Explanation Given Info: Q ( x ) = [ ( x − a 1 ) ( x − a 2 ) .... ( x − a n ) ] Formula Used: ith product formula Q ' ( x ) = ( a j − a 1 ) .. ( a j − a j − 1 ) ( a j − a j − 1 ) ... ( a j − a n ) Calculation: To differentiate Q ( x ) then,first take the logarithms of both side we have ln Q ( x ) = ln [ ( x − a 1 ) ( x − a 2 ) .... ( x − a n ) ] ln Q ( x ) = ln ( x − a 1 ) + ln ( x − a 2 ) + .... ln ( x − a n ) Now, d d x ln Q ( x ) = Q ' ( x ) Q ( x ) d d x ln Q ( x ) = 1 x − a 1 + 1 x − a 2 + .... 1 x − a n Multiplication by Q ( x ) on both side Q ' ( x ) = Q ( x ) [ 1 x − a 1 + 1 x − a 2 + .... 1 x − a n ] Q ' ( x ) = ( x − a 2 ) ( x − a 3 ) ..... ( x − a n ) + ( x − a 1 ) ( x − a 3 ) ..... ( x − a n ) + ... + ( x − a 1 ) ( x − a 2 ) Now apply ith product formula Q ' ( x ) has the ( x − a i ) Q ' ( x ) = ( a j − a 1 ) ... To determine (b) To find the partial fraction of P ( x ) = 2 x 2 − 1 and Q ( x ) = ( x + 1 ) ( x − 2 ) ( x − 3 )

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Question

Chapter 8.5, Problem 62E

To determine

(a)

To prove that $A_{j} = \frac{P (a_{j})}{Q' (a_{j})}$

To determine

(b)

To find the partial fraction of $P (x) = 2 x^{2} - 1$ and $Q (x) = (x + 1) (x - 2) (x - 3)$

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Chapter 8.5, Problem 61E

Chapter 8.6, Problem 1PQ

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00 (a) Starting with the geometric series Σ X^, find the sum of the series n = 0 00 Σηχη - 1, |x| < 1. n = 1 (b) Find the sum of each of the following series. 00 Σnx", n = 1 |x| < 1 (ii) n = 1 sin (c) Find the sum of each of the following series. (i) 00 Σn(n-1)x^, |x| <1 n = 2 (ii) 00 n = 2 n² - n 4n (iii) M8 n = 1 շո

(a) Use differentiation to find a power series representation for 1 f(x) = (4 + x)²* f(x) = 00 Σ n = 0 What is the radius of convergence, R? R = (b) Use part (a) to find a power series for f(x) = 1 (4 + x)³° f(x) = 00 Σ n = 0 What is the radius of convergence, R? R = (c) Use part (b) to find a power series for f(x) = x² (4 + x)³* 00 f(x) = Σ n = 2 What is the radius of convergence, R? R = Need Help? Read It Watch It SUBMIT ANSWER

answer for question 4 please

Chapter 8 Solutions

CALCULUS (CLOTH)

Ch. 8.1 - Prob. 1PQ Ch. 8.1 - Prob. 2PQ Ch. 8.1 - Prob. 3PQ Ch. 8.1 - Prob. 1E Ch. 8.1 - Prob. 2E Ch. 8.1 - Prob. 3E Ch. 8.1 - Prob. 4E Ch. 8.1 - Prob. 5E Ch. 8.1 - Prob. 6E Ch. 8.1 - Prob. 7E

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