Science:Math Exam Resources/Courses/MATH104/December 2013/Question 01 (i)/Solution 1

We use the point slope formula

$\frac{y-f(a)}{x-a}=f^{\prime }(a).$

Since $a=1$ we obtain $f(a)=\ln (1)=0$, while $f^{\prime }(x)=\frac{1}{x}$ implies $f^{\prime }(a)=1$. Hence

$\frac{y-0}{x-1}=1.$

Solving gives

$y=x-1$

Thus, the approximation we seek is given by

$y=1.25-1=0.25$.