Multivariable Calculus: Concepts and Contexts (with Tools for Enriching Calculus, Interactive Video Skillbuilder CD-ROM, and iLrn Homework Personal Tutor) PDF
Preview Multivariable Calculus: Concepts and Contexts (with Tools for Enriching Calculus, Interactive Video Skillbuilder CD-ROM, and iLrn Homework Personal Tutor)
Set your learning in motion with these valuable tools! Tools for Enriching™ Calculus CD-ROM (TEC) This CD-ROM helps you understand and visualize calculus by exploring concepts through interactive modules and animations. TEC icons in the text direct you to the appropriate module for exploration. TEC also has homework hints for specially marked exercises in each section. Interactive Video Skillbuilder CD-ROM This CD-ROM contains video instruction for every major concept in the text. It lets you review material that you may not have understood the first time you heard it, or if you missed class. In order to help you evaluate your progress, each section contains a 10-question Web quiz per section (the results of which can be emailed to the instructor) and a test for each chapter, with answers. iLrn™ Student Version Get expert help from your own tutor . . . online. Through iLrn, you have access to live online tutoring at vMentor™. The tutors at this free service will skillfully guide you through a problem using unique two-way audio and whiteboard features. The iLrn system also provides you with access to self-directed study via instructor-assigned, text-specific homework problems. Use the access code packaged with this text to get started today! vMentor is for use by proprietary, college, and university adopters only. If your textbook does not contain these useful tools, you can purchase them separately at http://series.brookscole.com/stewart REFERENCE PAGES A L G E B R A G E O M E T R Y ARITHMETIC OPERATIONS GEOMETRIC FORMULAS a共b ⫹ c兲 苷 ab ⫹ ac a ⫹ c 苷 ad ⫹ bc Formulas for area A, circumference C, and volume V: b d bd Triangle Circle Sector of Circle a ⫹ b c 苷 ba ⫹ bc bac 苷 ba ⫻ dc 苷 abdc 苷 2121 abbh sin rr2 12 r 2共 in radians兲 d EXPONENTS AND RADICALS a h r s x mx n 苷 x m⫹n x xmn 苷 x m⫺n A 苷 ¨ b CA 苷 2 r As 苷 ¨r r 共x m兲n 苷 x mn x⫺n 苷 x1n 共xy兲n 苷 x nyn 冉 x y冊n 苷 xyn Spher43e r 3 VCy苷linderr2h VCo苷ne13 r 2h x 1兾n 苷 sn x x m兾n 苷 sn x m 苷 (sn x )m r 2 sn xy 苷 sn x sn y 冑n x y 苷 sn xy r V 苷 r h h FACTORING SPECIAL POLYNOMIALS A 苷 4 r x 2 ⫺ y2 苷 共x ⫹ y兲共x ⫺ y兲 x 3 ⫹ y3 苷 共x ⫹ y兲共x 2 ⫺ xy ⫹ y2兲 x 3 ⫺ y3 苷 共x ⫺ y兲共x 2 ⫹ xy ⫹ y2兲 DISTANCE AND MIDPOINT FORMULAS BINOMIAL THEOREM Distance between P1共x1, y1兲 and P2共x2, y2兲: 共x ⫹ y兲2 苷 x 2 ⫹ 2xy ⫹ y2 共x ⫺ y兲2 苷 x 2 ⫺ 2xy ⫹ y2 共x ⫹ y兲3 苷 x 3 ⫹ 3x 2y ⫹ 3xy2 ⫹ y3 d 苷 s共x2 ⫺ x1兲2 ⫹ 共 y2 ⫺ y1兲2 共x ⫺ y兲3 苷 x 3 ⫺ 3x 2y ⫹ 3xy2 ⫺ y3 共x ⫹ y兲n 苷 x n ⫹ nx n⫺1y ⫹ n共n ⫺ 1兲 x n⫺2y2 Midpoint of P1P2: 冉 x1 ⫹2 x2 , y1 ⫹2 y2冊 2 ⫹ ⭈ ⭈ ⭈ ⫹ 冉n k冊x n⫺kyk ⫹ ⭈ ⭈ ⭈ ⫹ nxyn⫺1 ⫹ yn LINES where 冉n k冊 苷 n共n 1⫺ⴢ12兲 ⭈ⴢ ⭈3⭈ ⴢ共n⭈ ⭈⫺⭈ ⴢk k⫹ 1兲 Slope of line through P1共x1, y1兲 and P2共x2, y2兲: QUADRATIC FORMULA m 苷 xy2 2 ⫺ yx1 If ax 2 ⫹ bx ⫹ c 苷 0, then x 苷 ⫺b ⫾ s 2ba2 ⫺ 4ac . Point-slope equation of line through P1共x1, y1兲 with slope m: INEQUALITIES AND ABSOLUTE VALUE y ⫺ y1 苷 m共x ⫺ x1兲 If a ⬍ b and b ⬍ c, then a ⬍ c. Slope-intercept equation of line with slope m and y-intercept b: If a ⬍ b, then a ⫹ c ⬍ b ⫹ c. If a ⬍ b and c ⬎ 0, then ca ⬍ cb. y 苷 mx ⫹ b If a ⬍ b and c ⬍ 0, then ca ⬎ cb. If a ⬎ 0, then CIRCLES ⱍ x ⱍ 苷 a means x 苷 a or x 苷 ⫺a Equation of the circle with center 共h, k兲 and radius r: ⱍ x ⱍ ⬍ a means ⫺a ⬍ x ⬍ a ⱍ x ⱍ ⬎ a means x ⬎ a or x ⬍ ⫺a 共x ⫺ h兲2 ⫹ 共 y ⫺ k兲2 苷 r 2 1 Cut here and keep for reference REFERENCE PAGES T R I G O N O M E T R Y ANGLE MEASUREMENT FUNDAMENTAL IDENTITIES 1⬚ r苷adi 1an8s0 苷rad180⬚ 1 rad苷 180⬚ r ¨ r s ctasnc 苷 sin1 sceoct 苷 co1s 共s苷 inr radians兲 cot 苷co1s sin2 ⫹csoins 2 苷 1 tan RIGHT ANGLE TRIGONOMETRY 1 ⫹ tan2 苷 sec 2 1 ⫹ cot 2 苷 csc 2 苷 hoaypdpj 苷 ohapydpj ¨hyapdj opp stain共⫺ 兲 苷 ⫺stainn scions冉共⫺2兲⫺苷冊cos苷 cos adj opp cos冉 2 ⫺ 冊 苷 sin tan冉 2 ⫺ 冊 苷 cot TRIGONOMETRIC FUNCTIONS 苷 y r 苷 yr y TsHinE A LA苷Ws iOn FB S苷INEsiSn C B a 苷 x 苷 r r (x, y) a b c C r x c 苷 xy 苷 yx ¨ x TaH2 E苷 LbA2W⫹ OcF2 ⫺CO2SbIcN cEoSs A b GsiRn APHS OF THE TcRsIcG ONOMETRIC FUNCTIONS b 2 苷 a 2 ⫹ c 2 ⫺ 2ac cos B y y y y=tan x c 2 苷 a 2 ⫹ b 2 ⫺ 2ab cos C A cos y=sin x sec y=cos x 1 1 π 2π 2π ADDITION AND SUBTRACTION FORMULAS tan cot x π 2π x π x sin共x ⫹ y兲 苷 sin x cos y ⫹ cos x sin y _1 _1 sin共x ⫺ y兲 苷 sin x cos y ⫺ cos x sin y cos共x ⫹ y兲 苷 cos x cos y ⫺ sin x sin y siyn y=csc x csc y y=sec x y y=cot x cos共x ⫺ y兲 苷 cos x cos y ⫹ sin x sin y tan x ⫹ tan y tan共x ⫹ y兲 苷 co1s sec 1 1 ⫺ tan x tan y tan x ⫺ tan y tan π 2cπot x π 2π x π 2π x tan共x ⫺ y兲 苷 1 ⫹ tan x tan y _1 _1 DOUBLE-ANGLE FORMULAS sin 2x 苷 2 sin x cos x TRIGONOMETRIC FUNCTIONS OF IMPORTANT ANGLES cos 2x 苷 cos2x ⫺ sin2x 苷 2 cos2x ⫺ 1 苷 1 ⫺ 2 sin2x radians sin cos tan 2 tan x 0⬚ 0 0 1 0 tan 2x 苷 1 ⫺ tan2x 30⬚ 兾6 1兾2 s3兾2 s3兾3 45⬚ 兾4 s2兾2 s2兾2 1 HALF-ANGLE FORMULAS 960⬚ 兾32 s31兾2 1兾02 s—3 sin2x 苷 1 ⫺ c2os 2x cos2x 苷 1 ⫹ c2os 2x 2 REFERENCE PAGES S P E C I A L F U N C T I O N S a POWER FUNCTIONS f 共x兲 苷 x (i) f 共x兲 苷 xn, n a positive integer y y y=x$ (1, 1) y=x^ y=x# y=≈ y=x% (_1, 1) (1, 1) 0 x 0 x (_1, _1) n even n odd 1兾n n (i i) f 共x兲 苷 x 苷 sx, n a positive integer y y (1, 1) (1, 1) 0 x 0 x ƒ=œ# x„ 1 ⫺1 (ii i) f 共x兲 苷 x 苷 y x y=∆ 1 0 x 1 INVERSE TRIGONOMETRIC FUNCTIONS y π 2 ⫺1 arcsin x 苷 sin x 苷 y &? sin y 苷 x and ⫺ 艋 y 艋 2 2 ⫺1 0 lim tan x 苷 ⫺ x l ⫺⬁ 2 arccos x 苷 cos⫺1x 苷 y &? cos y 苷 x and 0 艋 y 艋 x ⫺1 lim tan x 苷 π x l ⬁ 2 ⫺1 _ arctan x 苷 tan x 苷 y &? tan y 苷 x and ⫺ ⬍ y ⬍ 2 2 2 y=tan–!x=arctan x 3 Cut here and keep for reference ƒ=œx„ RREEFFEERRENCE PPAAGGEESS S P E C I A L F U N C T I O N S EXPONENTIAL AND LOGARITHMIC FUNCTIONS y loga x 苷 y &? ay 苷 x y=´ y=x ln x 苷 loge x, where ln e 苷 1 ln x 苷 y &? ey 苷 x 1 y=ln x 0 Cancellation Equations Laws of Logarithms 1 x l oga共ax兲 苷 x aloga x 苷 x 1. loga共xy兲 苷 loga x ⫹ loga y l n共ex兲 苷 x eln x 苷 x 2. loga冉 yx冊 苷 loga x ⫺ loga y x lim ⫺⬁ ex 苷 0 xl ilm ⬁ e x 苷 ⬁ 3. loga共xr兲 苷 r loga x xl lim 0⫹ ln x 苷 ⫺⬁ xl ilm ⬁ ln x 苷 ⬁ ” 21 ’® ” 41 ’® y 10® 4® e® 2® y y=log™ x 1.5® y=ln x 1 y=log∞ x y=log¡¸ x 1® 0 1 x 0 x Exponential functions Logarithmic functions HYPERBOLIC FUNCTIONS y y=cosh x e x ⫺ e⫺x 1 sinh x 苷 2 csch x 苷 sinh x y=tanh x e x ⫹ e⫺x 1 cosh x 苷 2 sech x 苷 cosh x x sinh x cosh x tanh x 苷 coth x 苷 cosh x sinh x y=sinh x INVERSE HYPERBOLIC FUNCTIONS y 苷 sinh⫺1x &? sinh y 苷 x s inh⫺1x 苷 ln(x ⫹ sx2 ⫹ 1) y 苷 cosh⫺1x &? c osh y 苷 x and y 艌 0 c osh⫺1x 苷 ln(x ⫹ sx2 ⫺ 1) y 苷 tanh⫺1x &? t anh y 苷 x t anh⫺1x 苷 12 ln冉1 1 ⫹⫺ x冊 4 REFERENCE PAGES D I F F E R E N T I A T I O N R U L E S GENERAL FORMULAS d d 1. 共c兲 苷 0 2. 关cf 共x兲兴 苷 cf ⬘共x兲 dx dx d d 3. 关 f 共x兲 ⫹ t共x兲兴 苷 f ⬘共x兲 ⫹ t⬘共x兲 4. 关 f 共x兲 ⫺ t共x兲兴 苷 f ⬘共x兲 ⫺ t⬘共x兲 dx dx d d f 共x兲 t共x兲f ⬘共x兲 ⫺ f 共x兲t⬘共x兲 5. dx 关 f 共x兲t共x兲兴 苷 f 共x兲t⬘共x兲 ⫹ t共x兲f ⬘共x兲 (Product Rule) 6. dx 冋 t共x兲 册 苷 关t共x兲兴2 (Quotient Rule) 7. d f 共t共x兲兲 苷 f ⬘共t共x兲兲t⬘共x兲 (Chain Rule) 8. d 共x n 兲 苷 nx n⫺1 (Power Rule) dx dx EXPONENTIAL AND LOGARITHMIC FUNCTIONS 9. d 共e x 兲 苷 e x 10. d 共a x 兲 苷 a x ln a dx dx d 1 d 1 11. dx ln ⱍ x ⱍ 苷 x 12. dx 共loga x兲 苷 x ln a TRIGONOMETRIC FUNCTIONS 13. d 共sin x兲 苷 cos x 14. d 共cos x兲 苷 ⫺sin x 15. d 共tan x兲 苷 sec2x dx dx dx 16. d 共csc x兲 苷 ⫺csc x cot x 17. d 共sec x兲 苷 sec x tan x 18. d 共cot x兲 苷 ⫺csc2x dx dx dx INVERSE TRIGONOMETRIC FUNCTIONS 19. ddx 共sin⫺1x兲 苷 s1 1⫺ x 2 20. ddx 共cos⫺1x兲 苷 ⫺ s1 1⫺ x 2 21. ddx 共tan⫺1x兲 苷 1 ⫹1 x 2 22. ddx 共csc⫺1x兲 苷 ⫺ xsx12 ⫺ 1 23. ddx 共sec⫺1x兲 苷 xsx12 ⫺ 1 24. ddx 共cot⫺1x兲 苷 ⫺ 1 ⫹1 x 2 HYPERBOLIC FUNCTIONS 25. d 共sinh x兲 苷 cosh x 26. d 共cosh x兲 苷 sinh x 27. d 共tanh x兲 苷 sech2x dx dx dx 28. d 共csch x兲 苷 ⫺csch x coth x 29. d 共sech x兲 苷 ⫺sech x tanh x 30. d 共coth x兲 苷 ⫺csch2x dx dx dx INVERSE HYPERBOLIC FUNCTIONS 31. ddx 共sinh⫺1x兲 苷 s1 1⫹ x 2 32. ddx 共cosh⫺1x兲 苷 sx 21⫺ 1 33. ddx 共tanh⫺1x兲 苷 1 ⫺1 x 2 34. ddx 共csch⫺1x兲 苷 ⫺ ⱍ x ⱍsx12 ⫹ 1 35. ddx 共sech⫺1x兲 苷 ⫺ xs11⫺ x 2 36. ddx 共coth⫺1x兲 苷 1 ⫺1 x 2 5 Cut here and keep for reference M U L T I V A R I A B L E Calculus and the Image not available due to copyright restrictions Architecture of Curves The cover photograph shows the Walt Disney Concert Hall in Los Angeles, designed and built 1992–2003 by Frank Gehry and Associates. It is a daring building, a layered composition of curved surfaces in the form of billowing sails with brushed stain- less steel cladding. The highly complex structures that Frank Gehry designs would be impossible to build with- out the computer. The CATIA software that his architects and engineers use to produce the com- Images not available due to copyright restrictions puter models is based on principles of calculus— fitting curves by matching tangent lines, making sure the curvature isn’t too large, and control- ling parametric surfaces. “Consequently,” says Gehry, “we have a lot of freedom. I can play with shapes.” The process starts with Gehry’s initial sketches, which are translated into a succession of physical models. (Hundreds of different physical models were constructed during the design of the building, first with basic wooden blocks and then evolving into more sculptural forms.) Then an engineer uses a digitizer to record the coordinates of a series of points on a physical model. The digitized points are fed into a computer and the CATIA software is used to link these points with smooth curves. (It joins curves so that their tangent lines coincide.) The architect has considerable freedom in creating these curves, guided by displays of the curve, its derivative, and its curvature. Then the Image not available due to copyright restrictions curves are connected to each other by a parametric The CATIA program was developed in France surface, and again the architect can do so in many by Dassault Systèmes, originally for designing possible ways with the guidance of displays of the geo- airplanes, and was subsequently employed in the metric characteristics of the surface. automotive industry. Frank Gehry, because of his The CATIA model is then used to produce another complex sculptural shapes, is the first to use it in physical model, which, in turn, suggests modifications architecture. It helps him answer his question, and leads to additional computer and physical models. “How wiggly can you get and still make a building?”
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