find the linear approximation of ( f(x)=ln x ) at ( x = 1 ) and use it to estimate ( ln (1.26) ).\n(…

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

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

Answer

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)