James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

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6 ■ LIES MY CALCULATOR AND COMPUTER TOLD MEANSWERSSClick here for solutions.131. 3. 0 5. (a) 4.82549424 10 119. (b) 1, x 3 3x 4 0Stewart: Calculus: Early Transcendentals, Seventh Edition. ISBN: 0538497904. (c) 2012 Brooks/Cole. All rights reserved.Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
LIES MY CALCULATOR AND COMPUTER TOLD ME ■ 7SOLUTIONS1. The computer results are from Maple, with Digits:=16. The last column shows the values of the sixth-degreeTaylor polynomial for () =csc 2 2 near =0. Note that the second arrangement of Taylor’s polynomial iseasier to use with a calculator: 6 () = 1 3 + 1 15 2 + 2289 4 + 1675 6 = 1675 2 + 2289 2 + 1 15 2 + 1 3 () calculator () computer () Taylor01 033400107 03340010596845 033400106001 0333341 033334000010 0333340000001 03334 0333333400 03333334000001 034 03333334 033333333000001 20 033334 0333333330000001 100 or 200 0334 03333333300000001 10,000 or 20,000 04 033333333000000001 1,000,000 0 033333333We see that the calculator results start to deteriorate seriously at =00001,andforsmaller, they are entirelymeaningless. The different results “100 or 200” etc. depended on whether we calculated [(sin ) 2 ] 1 or[(sin ) 1 ] 2 . With Maple, the result is off by more than 10% when =00000001 (compare with the calculatorresult!) A detailed analysis reveals that the values of the function are always greater than 1 , but the computer3eventually gives results less than 1 3 .The polynomial 6 () was obtained by patient simplication of the expression for (), starting withsin 2 () = 1 (2)2(1 cos 2),wherecos 2 =1 + (2)4 ··· (2)10 + 2 12 (). Consequently, the exact2! 4!10!value of the limit is 6(0) = 1 3. It can also be obtained by several applications of l’Hospital’s Rule to the expression() = 2 sin 2 2 sin 2 with intermediate simplications.Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.3. From () = 25(We may assume 0; Why?), we have ln () =25ln ln(10001) and(10001) 0 ()() = 25 ln(10001). This derivative, as well as the derivative 0 () itself, is positive for0 0 =( 0 )=25 249,971015, and negative for 0. Hence the maximum value of () isln(10001) 250, a number too large to be calculated directly. Using decimal logarithms,(10001) 0log 10 ( 0 ) 12408987757,sothat( 0 ) 1229922 × 10 124 . The actual value of the limit is lim () =0;it would be wasteful and inelegant to use l’Hospital’s Rule twenty-ve times since we can transform () into 25() =, and the inside expression needs just one application of l’Hospital’s Rule to give 0.(10001) 25Stewart: Calculus: Early Transcendentals, Seventh Edition. ISBN: 0538497904. (c) 2012 Brooks/Cole. All rights reserved.
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calculusEarly Transcendentalseighth
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Calculus: Early Transcendentals, Ei
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xxPrefaceJoseph Stampfli, Indiana U
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Cengage Customizable YouBookYouBook
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Calculators, Computers, andOther Gr
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Diagnostic TestsSuccess in calculus
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2 a preview of calculusA¡A∞AA£A
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A10appendix A Numbers, Inequalities
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A12appendix B Coordinate Geometry a
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A18appendix C Graphs of Second-Degr
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A20appendix C Graphs of Second-Degr
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A22appendix c Graphs of Second-Degr
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A24appendix D TrigonometryAnglesAng
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A26appendix D Trigonometryhypotenus
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A28appendix D TrigonometryTrigonome
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A30appendix D Trigonometry18a 18b18
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A32appendix D TrigonometryExercises
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A34appendix E Sigma NotationA conve
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A36appendix E Sigma NotationSOLUTIO
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A38appendix E Sigma NotationExercis
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A40appendix F Proofs of TheoremsSin
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A42appendix F Proofs of Theorems3
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A44appendix F Proofs of TheoremsDWe
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A46appendix F Proofs of TheoremsPro
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A48appendix F Proofs of TheoremsSec
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A50appendix G The Logarithm Defined
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A58appendix h Complex NumbersSOLUTI
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A60appendix h Complex NumbersThe po
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A62appendix h Complex NumbersFrom t
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A64appendix h Complex NumbersExerci
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A66 Appendix I Answers to Odd-Numbe
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A100 Appendix I Answers to Odd-Numb
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Page 1354 and 1355: A140indexBrahe, Tycho, 875branches
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Page 1358 and 1359: A144indexfunction(s) (continued)gra
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Page 1364 and 1365: A150indexseries (continued)harmonic
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Page 1368 and 1369: Chapter 8Concept Check AnswersCut h
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Page 1382 and 1383: Chapter 14Concept Check Answers (co
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Page 1396 and 1397: 2 ■ LIES MY CALCULATOR AND COMPUT
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James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

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