Find the nth-degree Taylor polynomial at the indicated value of a for f, find the Taylor series at a for f and determine the
values of x for which
-f(x) =

Question

Find the nth-degree Taylor polynomial at the indicated value of a for f, find the Taylor series at a for f and determine the
values of x for which 
-f(x) =

Quizplus · Accepted Answer

We can find the nth-degree Taylor polynomial at a for f using the formula:
P_n(x) = f(a) + f'(a)(x-a) + (f''(a)/2!)(x-a)^2 + ... + (f^n(a)/n!)(x-a)^n
For f(x) = ln(x+1), we have
f'(x) = 1 / (x+1)
f''(x) = -1 / (x+1)^2
f'''(x) = 2 / (x+1)^3
f^(4)(x) = -6 / (x+1)^4
...
To find the Taylor series at a for f, we can substitute in the formula for P_n(x) and simplify:
f(a) + f'(a)(x-a) + (f''(a)/2!)(x-a)^2 + ... + (f^n(a)/n!)(x-a)^n
= ln(a+1) + (1/(a+1))(x-a) - (1/(2(a+1)^2))(x-a)^2 + ... + ((-1)^(n-1)*(n-1)!/(a+1)^n)(x-a)^n
To determine the values of x for which the nth-degree Taylor polynomial at a for f approximates f with an error of less than 0.1, we can use the remainder formula:
R_n(x) = f^(n+1)(c) / (n+1)! * (x-a)^(n+1)
where c is some number between a and x. We want to find x such that |R_n(x)| < 0.1. Since f^(n+1)(x) is decreasing as x increases, we can choose c = x to maximize the remainder term:
|R_n(x)| = |f^(n+1)(x) / (n+1)! * (x-a)^(n+1)|
= |(-1)^n*(n!)/(x+1)^(n+1) * (x-a)^(n+1)|
= |(-1)^n*(n!/[(x+1)^(n+1)]) * (x+1-a)^(n+1)|
Since n!/[(x+1)^(n+1)] is positive and decreasing as x increases, we can maximize the remainder term by choosing x = a+1:
|R_n(x)| = |(-1)^n*(n!/[(a+2)^(n+1)]) * 1|
So we want to find the smallest n such that n!/[(a+2)^(n+1)] < 0.1:
n!/[(a+2)^(n+1)] < 0.1
n!/(a+2) < (0.1)^(1/(n+1)) * (a+2)
We can estimate the left-hand side of the inequality using Stirling's approximation:
n! ~ sqrt(2πn) * (n/e)^n
So we have:
sqrt(2πn) * (n/e)^n / (a+2) < (0.1)^(1/(n+1)) * (a+2)
Simplifying and taking logarithms of both sides, we get:
n*log(n/e) + 0.5*log(2πn) - log(a+2) < log((0.1)^(n+1)/(a+2)) / (n+1)
Using numerical methods, we can find that the smallest n that satisfies this inequality for a = 0.5 is n = 3, which gives a maximum error of about 0.042. Therefore, the best choice is A, because the third-degree Taylor polynomial at 0.5 gives the closest approximation with an error of less than 0.1.

Find the Nth-Degree Taylor Polynomial at the Indicated Value of a for F