Question

Consider the following function f(x) = ln(1 + 2x), a = 5, n = 3, 4.8 ≤ x ≤ 5.2.

(a) Approximate f by a Taylor polynomial with degree n at the number a. t₃(x) =
a) 1 + 2(x-5) + ½(x-5)² - 2/3(x-5)³
b) 1 + (x-5) + (x-5)² - 1/3(x-5)³
c) 1 + 2(x-5) + (x-5)² - 2/3(x-5)³
d) 1 + (x-5) + ½(x-5)² - 1/3(x-5)³

Answer 1

The correct Taylor polynomial of degree 3 for f(x) = ln(1 + 2x) at a = 5 is option (a): t₃(x) = 1 + 2(x-5) + ½(x-5)² - ⅓(x-5)³.

Step-by-step explanation:

The Taylor polynomial of degree 3 for the function f(x) = ln(1 + 2x) at a = 5 is given by t₃(x). To find this polynomial, we calculate the derivatives of f at x = a and formulate the Taylor polynomial based on these derivatives. Using the general formula for the nth degree Taylor polynomial at a, the correct Taylor polynomial for the given function at a = 5 is:

t₃(x) = 1 + 2(x-5) + ½(x-5)² - ⅓(x-5)³

This corresponds to option (a) from the provided choices. The first derivative of f at x = 5 gives the linear term, the second derivative gives the quadratic term, and the third derivative provides the cubic term in the polynomial.