Activity 2.3.3.
Use the quotient rule to answer each of the questions below. Throughout, be sure to carefully label any derivative you find by name. That is, if you’re given a formula for \(f(x)\text{,}\) clearly label the formula you find for \(f'(x)\text{.}\) It is not necessary to algebraically simplify any of the derivatives you compute.
(a)
(b)
(c)
Determine the slope of the tangent line to the curve \(\displaystyle R(x) = \frac{x^2 - 2x - 8}{x^2 - 9}\) at the point where \(x = 0\text{.}\)
(d)
When a camera flashes, the intensity \(I\) of light seen by the eye is given by the function
\begin{equation*}
I(t) = \frac{100t}{e^t}\text{,}
\end{equation*}
where \(I\) is measured in candles and \(t\) is measured in milliseconds. Compute \(I'(0.5)\text{,}\) \(I'(2)\text{,}\) and \(I'(5)\text{;}\) include appropriate units on each value; and discuss the meaning of each.
