Question

find the linear approximation of ( f(x)=ln x ) at ( x = 1 ) and use it to estimate ( ln (1.26) ).
( l(x)=square )
( ln 1.26 approx square )

Explanation:

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

The function is \(f(x)=\ln x\). The derivative \(f^{\prime}(x)=\frac{1}{x}\).

Step2: Use the linear approximation formula

The linear - approximation formula is \(L(x)=f(a)+f^{\prime}(a)(x - a)\). Here \(a = 1\), \(f(1)=\ln(1)=0\), and \(f^{\prime}(1)=\frac{1}{1}=1\). So \(L(x)=0 + 1\times(x - 1)=x - 1\).

Step3: Estimate \(\ln(1.26)\)

Substitute \(x = 1.26\) into \(L(x)\). Then \(L(1.26)=1.26-1\).

Answer:

\(L(x)=x - 1\); \(\ln(1.26)\approx0.26\)