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)=square )

Explanation:

Step1: Find the second - derivative of \(f(x)\)

The Taylor polynomial of order \(n\) for a function \(f(x)\) about \(a = 0\) is given by \(P_{n}(x)=\sum_{k = 0}^{n}\frac{f^{(k)}(0)}{k!}x^{k}\), where \(f^{(k)}(x)\) is the \(k\) - th derivative of \(f(x)\).
We know \(f(x)=\ln(1 + 4x)\), \(f^{\prime}(x)=\frac{4}{1+4x}\), \(f^{\prime\prime}(x)=\frac{-16}{(1 + 4x)^{2}}\)
Evaluate at \(x = 0\): \(f(0)=\ln(1)=0\), \(f^{\prime}(0)=4\), \(f^{\prime\prime}(0)=-16\)

Step2: Calculate \(P_{2}(x)\)

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Answer:

\(P_{2}(x)=4x - 8x^{2}\)