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# exam1solutions - MATH1502 E sections Calculus II TEST 1...

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MATH1502, E sections - Calculus II TEST 1 - January 29 - 2009 NAME : ___________________________________ STUDENT NUMBER :_______________________ GROUP (e.g. E3) :_______________________ TEACHING ASSISTANT :_____________________ Write your solutions to the questions on this testpaper - you may use both sides of each sheet of paper. There are 53 marks on this paper. Full marks (100%) is 50 marks. You may NOT use a calculator or any notes. Question Points Ex 1 15 2 7 3 25 4 6 Total 53 ) 50 Question 1 Let f ( x ) = ln(1 2 x ) ln(1 + x ) : (a) Compute the 3 rd degree Taylor polynomial P 3 ( x ) of f (about 0 ). (6 marks) (b) Estimate the maximum error of j R 3 ( x ) j = j f ( x ) P 3 ( x ) j for x 2 & 0 ; 1 4 ± : (Hint: use 1 2 c ± 1 2 for c between 0 and 1 4 ) : (6 marks) (c) Use (a) to write down (without proof) the 6 th degree Taylor polynomial P 6 (about 0 ) to g ( x ) = ln ² 1 2 x 2 1 + 2 x 2 ³ ln ² 1 + x 2 1 x 2 ³ : (3 marks) Solutions 1

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(a) f ( x ) = ln(1 2 x ) ln(1 + x ) ) f (0) = ln 1 ln 1 = 0; f 0 ( x ) = 2 1 2 x 1 1 + x ) f 0 (0) = 2 1 = 3; f 00 ( x ) = 4 (1 2 x ) 2 + 1 (1 + x ) 2 ) f 00 (0) = 4 + 1 = 3; f 000 ( x ) = 16 (1 2 x ) 3 2 (1 + x ) 3 ) f 000 (0) = 16 2 = 18; (4 marks) f (4) ( x ) = 96 (1 2 x ) 4 + 6 (1 + x ) 4 : So P 3 ( x ) = f (0) + f 0 (0) x + f 00 (0) x 2 2! + f 000 (0) x 3 3! = 0 3 x 3 x 2 2! 18
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exam1solutions - MATH1502 E sections Calculus II TEST 1...

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