Science:Math Exam Resources/Courses/MATH110/April 2014/Question 09 (b)/Solution 1

To know whether the linear approximation is an over or underestimate we consider the second derivative at the point $x=0.9$:

${\begin{aligned}f(x)&=\ln(x)\\f'(x)&={\frac {1}{x}}\\f''(x)&=-{\frac {1}{x^{2}}}\end{aligned}}$

Since $f''(0.9)<0$ the function is concave down at this point, and hence the linear approximation $L(x)$ is an overestimate of the $\ln(0.9)$.