(a) Explanation Formula used: Taylor polynomial: Let n th degree Taylor polynomial of f at a is denoted by T n ( x ) and is defined as, T n ( x ) = ∑ i = 0 n f ( i ) ( a ) i ! ( x − a ) i = f ( a ) + f ′ ( a ) 1 ! ( x − a ) + f ″ ( a ) 2 ! ( x − a ) 2 + ⋯ + f ( n ) ( a ) n ! ( x − a ) n And f is the sum of its Taylor series as f ( x ) = ∑ n = 0 ∞ f ( n ) ( a ) n ! ( x − a ) n . Calculation: The given function is f ( x ) = sin x and centered at a = 0 . The first derivative of f ( x ) is f ′ ( x ) = cos x and the corresponding value at a = 0 is f ′ ( 0 ) = cos 0 = 1 . The second derivative is f ″ ( x ) = − sin x and the corresponding value at a = 0 is f ″ ( 0 ) = − sin 0 = 0 . The third derivative is f ‴ ( x ) = − cos x and the corresponding value at a = 0 is f ‴ ( 0 ) = − cos 0 = − 1 . The fourth derivative is f ( 4 ) ( x ) = sin x and the corresponding value at a = 0 is f ( 4 ) ( 0 ) = sin 0 = 0 . The fifth derivative is f ( 5 ) ( x ) = cos x and the corresponding value at a = 0 is f ( 5 ) ( 0 ) = cos 0 = 1 (b) To determine To evaluate: The function f and its polynomials at x = π 4 , π 2 and π . (c) To determine To explain: The reason on how the Taylor polynomials converges to f ( x ) .

8th Edition

ISBN: 9780176743826

Author: Stewart

Publisher: VST

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Chapter 11.11, Problem 1E

(a)

To determine

To find: The Taylor polynomials up to degree 5 for f(x)=sinx centered at a=0 and graph f and the polynomials.

(b)

To determine

To evaluate: The function f and its polynomials at x=π4,π2 and π.

(c)

To determine

To explain: The reason on how the Taylor polynomials converges to f(x).

Expert Solution & Answer

Chapter 11 Solutions

EBK SINGLE VARIABLE CALCULUS: EARLY TRA

Ch. 11.1 - (a) What is a sequence? (b) What does it mean to...Ch. 11.1 - (a) What is a convergent sequence? Give two...Ch. 11.1 - List the first five terms of the sequence. 3....Ch. 11.1 - List the first five terms of the sequence. 4....Ch. 11.1 - List the first five terms of the sequence. 5....Ch. 11.1 - List the first five terms of the sequence. 6....Ch. 11.1 - List the first five terms of the sequence. 7....Ch. 11.1 - List the first five terms of the sequence. 8....Ch. 11.1 - Prob. 9E Ch. 11.1 - List the first five terms of the sequence. 10. a1...

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The Taylor polynomials up to degree 5 for f ( x ) = sin x centered at a = 0 and graph f and the polynomials.

Solution Summary: The author explains how to find the Taylor polynomials up to degree 5 for f(x)=mathrmsinx centered at a=0 and

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To find: The Taylor polynomials up to degree 5 for f(x)=sinx centered at a=0 and graph f and the polynomials.

To evaluate: The function f and its polynomials at x=π4,π2 and π.

To explain: The reason on how the Taylor polynomials converges to f(x).

Chapter 11 Solutions