Question

for the function f(x) = ln(1 + 8x), find the taylor polynomials of orders 0, 1, 2, and 3 generated by f at a = 0. p₀(x) = □

Explanation:

Step1: Recall Taylor Polynomial Order 0

The Taylor polynomial of order \( n = 0 \) for a function \( f(x) \) at \( a = 0 \) (Maclaurin polynomial of order 0) is given by \( P_0(x)=f(0) \).
First, find \( f(0) \) for \( f(x)=\ln(1 + 8x) \). Substitute \( x = 0 \) into \( f(x) \): \( f(0)=\ln(1+8\times0)=\ln(1) = 0 \).

Step2: Form \( P_0(x) \)

Since \( P_0(x)=f(0) \), and \( f(0) = 0 \), we have \( P_0(x)=0 \).

Answer:

\( P_0(x)=\boxed{0} \)