a. Explanation Given information: Plot f ( x ) = 1 1 + x together with the Taylor polynomials T n at a = 1 for 1 ≤ n ≤ 4 on the interval [ − 2 , 8 ] . Formula used: The Taylor series is T ( x ) = ∑ n = 0 ∞ f ( n ) ( c ) n ! ( x − c ) n Where f ( x ) is represented by a power series centred at c in an interval | x − c | < R with R > 0 . Calculation: Consider the given function. f ( x ) = 1 x + 1 ⇒ f ( 1 ) = 1 2 Now use successive differentiation and substitute x = 1 . f ' ( x ) = − 1 ( x + 1 ) 2 ⇒ f ' ( 1 ) = − 1 4 f ' ' ( x ) = 2 ( x + 1 ) 3 ⇒ f ' ' ( 1 ) = 1 4 f ' ' ' ( x ) = − 6 ( x + 1 ) 4 ⇒ f ' ' ' ( 1 ) = − 3 8 f i v ( x ) = 24 ( x + 1 ) 5 ⇒ f i v ( 1 ) = 3 4 The Taylor series of f ( x ) which is centred at c is given by T ( x ) = ∑ n = 0 ∞ f ( n ) ( c ) n ! ( x − c ) n The Taylor series of f ( x ) = 1 x + 1 which is centred at c = 1 is given by T ( x ) = ∑ n = 0 ∞ f ( n ) ( 1 ) n ! ( x − 1 ) n T ( x ) = f ( 0 ) ( 1 ) 0 ! ( x − 1 ) 0 + f 1 ( 1 ) 1 ! ( x − 1 ) 1 + f 2 ( 1 ) 2 ! ( x − 1 ) 2 + f 3 ( 1 ) 3 ! ( x − 1 ) 3 + .. b. To determine To explain: The happening for x < − 1. c. To determine To plot : T 30 together with f on [ − 2 , 8 ] and to find an interval over which T 30 appear to approximate f closely.

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Author: Rogawski

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Chapter 11.7, Problem 36E

To determine

To find:The interval over which T4 appear to approximate f closely.

To determine

To explain:The happening for x<−1.

To determine

To plot: T30 together with f on [−2,8] and to find an interval over which T30 appear to approximate f closely.

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Chapter 11.7, Problem 35E

Chapter 11.7, Problem 37E

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Chapter 11 Solutions

CALCULUS (CLOTH)

Ch. 11.1 - Prob. 1PQ Ch. 11.1 - Prob. 2PQ Ch. 11.1 - Prob. 3PQ Ch. 11.1 - Prob. 4PQ Ch. 11.1 - Prob. 5PQ Ch. 11.1 - Prob. 1E Ch. 11.1 - Prob. 2E Ch. 11.1 - Prob. 3E Ch. 11.1 - Prob. 4E Ch. 11.1 - Prob. 5E

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The interval over which T 4 appear to approximate f closely.

Solution Summary: The author explains that the interval over which T_4 appear to approximate f closely is [0.1,2].

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To find:The interval over which T4 appear to approximate f closely.

To explain:The happening for x<−1.

To plot: T30 together with f on [−2,8] and to find an interval over which T30 appear to approximate f closely.

Want to see the full answer?

Chapter 11 Solutions