Explanation Given: The function f ( x ) is, f ( x ) = { e − 1 x 2 for x ≠ 0 0 for x = 0 Proof: The first derivative of the function f ( x ) is, f ′ ( x ) = − ( − 2 x 3 ) e − 1 x 2 = 2 e − 1 x 2 x 3 Compare the above equation with f ( k ) ( x ) = P ( x ) e − 1 x 2 x r to obtain the value of P ( x ) , r and n as P ( x ) = 2 r = 3 n = 1 Consider f ( k ) ( x ) = P ( x ) e − 1 x 2 x r , then f ( k + 1 ) ( x ) can be calculated as, f ( k + 1 ) ( x ) = x r ( P ′ ( x ) e − 1 x 2 + P ( x ) ( 2 x 3 ) e − 1 x 2 ) − P ( x ) e − 1 x 2 ( r x r − 1 ) ( x r ) 2 = e − 1 x 2 ( x r ( P ′ ( x ) + P ( x ) 2 x 3 ) − P ( x ) r x r − 1 ) x 2 r = e − 1 x 2 ( x r ( x 3 P ′ ( x ) + 2 P ( x ) − x 3 P ( x ) r x − 1 ) x 2 r x 3 ) = e − 1 x 2 ( x r ( x 3 P ′ ( x ) + P ( x ) ( 2 − r x 2 ) ) x r x r + 3 ) Simplify the above equation as, f (

4th Edition

ISBN: 9781319050740

Author: Jon Rogawski, Colin Adams, Robert Franzosa

Publisher: W. H. Freeman

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Chapter 4.5, Problem 77E

To determine

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That for k≥1 and x≠0, f(k)(x)=P(x)e−1x2xr is true.

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Chapter 4.5, Problem 76E

Chapter 4.5, Problem 78E

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Chapter 4 Solutions

Calculus: Early Transcendentals

Ch. 4.1 - Prob. 1PQ Ch. 4.1 - Prob. 2PQ Ch. 4.1 - Prob. 3PQ Ch. 4.1 - Prob. 4PQ Ch. 4.1 - Prob. 1E Ch. 4.1 - Prob. 2E Ch. 4.1 - Prob. 3E Ch. 4.1 - Prob. 4E Ch. 4.1 - Prob. 5E Ch. 4.1 - Prob. 6E

Ch. 4.1 - Prob. 7E Ch. 4.1 - Prob. 8E Ch. 4.1 - Prob. 9E Ch. 4.1 - Prob. 10E Ch. 4.1 - Prob. 11E Ch. 4.1 - Prob. 12E Ch. 4.1 - Prob. 13E Ch. 4.1 - Prob. 14E Ch. 4.1 - Prob. 15E Ch. 4.1 - Prob. 16E Ch. 4.1 - Prob. 17E Ch. 4.1 - Prob. 18E Ch. 4.1 - Prob. 19E Ch. 4.1 - Prob. 20E Ch. 4.1 - Prob. 21E Ch. 4.1 - Prob. 22E Ch. 4.1 - Prob. 23E Ch. 4.1 - Prob. 24E Ch. 4.1 - Prob. 25E Ch. 4.1 - Prob. 26E Ch. 4.1 - Prob. 27E Ch. 4.1 - Prob. 28E Ch. 4.1 - Prob. 29E Ch. 4.1 - Prob. 30E Ch. 4.1 - Prob. 31E Ch. 4.1 - Prob. 32E Ch. 4.1 - Prob. 33E Ch. 4.1 - Prob. 34E Ch. 4.1 - Prob. 35E Ch. 4.1 - Prob. 36E Ch. 4.1 - Prob. 37E Ch. 4.1 - Prob. 38E Ch. 4.1 - Prob. 39E Ch. 4.1 - Prob. 40E Ch. 4.1 - Prob. 41E Ch. 4.1 - Prob. 42E Ch. 4.1 - Prob. 43E Ch. 4.1 - Prob. 44E Ch. 4.1 - Prob. 45E Ch. 4.1 - Prob. 46E Ch. 4.1 - Prob. 47E Ch. 4.1 - Prob. 48E Ch. 4.1 - Prob. 49E Ch. 4.1 - Prob. 50E Ch. 4.1 - Prob. 51E Ch. 4.1 - Prob. 52E Ch. 4.1 - Prob. 53E Ch. 4.1 - Prob. 54E Ch. 4.1 - Prob. 55E Ch. 4.1 - Prob. 56E Ch. 4.1 - 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