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.

1. Let $f(x) = 5 \sec(x) - 2\csc(x)$. Find the slope of the tangent line to $f$ at the point where $x =\frac{\pi}{3}$.

2. Let $p(z) = z^2\sec(z) - z\cot(z)$. Find the instantaneous rate of change of $p$ at the point where $z = \frac{\pi}{4}$.

3. Let $h(t) = \displaystyle \frac{\tan (t)}{t^2+1} - 2e^t \cos(t)$. Find $h'(t)$.

4. Let $g(r) = \displaystyle \frac{r \sec(r) }{5^r}$. Find $g'(r)$.

5. 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 $$\begin{gathered} s(t) = \frac{15 \sin(t)}{e^t} \end{gathered}$$. 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 $ds/dt$, state the units on this function, and explain what it tells you about the object's motion. Finally, compute and interpret $s'(2)$.