Assignment4(1804)(v2-22-3-11)

Assignment4(1804)(v2-22-3-11) - (a) f ( x ) = sin 2 x √ 2...

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A4/MATH1804/2010-11/2nd THE UNIVERSITY OF HONG KONG DEPARTMENT OF MATHEMATICS MATH1804 University Mathematics A Assignment 4 Due date : Mar 29, 2011 before 17:00. Remember to write down your Name , Uni. number and Tutorial Group number . You are welcome to see the instructor or the demonstrators if you have any difficulties. See “Course information” at http://hkumath.hku.hk/course/MATH1804 for availabilities. Please drop your work in the assignment box marked MATH1804 on the 4th floor of Run Run Shaw Building. No late work will be accepted. 1. Let f ( x ) be a function satisfying f (1) = 2 , f 0 (1) = - 1 , f 00 (1) = 0 and f 000 (1) = - 2 . (a) Use the third Taylor polynomial P 3 ( x ) of f ( x ) at x = 1 to approximate f (1 . 1) . (b) If f (4) ( x ) = - ( x 2 + 1) , determine an upper bound for the error committed. 2. Find an approximate value for sin 36 so that the error is less than 0 . 0002 . [ Suggestion : Let f ( x ) = sin x . Then sin 36 = f ( π/ 5) . Consider the n -th Taylor polyno- mial of f at x = π/ 6 .] 3. Find f 0 ( x ) for the following functions: (a) f ( x ) = 3 3 x 2 - 2 (b) f ( x ) = e x - e - x e x + e - x (c) f ( x ) = log 5 (5 x 2 ) 4. Use logarithmic differentiation to find the derivative of the following functions:
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Unformatted text preview: (a) f ( x ) = sin 2 x √ 2 x 2 + 3 (b) f ( x ) = (3 x ) cos x 5. Compute the following limits: lim x → + ∞ ± 1 + 1 x ² 2 x , lim x → + ∞ ± 1 + 3 x ² 4 x , lim x → + ∞ ± 1 + 1 5 x ² 2 x 1 6. Find the second Taylor polynomial P 2 ( x ) of the following functions at x = 0 : ( a ) f ( x ) = e x sin x ( b ) f ( x ) = ln(cos x + 1) 7. (Revision on Indefinite Integrals) Apply the basic integration formulas to find the following indefinite integrals: (a) Z x 2-3 √ x dx (b) Z ( √ 2 x + e 2 x-3) dx (c) Z 1 + sin 2 x sin 2 x dx 8. (Fundamental Theorem of Calculus) (a) (i) Find d dx ± 1 3 (sin x + 1) 3 ² . (ii) Use the Fundamental Theorem of Calculus to find Z π/ 2 (sin x + 1) 2 cos xdx . (b) Use the Fundamental Theorem of Calculus to find: (i) Z 2-2 | x | dx . (ii) Z π/ 3 4 sec u tan udu . (iii) d dx Z √ x sin( t 2 ) dt . (iv) d dx Z 3 x √ x e t t dt . 2...
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Assignment4(1804)(v2-22-3-11) - (a) f ( x ) = sin 2 x √ 2...

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