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Assignment4(1804)

# Assignment4(1804) - x 4 5 4 Compute the following limits...

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A4/MATH1804/2009-10/2nd THE UNIVERSITY OF HONG KONG DEPARTMENT OF MATHEMATICS MATH1804 University Mathematics A Assignment 4 Due date : Mar 26, 2010 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://147.8.101.93/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. Find an approximate value for cos 36 so that the error is less than 0 . 0002 . [ Suggestion : Let f ( x ) = cos x . Then cos 36 = f ( π/ 5) . Consider the n -th Taylor polyno- mial of f at x = π/ 6 .] 2. Find f 0 (0) for the following functions: (a) f ( x ) = ln (3 x + 2) 3 x + 2 (b) f ( x ) = 3 4 x 2 - 2 (c) f ( x ) = e x - e - x e x + e - x 3. Use logarithmic differentiation to find the derivative of the following functions: (a) f ( x ) = x 4 (5 + 3 x ) x 2 + 1 (b) f ( x ) = 1 + x

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Unformatted text preview: x ) 4 / 5 4. Compute the following limits: lim x → + ∞ ± 1 + 1 x ² 4 x , lim x → + ∞ ± 1 + 2 x ² 3 x , lim x → + ∞ ± 1 + 1 2 x ² 3 x 5. Find the second Taylor polynomial P 2 ( x ) of the following functions at the given point: ( a ) f ( x ) = 5 x 2 at x = 1 , ( b ) f ( x ) = e sin x at x = 0 , ( c ) f ( x ) = ln(cos x ) at x = 0 1 6. Find the following indeﬁnite integrals: (a) Z 1 x 2-1 x dx (b) Z 1 + 2 x 3 x dx (c) Z √ x + 3 cos x + e x dx 7. (a) (i) Find d dx ± 1 3 sin 3 x ² . (ii) Use the Fundamental Theorem of Calculus to ﬁnd Z π/ 2 sin 2 x cos xdx . (b) Use the Fundamental Theorem of Calculus to ﬁnd d dx Z 4 x x 2 cos tdt . 8. Deﬁne a function F ( x ) = Z x e-t 2 dt (a) Show that F ( x ) is strictly increasing on R . (b) Find all points of inﬂection of F ( x ) . 2...
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Assignment4(1804) - x 4 5 4 Compute the following limits...

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