Factor: $A'(t) = 1000 e^{-0.03t} [ -0.03 \sin(0.1\pi t) + 0.1\pi \cos(0.1\pi t) ]$.

February 22, 2026

$Factor: $A'(t) = 1000 e^{-0.03t} [ -0.03 \sin(0.1\pi t) + 0.1\pi \cos(0.1\pi t) ]$.$

Slope = $rac{-2}{-1} = 2$. The reflected ray has slope 2 relative to the horizontal.
A micropaleontologist observes that the abundance $A(t)$ of a microfossil species follows $A(t) = 1000 e^{-0.03t} \sin(0.1\pi t)$, where $t$ is time in thousands of years. At what time $t > 0$ does the first local maximum occur?
To find maxima, take derivative: $A'(t) = 1000 [ -0.03 e^{-0.03t} \sin(0.1\pi t) + e^{-0.03t} (0.1\pi) \cos(0.1\pi t) ]$.
Set derivative to zero: $-0.03 \sin(0.1\pi t) + 0.1\pi \cos(0.1\pi t) = 0$.
$rac{\sin}{\cos} = rac{0.1\pi}{0.03} \Rightarrow an(0.1\pi t) = rac{10\pi}{3} pprox rac{31.416}{3} pprox 10.472$.
Then $0.1\pi t = rctan(10.472) pprox 1.471$ radians.

Related Articles