We are given `f(x)=ln x` and we are interested in approximating the value of `f(x)` near `x=1` :
(3) Thus the slope of the line tangent to the curve at (1,0) is `f'(1)=1`
(4) The equation of the line tangent to `f(x)` at (1,0) is:
`y-0=1(x-1) "or" y=x-1`
For `x` close to 1, the output for this line is approximately the same as the output for `f(x)` . If `L(x)=x-1` then `L(x)~~f(x)` for all x close to 1-- the closer to 1 the better the approximation.
(5) So `L(x)=x-1` and `f(1.12)~~L(1.12)=.12`
** The actual value for `ln(1.12)~~0.113328685307`