Question

A Cepheid variable star is a star whose brightness alternately increases and decreases. The most easily visible such star is Delta Cephei, for which the interval between times of maximum brightness is \(5.4\) days. The average brightness of this star is \(4.0\) and its brightness changes by \( \pm 0.3\). In view of these data, the brightness of Delta Cephei at time \(t\), where \(t\) is measured in days, has been modelled by the function

\(B\left( t \right) = 4.0 + 0.35\sin \left( {\frac{{2\pi t}}{{5.4}}} \right)\)

(a) Find the rate of change of the brightness after \(t\) days.

(b) Find, correct to two decimal places, the rate of increase after one day.

Short answer

(a) Thus, the required rate of change is \(\frac{{7\pi }}{{54}}\cos \frac{{2\pi t}}{{5.4}}\).

(b) Thus, the required rate of increase is \(0.16\).

Step-by-step solution

The chain rule

If \(g\) is differentiable at \(x\) and \(f\) is differentiable at \(g\left( x \right)\), then the composite function \(F = f \circ g\) defined by \(F\left( x \right) = f\left( {g\left( x \right)} \right)\) is differentiable at \(x\) and \(F'\) is given by the product \(F'\left( x \right) = f'\left( {g\left( x \right)} \right) \cdot g'\left( x \right)\). In Leibniz notation, if \(y = f\left( u \right)\) and \(u = g\left( x \right)\) are both differentiable functions, then \(\frac{{dy}}{{dx}} = \frac{{dy}}{{du}} \cdot \frac{{du}}{{dx}}\).

(a) Step 2: The rate of change of brightness

Differentiate the brightness function with respect to \(t\) as follows:

\(\begin{aligned}B'\left( t \right) &= 0 + 0.35\cos \left( {\frac{{2\pi t}}{{5.4}}} \right)\frac{d}{{dt}}\left( {\frac{{2\pi t}}{{5.4}}} \right)\\ &= 0.35\cos \left( {\frac{{2\pi t}}{{5.4}}} \right)\left( {\frac{{2\pi }}{{5.4}}} \right)\\ &= \frac{{7\pi }}{{54}}\cos \left( {\frac{{2\pi t}}{{5.4}}} \right)\end{aligned}\)

The rate of change of brightness is \(\frac{{7\pi }}{{54}}\cos \frac{{2\pi t}}{{5.4}}\).

(b) Step 3: The rate of change at \(t = 1\)

Put \(t = 1\) in \(B'\left( t \right)\) and solve as follows:

\(\begin{aligned}{l}B'\left( t \right) &= \frac{{7\pi }}{{54}}\cos \left( {\frac{{2\pi t}}{{5.4}}} \right)\\B'\left( 1 \right) &= \frac{{7\pi }}{{54}}\cos \left( {\frac{{2\pi }}{{5.4}}} \right)\\B'\left( 1 \right) &\approx 0.16\end{aligned}\)

Therefore, the rate of increase after one day is \(0.16\).