Activity 2.5.4

Use known derivative rules, including the chain rule, as needed to respond to each of the following prompts.

1. Find an equation for the tangent line to the curve $y= \sqrt{e^x + 3}$ at the point where $x=0$.

2. If $\displaystyle s(t) = \frac{1}{(t^2+1)^3}$ represents the position function of a particle moving horizontally along an axis at time $t$ (where $s$ is measured in inches and $t$ in seconds), find the particle's instantaneous velocity at $t=1$. Is the particle moving to the left or right at that instant?

3. At sea level, air pressure is 30 inches of mercury. At an altitude of $h$ feet above sea level, the air pressure, $P$, in inches of mercury, is given by the function $P = 30 e^{-0.0000323 h}$. Compute $dP/dh$ and explain what this derivative function tells you about air pressure, including a discussion of the units on $dP/dh$. In addition, determine how fast the air pressure is changing for a pilot of a small plane passing through an altitude of $1000$ feet.

4. Suppose that $f(x)$ and $g(x)$ are differentiable functions and that the following information about them is known:

   $x$	$f(x)$	$f'(x)$	$g(x)$	$g'(x)$
   $-1$	$2$	$-5$	$-3$	$4$
   $2$	$-3$	$4$	$-1$	$2$

   If $C(x)$ is a function given by the formula $f(g(x))$, determine $C'(2)$. In addition, if $D(x)$ is the function $f(f(x))$, find $D'(-1)$.