12. -/1 points if f(t) = sec(t), find f(π/4). f(π/4) =

Explanation:

Step1: Find first - derivative

We know that if $y = \sec(t)$, then $y'=\sec(t)\tan(t)$ by the derivative formula $\frac{d}{dt}\sec(t)=\sec(t)\tan(t)$.

Step2: Find second - derivative

Using the product rule $(uv)' = u'v + uv'$, where $u=\sec(t)$ and $v = \tan(t)$. $u'=\sec(t)\tan(t)$ and $v'=\sec^{2}(t)$. So $y''=\sec(t)\tan(t)\cdot\tan(t)+\sec(t)\cdot\sec^{2}(t)=\sec(t)\tan^{2}(t)+\sec^{3}(t)$.

Step3: Evaluate at $t=\frac{\pi}{4}$

We know that $\sec(\frac{\pi}{4})=\sqrt{2}$ and $\tan(\frac{\pi}{4}) = 1$. Substitute into $y''$: $y''(\frac{\pi}{4})=\sqrt{2}\times1^{2}+\left(\sqrt{2}\right)^{3}=\sqrt{2}+ 2\sqrt{2}=3\sqrt{2}$.

Answer:

$3\sqrt{2}$