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

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

This preview shows pages 1–2. Sign up to view the full content.

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:

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

## This note was uploaded on 04/19/2011 for the course MATH 1804 taught by Professor Ng during the Spring '11 term at HKU.

### Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online