheCk answers (continued) (a) How you change from rectangular coordinates to cylindrical coordinates in a triple integral?  h2sd u2sr cos , r sin d ␣ h1sd u1sr cos , r sin d yyy f sx, y, zd dV − y y y E f sr cos , r sin , zd r dz dr d where | E − h sr, , zd ␣ < < , h1sd < r < h 2sd, u1sr cos , r sin d < z < u 2sr cos , r sin d j Thus we replace x by r cos , y by r sin , dV by r dz dr d, and use appropriate limits of integration (b) How you change from rectangular coordinates to spherical coordinates in a triple integral? d  t2s, d c ␣ t1s, d yyy f sx, y, zd dV − y y y E f s sin cos , sin sin , cos d sin d d d | E − 5s, , d ␣ < < , c < < d, t1s, d < < t 2s, d j where Thus we replace x by sin cos , y by sin sin , z by cos , dV by sin d d d, and use appropriate limits of integration (c) In what situations would you change to cylindrical or spherical coordinates? We may want to change from rectangular to cylindrical or spherical coordinates in a triple integral if the region E of integration is more easily described in cylindrical or spherical coordinates Regions that involve symmetry about the z-axis are often simpler to describe using cylindrical coordinates, and regions that are symmetrical about the origin are often simpler in spherical coordinates Also, sometimes the integrand is easier to integrate using cylindrical or spherical coordinates 10 (a) If a transformation T is given by x − tsu, vd, y − hsu, vd, what is the Jacobian of T ? Z Z −x −sx, yd −u − −su, vd −y −u −x −x −y −v −x −y − −y −u −v −v −u −v (b) How you change variables in a double integral? We change from an integral in x and y to an integral in u and v by expressing x and y in terms of u and v and writing dA − Z −sx, yd −su, vd Z du dv Thus, under the appropriate conditions, yy f sx, yd dA − yy f sxsu, vd, ysu, vdd R S Z −sx, yd −su, vd Z du dv where R is the image of S under the transformation (c) How you change variables in a triple integral? Similarly to the case of two variables in part (b), yyy f sx, y, zd dV − yyy f sxsu, v, wd, ysu, v, wd, zsu, v, wdd R where S −sx, y, zd − −su, v, wd −x −u −y −u −z −u −x −v −y −v −z −v Z −sx, y, zd −su, v, wd Z du dv dw −x −w −y −w −z −w is the Jacobian Copyright 2016 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it chaPTEr 16 ConCept CheCk answers Cut here and keep for reference What is a vector ield? Give three examples that have physical meaning (d) Write the deinitions of the line integrals along C of a scalar function f with respect to x, y, and z A vector ield is a function that assigns a vector to each point in its domain f sx, y, zd dx − lim y f sx, y, zd dy − lim y f sx, y, zd dz − lim C A vector ield can represent, for example, the wind velocity at any location in space, the speed and direction of the ocean current at any location, or the force vector of the earth’s gravitational ield at a location in space C C (a) What is a conservative vector ield? The function f in part (a) is called a potential function for F If C is given by the parametric equations x − xstd, y − ystd, a < t < b, we divide the parameter interval fa, bg into n subintervals fti21, ti g of equal width The ith subinterval corresponds to a subarc of C with length Dsi Then f sx, y, zd dy − y f sxstd, ystd, zstdd y9std dt y f sx, y, zd dz − y f sxstd, ystd, zstdd z9std dt b C a b a If F is a continuous vector ield and C is given by a vector function rstd, a < t < b, then n o f sx *i , y*i d Dsi n l ` i−1 f sx, yd ds − lim y C f sx, yd ds − y f sxstd, ystdd b a ỴS D S D dx dt dy dt F ؒ dr − y Fsrstdd ؒ r9std dt − b a a y ỴS D S D S D dx dt dy dt dz dt The center of mass is sx, y d, where x− m y x sx, yd ds y− m y y sx, yd ds C C F ؒ dr − y P dx Q dy R dz C State the Fundamental Theorem for Line Integrals dt (c) Write expressions for the mass and center of mass of a thin wire shaped like a curve C if the wire has linear density function sx, yd The mass is m − yC sx, yd ds F ؒ T ds (c) If F − kP, Q, R l, what is the connection between the line integral of F and the line integrals of the component functions P, Q, and R? dt C b C It represents the work done by F in moving a particle along the curve C f sx, y, zd ds − y f sxstd, ystd, zstdd y (b) If F is a force ield, what does this line integral represent? Similarly, if C is a smooth space curve, then C y b a (a) Deine the line integral of a vector ield F along a smooth curve C given by a vector function rstd (b) How you evaluate such a line integral? y f sx, y, zd dx − y f sxstd, ystd, zstdd x9std dt C where sx *i , y*i d is any sample point in the ith subarc C n o f sx*i , yi*, zi*d Dzi n l ` i−1 y C (a) Write the deinition of the line integral of a scalar function f along a smooth curve C with respect to arc length y n o f sx*i , yi*, zi*d Dyi n l ` i−1 (e) How you evaluate these line integrals? (b) What is a potential function? C o f sx*i , yi*, zi*d Dx i n l ` i−1 (We have similar results when f is a function of two variables.) A conservative vector ield F is a vector ield that is the gradient of some scalar function f , that is, F − =f y n y If C is a smooth curve given by rstd, a < t < b, and f is a differentiable function whose gradient vector = f is continuous on C, then y C = f ؒ dr − f srsbdd f srsadd (a) What does it mean to say that yC F ؒ dr is independent of path? yC F ؒ dr is independent of path if the line integral has the same value for any two curves that have the same initial points and the same terminal points (b) If you know that yC F ؒ dr is independent of path, what can you say about F? We know that F is a conservative vector ield, that is, there exists a function f such that =f − F (continued) Copyright 2016 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it chaPTEr 16 ConCept CheCk answers (continued) State Green’s Theorem (b) Write an expression for the area of a parametric surface Let C be a positively oriented, piecewise-smooth, simple closed curve in the plane and let D be the region bounded by C If P and Q have continuous partial derivatives on an open region that contains D, then y C P dx Q dy − yy S −Q −P −x −y D If S is a smooth parametric surface given by rsu, vd − xsu, vd i ysu, vd j zsu, vd k where su, vd [ D and S is covered just once as su, vd ranges throughout D, then the surface area of S is dA | D Write expressions for the area enclosed by a curve C in terms of line integrals around C (c) What is the area of a surface given by an equation z − tsx, yd? ᭺ x dy − 2y ᭺ y dx − ᭺ x dy y dx A−y 2y C C AsSd − C −R −Q −y −z D S i1 −P −R −z −x D S j1 −Q −P −x −y D k 11 −z −y dA m n yy f sx, y, zd dS − m,lim o o f sP*d DS nl` ij −P −Q −R 1 −=ؒF −x −y −z (c) If F is a velocity ield in luid low, what are the physical interpretations of curl F and div F? At a point in the luid, the vector curl F aligns with the axis about which the luid tends to rotate, and its length measures the speed of rotation; div F at a point measures the tendency of the luid to low away (diverge) from that point 10 If F − P i Q j, how you determine whether F is conservative? What if F is a vector ield on R3 ? If P and Q have continuous irst-order derivatives and −P −Q − , then F is conservative −y −x If F is a vector ield on R whose component functions have continuous partial derivatives and curl F − 0, then F is conservative 11 (a) What is a parametric surface? What are its grid curves? where DSij is the area of the patch Sij and Pij* is a sample point from the patch (S is divided into patches in such a way that ensures that DSij l as m, n l `.) (b) How you evaluate such an integral if S is a parametric surface given by a vector function rsu, vd? yy f sx, y, zd dS − yy f srsu, vdd | r u S rsu, vd − xsu, vd i ysu, vd j zsu, vd k of two parameters u and v Equivalent parametric equations are y − ysu, vd z − zsu, vd The grid curves of S are the curves that correspond to holding either u or v constant | rv dA D where D is the parameter domain of S (c) What if S is given by an equation z − tsx, yd? yy f sx, y, zd dS S − yy f sx, y, tsx, ydd D ỴS D S D −z −x −z −y 1 dA (d) If a thin sheet has the shape of a surface S, and the density at sx, y, zd is sx, y, zd, write expressions for the mass and center of mass of the sheet The mass is m− yy sx, y, zd dS S A parametric surface S is a surface in R described by a vector function ij i−1 j−1 S (b) Deine div F x − xsu, vd −z −x We divide S into “patches” Sij Then −= 3F div F − Ỵ S D S D 12 (a) Write the deinition of the surface integral of a scalar function f over a surface S (a) Deine curl F S yy D Suppose F is a vector ield on R3 curl F − | AsSd − yy ru rv dA D The center of mass is sx, y, z d, where m yy x sx, y, zd dS y− m yy y sx, y, zd dS z− m yy z sx, y, zd dS x− S S S (continued) Copyright 2016 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it chaPTEr 16 ConCept CheCk answers (continued) Cut here and keep for reference 13 (a) What is an oriented surface? Give an example of a nonorientable surface An oriented surface S is one for which we can choose a unit normal vector n at every point so that n varies continuously over S The choice of n provides S with an orientation A Möbius strip is a nonorientable surface (It has only one side.) (b) Deine the surface integral (or lux) of a vector ield F over an oriented surface S with unit normal vector n (c) How you evaluate such an integral if S is a parametric surface given by a vector function rsu, vd? u S rv d dA D We multiply by 21 if the opposite orientation of S is desired (d) What if S is given by an equation z − tsx, yd? If F − kP, Q, Rl, yy F ؒ dS − yy S D S 2P Let S be an oriented piecewise-smooth surface that is bounded by a simple, closed, piecewise-smooth boundary curve C with positive orientation Let F be a vector ield whose components have continuous partial derivatives on an open region in R that contains S Then y C F ؒ dr − yy curl F ؒ dS 15 State the Divergence Theorem S yy F ؒ dS − yy F ؒ sr 14 State Stokes’ Theorem S yy F ؒ dS − yy F ؒ n dS S for the upward orientation of S; we multiply by 21 for the downward orientation D −t −t 2Q R dA −x −y Let E be a simple solid region and let S be the boundary surface of E, given with positive (outward) orientation Let F be a vector ield whose component functions have continuous partial derivatives on an open region that contains E Then yy F ؒ dS − yyy div F dV S E 16 In what ways are the Fundamental Theorem for Line Integrals, Green’s Theorem, Stokes’ Theorem, and the Divergence Theorem similar? In each theorem, we integrate a “derivative” over a region, and this integral is equal to an expression involving the values of the original function only on the boundary of the region Copyright 2016 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Copyright 2016 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it chaPTEr 17 ConCept CheCk answers Cut here and keep for reference (a) Write the general form of a second-order homogeneous linear differential equation with constant coeficients ay0 by9 cy − where a, b, and c are constants and a ± (b) Write the auxiliary equation ar br c − (c) How you use the roots of the auxiliary equation to solve the differential equation? Write the form of the solution for each of the three cases that can occur If the auxiliary equation has two distinct real roots r1 and r 2, the general solution of the differential equation is y − c1 e r x c2 e r x If the roots are real and equal, the solution is y − c1 e rx c2 xe rx where r is the common root If the roots are complex, we can write r1 − ␣ i and r − ␣ i, and the solution is y − e ␣ xsc1 cos x c2 sin xd (a) What is an initial-value problem for a second-order differential equation? An initial-value problem consists of inding a solution y of the differential equation that also satisies given conditions ysx d − y0 and y9sx d − y1, where y0 and y1 are constants (b) What is a boundary-value problem for such an equation? A boundary-value problem consists of inding a solution y of the differential equation that also satisies given boundary conditions ysx d − y0 and ysx1 d − y1 (a) Write the general form of a second-order nonhomogeneous linear differential equation with constant coeficients ay0 by9 cy − Gsxd, where a, b, and c are constants and G is a continuous function (b) What is the complementary equation? How does it help solve the original differential equation? The complementary equation is the related homogeneous equation ay0 by9 cy − If we ind the general solution yc of the complementary equation and yp is any particular solution of the nonhomogeneous differential equation, then the general solution of the original differential equation is ysxd − ypsxd ycsxd (c) Explain how the method of undetermined coeficients works To determine a particular solution yp of ay0 by9 cy − Gsxd, we make an initial guess that yp is a general function of the same type as G If Gsxd is a polynomial, choose yp to be a general polynomial of the same degree If Gsxd is of the form Ce k x, choose ypsxd − Ae k x If Gsxd is C cos k x or C sin k x, choose ypsxd − A cos k x B sin k x If Gsxd is a product of functions, choose yp to be a product of functions of the same type Some examples are: ypsxd Gsxd x e 2x sin 2x xe 2x Ax Bx C Ae 2x A cos 2x B sin 2x sAx Bde 2x We then substitute yp, yp9, and yp0 into the differential equation and determine the coeficients If yp happens to be a solution of the complementary equation, then multiply the initial trial solution by x (or x if necessary) If Gsxd is a sum of functions, we ind a particular solution for each function and then yp is the sum of these The general solution of the differential equation is ysxd − ypsxd ycsxd (d) Explain how the method of variation of parameters works We write the solution of the complementary equation ay0 by9 cy − as ycsxd − c1 y1sxd c2 y2sxd, where y1 and y2 are linearly independent solutions We then take ypsxd − u1sxd y1sxd u2sxd y2sxd as a particular solution, where u1sxd and u2sxd are arbitrary functions After computing yp9, we impose the condition that u91 y1 u92 y2 − (1) and then compute yp0 Substituting yp, yp9, and yp0 into the original differential equation gives asu91 y91 u92 y92 d − G (2) We then solve equations (1) and (2) for the unknown functions u91 and u92 If we are able to integrate these functions, then a particular solution is ypsxd − u1sxd y1sxd u2sxd y2sxd and the general solution is ysxd − ypsxd ycsxd Discuss two applications of second-order linear differential equations The motion of an object with mass m at the end of a spring is an example of simple harmonic motion and is described by the second-order linear differential equation m d 2x kx − dt (continued) Copyright 2016 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Chapter 17 ConCept CheCk answers (continued) where k is the spring constant and x is the distance the spring is stretched (or compressed) from its natural length If there are external forces acting on the spring, then the differential equation is modiied Second-order linear differential equations are also used to analyze electrical circuits involving an electromotive force, a resistor, an inductor, and a capacitor in series See the discussion in Section 17.3 for additional details How you use power series to solve a differential equation? We irst assume that the differential equation has a power series solution of the form ` y− o cn x n − c0 c1 x c2 x c3 x ∙ ∙ ∙ n−0 Differentiating gives ` y9 − ` o ncn x n21 − n−0 o sn 1dcn11 x n n−1 and ` y0 − ` o nsn 1dcn x n22 − n−0 o sn 2dsn 1dcn12 x n n−2 We substitute these expressions into the differential equation and equate the coeficients of x n to ind a recursion relation involving the constants cn Solving the recursion relation gives a formula for cn and then ` y− o cn x n n−0 is the solution of the differential equation Copyright 2016 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Copyright 2016 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it ... Library of Congress Control Number: 2014951195 Student Edition: ISBN: 97 8-1 -2 8 5-7 415 5-0 Loose-leaf Edition: ISBN: 97 8-1 -3 0 5-2 723 5-4 Cover Image: elisanth/123RF; tharrison/Getty Images Cengage Learning... Anderson, Jeffery A Cole, and Daniel Drucker ISBN 97 8-1 -3 0 5-2 723 9-2 Multivariable By Dan Clegg and Barbara Frank ISBN 97 8-1 -3 0 5-2 761 1-6 Includes worked-out solutions to all exercises in the text Printed... Jeffery A Cole, and Daniel Drucker ISBN 97 8-1 -3 0 5-2 724 2-2 Multivariable By Dan Clegg and Barbara Frank ISBN 97 8-1 -3 0 5-2 718 2-1 Provides completely worked-out solutions to all oddnumbered exercises