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Ma1502Te1-Fall07Solns

# Ma1502Te1-Fall07Solns - MATH1502 Calculus II TEST 1...

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MATH1502 - Calculus II TEST 1 - September 18 - 2007 NAME : ___________________________________ STUDENT NUMBER :_______________________ Write your solutions to the questions on this testpaper - you may use both sides of each sheet of paper. There are 55 marks on this paper. Full marks (100%) is 50 marks. You may NOT use a calculator or any notes. Question Points Ex 1 16 2 8 3 24 4 7 Total 55 ) 50 Question 1 Let f ( x ) = ln (1 + 2 x ) ° ln (1 + x ) : (i) Compute the 3 rd degree Taylor polynomial P 3 ( x ) of f (about 0 ) and also compute the Lagrange form of the remainder R 3 ( x ) . (8 marks) (ii) Estimate the maximum error of j R 3 ( x ) j = j f ( x ) ° P 3 ( x ) j for x in [0 ; 1] : (4 marks) (iii) Use (i) to write down (without proof) the 6 th degree Taylor polynomial P 6 (about 0 ) to g ( x ) = ln ° 1 + 2 x 2 1 ° 2 x 2 ± ° ln ° 1 + x 2 1 ° x 2 ± : (4 marks) Solutions (i) f ( x ) = ln (1 + 2 x ) ° ln (1 + x ) ) f (0) = ln 1 ° ln 1 = 0; 1

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f 0 ( x ) = 2 1 + 2 x ° 1 1 + x ) f 0 (0) = 1; f 00 ( x ) = ° 4 (1 + 2 x ) 2 + 1 (1 + x ) 2 ) f 00 (0) = ° 3; f 000 ( x ) = ° 4 ( ° 2) (2) (1 + 2 x ) 3 ° 2 (1 + x ) 3 = 16 (1 + 2 x ) 3 ° 2 (1 + x ) 3 ) f 000 (0) = 14; f (4) ( x ) = 16 ( ° 3) 2 (1 + 2 x ) 4 ° 2 ( ° 3) (1 + x ) 4 = ° 96 (1 + 2 x ) 4 + 6 (1 + x ) 4 : Then P 3 ( x ) = f (0) + f 0 (0) x + f 00 (0) x 2 2! + f 000 (0) x 3 3! = 0 + x ° 3 x 2 2! + 14 x 3 3! = x ° 3 2 x 2 + 7 3 x 3 : (8 marks) Graders: please give up to 5 marks for di/erentation, and one mark for writing down the de°nition of the Taylor polynomial, 2 marks for substituting correctly. (ii) The Lagrange form of the remainder is R 3 ( x ) = 1 4! f (4) ( c ) x 4 = ° 1 4!
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Ma1502Te1-Fall07Solns - MATH1502 Calculus II TEST 1...

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