Spring2003 Midterm

Spring2003 Midterm - ‘ University of Waterloo Waterloo...

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Unformatted text preview: ‘ University of Waterloo Waterloo, Ontario Mathematics 138 Mid-Term Test - Spring Term 2003 Duration: ‘2 hours Date: June 16. 2003 ONLY BASIC SCIENTIFIC CALCULATORS PERMITTED Initial“ Id. Numbe‘ Family Name: Signature: Instructors: (Check one) 01 D. Woolford 02 M. Shpigel 03 Y. Spigel 04 D. Harmsworth 000K Instructions: Title page/instructions - 1 1. Complete the information required on this page in the spaces provided. Check your section number in the appropriate box. 2. Attempt all questions, in the space provided. If you require more space, use the reverse of the preceding page . HHIIE 3. The marks for each question are in- dicated. Marks will be deducted for negligently presented work. Your grade will be influenced by how clearly you express your ideas, and by how well you organize your solu— tions. Justification should be provided by referring to definitions and theorems where appropriate. \r'. I 4. This examination has ten pages. The last page is for rough work. 5. A one-mark deduction may be ap- 65 (+3) plied for improper completion of this title page or for ignoring instruc- tions. Math 138 - Mid-Term Exam Spnug lerm 2W.) x {13] 1. Evaluate the following: (a) f cossrdr o _ Q ~11 V' f l// 1/ _ r/ x" ’/ //II ll/ , / I/V/ ~\ .v _v I" +1 1 1111: . . . . . (b) ——- dx (Notlce that thxs integral :5 Improper!) o v5 / ., \/ Page 2 of 10 3 Spring Term 2003 Math 138 - Mid-Term Exam ‘ (c; rm (1 —-2z)% // _, \. 'I’ ‘ K, ‘ 1’ n 5. . ’i' ’ " ., / /// // , / i‘ / / / x, / l// / / Z/ / {7/ l/ ,/ . I’// x/ // / / ,/ / / ,/ ,/ l/ . f/ Page 3 of 10 3') Math 138 - Mid-Term Exam uyuug, um. WW . . {10} 2. Show that. the sequence defined recursively by a, = an“ 2 VS + an is increasing and bounded. Explain why this means that the sequence must be com'ergem. and find its limit. / v Page 4 of 10 Maui 138 - Mid—Term Exam Spring lerm 2mm 0 [14} 3. Determine whether each of the following converges or diverges. For parts (d). (f). and (g), you should distinguish between absolute convergence and conditional convergence. ln answering these. you may refer to conclusions you have reached for other parts: for example. in your explanation of {d} you may wish to refer to (c). 4 {3+5nz}x la.) 0 n-+n nzv‘l 0c 3+5n2 (b) 2:! n2+n 3» x no 3 ~;. n 11—1 “7‘ (d) g 1) ,H at“ 7 \h .\ 3‘ f R; Page 5 of 10 Math 138 — Mid—Term Exam Spring Term EUU'J I. Z 371 h (e) n2 + 4 3‘ n=l , , “Ekijtd’ < gunman) W 0664 mm czgdm'l'rm of“ nit-IA a) 00 "_ 3n (0 gm) km hmvf‘?’ "’71, y/ "t (g) f; (—1) En— H 7 x Page 6 of 10 Mth 138 - Mid-Term Exam Spring 'lerm ZUUJ I x \X 1 . . [10} 4, (3} Find an approximation to the sum of the series 2 £3 correct to mthn 0001‘ n2] 1” I (b) Find an approximation to the sum of the series 2 (——1)""l$ ? correct to within n=1 0.001. - Page 7 of 10 ' Math 138 ~ Mid—Term Exam Spring Term 2003 WWW . DC. , ~ . ‘ _ . ~33"{1 + 15” [12] 5‘ Fmd the radlus and interval of convergence of the power series 2 l 71:0 9’ - 2 fix \ 3 1 .2:;\ ’ 1 Page 8 of 10 9.; Math 138 — Mid—Term Exam :pnug term .cuu.) ' {6] 6. Have that if 2 30"] converges. then 2111, converges. BONUS: A pane of a certain type of glass reflects half the incident light, absorbs one-fourth, and {31 transmits one—fourth. A window is made of two panes of this glass separated by a small space. What fraction of the incident Right I is transmitted by the double window? I (x § - _' _ r , “:1 ‘ 4 n. ' 7 ' \_ t“, ‘ ,. - 9 a A... x s ' / _,_, (.5 1 “:1 g » - .,> mm...“ ; 7 pk. . f V3 ‘ 9‘ a r ‘3 Page 9 of 10 '\ ,\ ( r~ ‘ ~ p V" “>3 A-I’ ...
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Spring2003 Midterm - ‘ University of Waterloo Waterloo...

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