Find the linear approximation of 𝑓(𝑥)=ln𝑥 at 𝑥=1 and use it to estimate ln(1.06)

Question

Find the linear approximation of 𝑓(𝑥)=ln𝑥 at 𝑥=1 and use it to estimate ln(1.06). please help

Answer 1

The linear approximation at the point x=1 is a line y=Ax+B where the slope A=f'(1)=1/1=1, since f'(x)=1/x. In order to find B we use that the line should pass through the point (1,ln1=0) which implies that 0=1+B. Therefore, the tangent line is given by the formula y=x-1. To approximate the value ln(1.06) we just evaluate the number y=1.06-1=0.06.