Second-Order Taylor Polynomial Remainder Term for f(x) = x^3 * ln(x), a = 1

[anon90]

(a) I found that second-order Taylor polynomial P2(x) for the function f(x) = x^3 * ln(x), a = 1 is (x - 1) + 5/2 * (x - 1) ^ 2.

(b) Use Taylor's Theorem to give the Remainder Term R2(x) = f(x) - P2(x), as a function of x and some z between 1 and x.

What I came up for (b) was (x ^ 3 * ln(x)) - ((x - 1) + 5/2 * (x - 1) ^ 2) but it doesn't seem to be right.

This is an online assignment. Therefore, (a) is shown to be right but (b) is not. Need help with part (b).

Any help is much appreciated. Thank you for your time and effort.

 

[Prove It]

(a) I found that second-order Taylor polynomial P2(x) for the function f(x) = x^3 * ln(x), a = 1 is (x - 1) + 5/2 * (x - 1) ^ 2.

(b) Use Taylor's Theorem to give the Remainder Term R2(x) = f(x) - P2(x), as a function of x and some z between 1 and x.

What I came up for (b) was (x ^ 3 * ln(x)) - ((x - 1) + 5/2 * (x - 1) ^ 2) but it doesn't seem to be right.

This is an online assignment. Therefore, (a) is shown to be right but (b) is not. Need help with part (b).

Any help is much appreciated. Thank you for your time and effort.

The Mean Value Theorem may help you to find the remainder term.