Single Variable Calculus, Chapter 3, 3.5, Section 3.5, Problem 77

A star whose brightness alternatively increases and decreases has a function of $B(t) = 4.0 + 0.35 \sin \displaystyle \left( \frac{2 \pi t}{5.4} \right)$ where $t$ is measured in days.

a.) Find the rate of charge of the brightness after $t$ days.


$\begin{equation} \begin{aligned} \frac{dB}{dt} =& \frac{d}{dt} (4) + 0.35 \frac{d}{dt} \left[ \sin \left( \frac{2 \pi t}{5.4} \right) \right] \cdot \frac{d}{dt} \left( \frac{2 \pi t}{5.4} \right) \\ \\ \frac{dB}{dt} =& 0.35 \cos \left( \frac{2 \pi t}{5.4} \right) \left( \frac{2 \pi}{5.4} \right) \\ \\ \frac{dB}{dt} =& \frac{7 \pi}{54} \cos \left( \frac{2 \pi t}{5.4} \right) \end{aligned} \end{equation}$




b.) Evaluate the rate of increase after one day.



$\begin{equation} \begin{aligned} @ t =& 1 \\ \\ \frac{dB}{dt} =& \frac{7 \pi}{54} \cos \left( \frac{2 \pi (1)}{5.4} \right) \\ \\ \frac{dB}{dt} =& 0.1613 \text{ brightness/day} \end{aligned} \end{equation}$