linearization and differentials: problem 1\n(1 point)\nfind the linear approximation of ( f(x)=ln x ) at ( x…

Explanation:

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

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

Step2: Evaluate (f(x)) and (f^\prime(x)) at (x = 1)

When (x = 1), (f(1)=\ln(1)=0), and (f^\prime(1)=\frac{1}{1}=1).

Step3: Use the linear - approximation formula (L(x)=f(a)+f^\prime(a)(x - a))

Here (a = 1), so (L(x)=f(1)+f^\prime(1)(x - 1)). Substituting (f(1)=0) and (f^\prime(1)=1) into the formula, we get (L(x)=0 + 1\times(x - 1)=x-1).

Step4: Estimate (\ln(1.19)) using the linear approximation

Let (x = 1.19) in (L(x)). Then (\ln(1.19)\approx L(1.19)). Since (L(x)=x - 1), when (x = 1.19), (L(1.19)=1.19-1=0.19).

Answer:

(L(x)=x - 1); (\ln(1.19)\approx0.19)