Assignment 4

# Assignment 4 - c . 4. [9 marks: 3 marks each] Evaluate the...

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DEPARTMENT OF MATHEMATICS MATHS 208 SC 2008 Assignment 4 Due: Tuesday 14 th October 1. [6 marks: 3 marks each] Use the ratio test to determine if the following series are convergent or divergent. (a) = 1 ! 2 n n n (b) = 1 3 ) ln( n n n . 2. [8 marks: 4 marks each] Find (i) the Taylor polynomials of degree 3 about the centre 0 = c for the following functions; and (ii) the error at 2 . 0 = x in using each Taylor polynomial to approximate the original functions. (a) x x f + = 1 ) ( (b) ) 1 ln( ) ( x x f + = . 3. [15 marks: 5 marks each] Find (i) the Taylor series for each of the following functions and (ii) the interval of convergence of each series, using the Ratio Test. (a) ) 2 cos( x about 0 = c (b) x 3 2 1 about 0 = c (c) ) 2 ln( x about 0 =
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Unformatted text preview: c . 4. [9 marks: 3 marks each] Evaluate the following: (a) dt t t ) ln( (b) d ) cos( ) ( sin 2 (c) dx x x + 2 1 2 . 5. (a) [3 marks] Show that 3 2 3 x y = is the solution of the initial value problem x y dx dy 2 = , 2 3 ) 1 ( = y . (b) [10 marks: 5 marks each] Find the general solution (whether explicit or implicit) of the following differential equations: (i) ) ln( ) 1 ( 1 t dt dy t y = + ; (ii) y x y dx dy ) 4 ( 2 2 + = . (c) [2 marks] Use your result from part (b) to find the solution to the following initial value problem: 2 ) ( , ) 4 ( 2 2 = + = y y x y dx dy . [Total: 53 marks]...
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## This note was uploaded on 10/17/2010 for the course DF aadd taught by Professor D during the Spring '10 term at Clayton.

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