Question

for the function ( f(x)=ln (1 + 4x) ), find the taylor polynomials of orders 0, 1, 2, and 3 generated by ( f ) at ( a = 0 ).
( p_{0}(x)=0 )
( p_{1}(x)=4x )
( p_{2}(x)=4x - 8x^{2} )
( p_{3}(x)=square )

Explanation:

Step1: Recall the Taylor polynomial formula

The Taylor polynomial of order \(n\) for a function \(f(x)\) about \(a = 0\) (Maclaurin polynomial) is \(P_n(x)=\sum_{k = 0}^{n}\frac{f^{(k)}(0)}{k!}x^{k}\), where \(f^{(k)}(x)\) is the \(k\) - th derivative of \(f(x)\).

First, find the derivatives of \(f(x)=\ln(1 + 4x)\):

• \(f(x)=\ln(1 + 4x)\), so \(f(0)=\ln(1+4\times0)=0\)
• \(f^{\prime}(x)=\frac{4}{1 + 4x}\), then \(f^{\prime}(0)=\frac{4}{1+4\times0}=4\)
• \(f^{\prime\prime}(x)=\frac{-16}{(1 + 4x)^{2}}\), so \(f^{\prime\prime}(0)=\frac{-16}{(1 + 0)^{2}}=-16\)
• \(f^{\prime\prime\prime}(x)=\frac{128}{(1 + 4x)^{3}}\), and \(f^{\prime\prime\prime}(0)=\frac{128}{(1+0)^{3}} = 128\)

Step2: Calculate \(P_3(x)\)

$$
LATEXBLOCK0
$$

Answer:

\(P_3(x)=4x - 8x^{2}+\frac{64}{3}x^{3}\)