Suppose you want to estimate ln(1.1). The n-th Taylor polynomial Tn(x) of the function f(x) =...

Transcribed Image Text:

Suppose you want to estimate ln(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 ln(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 In(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)¯”. Suppose you want to estimate ln(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 ln(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 In(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)¯”.