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Calculus Suppose you want to estimate In(1.1). The n-th Taylor polynomial Tn(x) of the function f(x) = ln(1 + x), centered around c = 0, can be used to find the approximate value of In(1.1) by setting x = 0.1 in Tn(2). Use the formula for the error term En(x) in Proposition 2.9 to de- termine the smallest value of n so that Tn (0.1) approximates ln(1.1) to within ±0.000001. In other words, you need to find a value of n so that En (0.1)| < 10-6 (and you need to show that your value of n works!). Hint: The n-th derivative of f(x) is given by the formula f(n)(x) = (-1)n-¹ (n − 1)!(1+x)¯n. -

Calculus Suppose you want to estimate In(1.1). The n-th Taylor polynomial Tn(x) of the function f(x) = ln(1 + x), centered around c = 0, can be used to find the approximate value of In(1.1) by setting x = 0.1 in Tn(2). Use the formula for the error term En(x) in Proposition 2.9 to de- termine the smallest value of n so that Tn (0.1) approximates ln(1.1) to within ±0.000001. In other words, you need to find a value of n so that En (0.1)| < 10-6 (and you need to show that your value of n works!). Hint: The n-th derivative of f(x) is given by the formula f(n)(x) = (-1)n-¹ (n − 1)!(1+x)¯n. -

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Calculus Suppose you want to estimate In(1.1). The n-th Taylor polynomial Tn(x) of the function f(x) = ln(1 + x), centered around c = 0, can be used to find the approximate value of In(1.1) by setting x = 0.1 in Tn(2). Use the formula for the error term En(x) in Proposition 2.9 to de- termine the smallest value of n so that Tn (0.1) approximates ln(1.1) to within ±0.000001. In other words, you need to find a value of n so that En (0.1)| < 10-6 (and you need to show that your value of n works!). Hint: The n-th derivative of f(x) is given by the formula f(n)(x) = (-1)n-1 (n − 1)!(1 + x)¯n. -
Transcribed Image Text:Calculus Suppose you want to estimate In(1.1). The n-th Taylor polynomial Tn(x) of the function f(x) = ln(1 + x), centered around c = 0, can be used to find the approximate value of In(1.1) by setting x = 0.1 in Tn(2). Use the formula for the error term En(x) in Proposition 2.9 to de- termine the smallest value of n so that Tn (0.1) approximates ln(1.1) to within ±0.000001. In other words, you need to find a value of n so that En (0.1)| < 10-6 (and you need to show that your value of n works!). Hint: The n-th derivative of f(x) is given by the formula f(n)(x) = (-1)n-1 (n − 1)!(1 + x)¯n. -
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