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

We use the point slope formula

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

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

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

Solving gives

$\displaystyle y=x-1$

Thus, the approximation we seek is given by

$\displaystyle y=1.25-1=0.25$.