(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 f ( x ) = ∑ n = 0 ∞ f ( n ) ( a ) n ! ( x − a ) n . Calculation: The given function is f ( x ) = ln ( 1 + 2 x ) , centered at a = 0 . Then, f ( 1 ) = ln 3 . The first derivative of f ( x ) is f ′ ( x ) = 2 1 + 2 x and the corresponding value at a = 0 is f ′ ( 1 ) = 2 3 . The second derivative is f ″ ( x ) = − 4 ( 1 + 2 x ) 2 and the corresponding value at a = 0 is f ″ ( 1 ) = − 4 9 . The third derivative is f ‴ ( x ) = 16 ( 1 + 2 x ) 3 and the corresponding value at a = 0 is f ‴ ( 0 ) = 16 27 (b) To determine To estimate: The accuracy of the approximation f ( x ) ≈ T n ( x ) when x lies in the interval 0.5 ≤ x ≤ 1.5 by using Taylor’s inequality. (c) To determine To check: The result in part (b) by graphing | R n ( x ) | .

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Chapter 11.11, Problem 18E

(a)

To determine

To approximate: The function f(x)=ln(1+2x) by a Taylor polynomial with degree 3 at a=1.

(b)

To determine

To estimate: The accuracy of the approximation f(x)≈Tn(x) when x lies in the interval 0.5≤x≤1.5 by using Taylor’s inequality.

(c)

To determine

To check: The result in part (b) by graphing |Rn(x)|.

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Chapter 11.11, Problem 17E

Chapter 11.11, Problem 19E

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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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To approximate: The function f ( x ) = ln ( 1 + 2 x ) by a Taylor polynomial with degree 3 at a = 1 .

Solution Summary: The author explains how to approximate the function f(x)=mathrmln

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To approximate: The function f(x)=ln(1+2x) by a Taylor polynomial with degree 3 at a=1.

To estimate: The accuracy of the approximation f(x)≈Tn(x) when x lies in the interval 0.5≤x≤1.5 by using Taylor’s inequality.

To check: The result in part (b) by graphing |Rn(x)|.

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Chapter 11 Solutions