Question

1. - / 5.55 points if f(x) = sec x, find f(π/6). f(π/6) =

Explanation:

Step1: Compute first derivative of \(f(x)\)

\(f'(x) = \sec x \tan x\)

Step2: Compute second derivative using product rule

\(f''(x) = \sec x \tan^2 x + \sec^3 x\)

Step3: Substitute \(x = \pi/6\)

\(\sec(\pi/6) = \frac{2}{\sqrt{3}}\), \(\tan(\pi/6) = \frac{1}{\sqrt{3}}\)
\(f''(\pi/6) = \frac{2}{\sqrt{3}} \cdot (\frac{1}{\sqrt{3}})^2 + (\frac{2}{\sqrt{3}})^3 = \frac{2}{\sqrt{3}} \cdot \frac{1}{3} + \frac{8}{3\sqrt{3}} = \frac{2 + 8}{3\sqrt{3}} = \frac{10}{3\sqrt{3}} = \frac{10\sqrt{3}}{9}\)

Answer:

\(\frac{10\sqrt{3}}{9}\)