sample4 - Math 1ZA3 1st Sample Test for Suggested Problems...

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Math 1ZA3 1st Sample Test for Suggested Problems #4 Name :___________________________________________ (Last Name) (First Name) Student Number: Tutorial Number: _____________________ ____________________ This test consists of 26 multiple choice questions worth 1 mark each (no part marks). All questions must be answered on the COMPUTER CARD with an HB PENCIL. Marks will not be deducted for wrong answers (i.e., there is no penalty for guessing). You are responsible for ensuring that your copy of the test is complete. Bring any discrepancy to the attention of the invigilator. Calculators are NOT allowed. 1. Which of the following series converge? (i) (ii)  8œ" 8œ" ∞∞ 8" # 8 " Ñ x 8 "†$†&†(† â †Ð#8"Ñ 8 Ð"Ñ (a) (b) (c) (d) (i) only (ii) only (i) and (ii) neither 2. Which of the following series are absolutely convergent? (i) (ii)  8œ" 8œ" 8 8# 8" 8$ 8 Ð"Ñ Ð"Ñ # $ 8 (a) (b) (c) (d) (i) only (ii) only (i) and (ii) neither 3. Find the interval of convergence of 8œ" 8 # 8Ð$B  #Ñ (a) (b) (c) (d) (e) Ð∞ß ∞Ñ Ò ß  Ó Ò ß  Ñ Ò"ß  Ó Ò"ß  Ñ "# " " $$ $ $ 4. If is convergent, what can you conclude about the convergence of the following 8œ! 8 8 -# series? (i) (ii) 8œ! 8œ! 88 -Ð# Ñ -Ð$ Ñ (a) (b) (c) convergent, nothing convergent, convergent nothing, divergent (d) (e) nothing, nothing convergent, divergent 5. Find the radius of convergence of 8œ" 8 Ð8  "ÑxÐ$B  "Ñ (a) (b) (c) (d) (e) ∞!"
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6. Find a power series representation of 0ÐBÑ œ B B tan " (a) (b) (c)  8œ! 8œ! 8œ! ∞∞ Ð"Ñ Ð"Ñ 8" #8" 8# #8" 8 #8 88 B B Ð"Ñ #8 B (d) (e) 8œ! 8œ! " #8" #8" #8# #8# Ð"Ñ BB 8 7. Find the sum of 8œ! 8# " 8" B (a) (b) (c) (d) (e) ln ln Ð"  BÑ B Ð"  BÑ B"B "B "B "B $# 8. Find the Taylor series for 0ÐBÑ œ " B $ centered at . +œ# (a) " # 8œ" "†%†(†â†Ð$8#Ñ '8 x $ 8  "  Ð"Ñ ÐB  #Ñ (b) " # 8œ" #†&†)†â†Ð$8"Ñ x $ 8 "  Ð"Ñ ÐB  #Ñ (c) " # 8œ" "†%†(†â†Ð$8#Ñ x $ 8" "  Ð"Ñ ÐB  #Ñ (d) " # 8œ" #†&†)†â†Ð$8"Ñ ' $ 8 "  Ð"Ñ ÐB  #Ñ (e) " # 8œ" #†&†)†â†Ð$8"Ñ x $ 8" "  Ð"Ñ ÐB  #Ñ 9. Suppose that a function has the following Maclaurin series: 0 Ð"Ñ B "†%†(†â†Ð$8#Ñ 8x$ 8œ! 8$ 8 8 Find the radius of convergence. (a) (b) (c) (d) (e) $ $ $" ! " $
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10. Find the sum of the series " /  â /// #x $x %x #$% (a) (b) (c) (d) (e) // " Ð / ÑÐ / Ñ / / cos sin 11. Use series to evaluate the following limit lim sin BÄ! "! * #) B B Ð B Ñ B (a) (b) (c) (d) (e) """"" #$'*# ) 12. Consider the Taylor polynomial for at . Use Taylor's inequality to X ÐBÑ 0ÐBÑ œ B + œ " $ $Î% estimate the accuracy of the approximation when lies in the interval 0ÐBÑ ¸ X ÐBÑ B $ !Þ) Ÿ B Ÿ "Þ# . (a) (b) (c) (d) "$& & "$& & "$& & "& & %x #! % %x #! ' %x "! % $x#! % "$Î% "$Î% "$Î% *Î% %%%$  (e) "& & $x#! ' *Î% $ 13. v Consider the point and the vector . Find a point such that TÐ"ß#ß#Ñ œØ"ß$ß"Ù U TU Ä has the opposite direction to . v (a) (b) (c) (d) (e) Ð"ß #ß "Ñ Ð"ß )ß %Ñ Ð#ß "ß "Ñ Ð!ß &ß $Ñ Ð%ß (ß "Ñ 14. Find the parametric equations of the line passing through the points and TÐ$ß "ß %Ñ UÐ$ß "ß &Ñ .
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sample4 - Math 1ZA3 1st Sample Test for Suggested Problems...

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