# How do I find the average rate of change of f(x) = sec x from 0 to pi/4?

Sep 19, 2014

Average rate of change = $\frac{f \left(b\right) - f \left(a\right)}{b - a}$, where $b$ is the upper bound and $a$ is the lower bound.

$f \left(x\right) = \sec x$

$\frac{f \left(\frac{\pi}{4}\right) - f \left(0\right)}{\frac{\pi}{4} - 0} = \frac{\sec \left(\frac{\pi}{4}\right) - \sec \left(0\right)}{\frac{\pi}{4}} = \frac{\sqrt{2} - 1}{\frac{\pi}{4}} = \frac{0.4142}{0.7854} = 0.5274$

Things to remember:

Review the unit circle

$\frac{\pi}{4} = 45$, degrees and is a special triangle, $45 , 45 , 90 : 1 , 1 , \sqrt{2}$

$\cos \left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}$

$\sec \left(\frac{\pi}{4}\right) = \frac{1}{\cos \left(\frac{\pi}{4}\right)} = \frac{1}{\frac{1}{\sqrt{2}}} = \frac{1}{1} \cdot \frac{\sqrt{2}}{1} = \frac{\sqrt{2}}{1} = \sqrt{2}$

$\cos \left(0\right) = 1$

$\sec \left(0\right) = \frac{1}{\cos \left(0\right)} = \frac{1}{1} = 1$