Activity 2.4.4.
Respond to each of the following prompts. Where a derivative is requested, be sure to label the derivative function with its name using proper notation.
(a)
Let \(f(x) = 5 \sec(x) - 2\csc(x)\text{.}\) Find the slope of the tangent line to \(f\) at the point where \(x =\frac{\pi}{3}\text{.}\)
(b)
Let \(p(z) = z^2\sec(z) - z\cot(z)\text{.}\) Find the instantaneous rate of change of \(p\) at the point where \(z = \frac{\pi}{4}\text{.}\)
(c)
(d)
(e)
When a mass hangs from a spring and is set in motion, the object’s position oscillates in a way that the size of the oscillations decrease. This is usually called a damped oscillation. Suppose that for a particular object, its displacement from equilibrium (where the object sits at rest) is modeled by the function
\begin{equation*}
s(t) = \frac{15 \sin(t)}{e^t}\text{.}
\end{equation*}
Assume that \(s\) is measured in inches and \(t\) in seconds. Sketch a graph of this function for \(t \ge 0\) to see how it represents the situation described. Then compute \(ds/dt\text{,}\) state the units on this function, and explain what it tells you about the object’s motion. Finally, compute and interpret \(s'(2)\text{.}\)
