Chapter 5 - Trigonometric Functions - Section 5.5 Graphs of the Tangent, Cotangent, Cosecant, and Secant Functions - 5.5 Assess Your Understanding - Page 442: 42

$\frac{4\sqrt 3-6}{\pi}$.

1. Given $f(x)=sec(x)$, we have $f(0)=sec(0)=1$ and $f(\frac{\pi}{6})=sec(\frac{\pi}{6})=\frac{2\sqrt 3}{3}$, 2. The rate of change required is given by $\frac{f(\frac{\pi}{6})-f(0)}{\frac{\pi}{6}-0}=\frac{\frac{2\sqrt 3}{3}-1}{\frac{\pi}{6}}=\frac{4\sqrt 3-6}{\pi}$.