Exercise 7 — Active Calculus

When an aircraft attempts to climb as rapidly as possible, its climb rate (in feet per minute) decreases as altitude increases, because the air is less dense at higher altitudes. Table provides performance data for a certain single engine aircraft, giving its climb rate at various altitudes, where $c(h)$ denotes the climb rate of the airplane at an altitude $h$ .

Data for the climbing aircraft.
$h$ (feet)	$0$	$1000$	$2000$	$3000$	$4000$	$5000$
$c$ (ft/min)	$925$	$875$	$830$	$780$	$730$	$685$

$h$ (feet)	$6000$	$7000$	$8000$	$9000$	$10{,}000$
$c$ (ft/min)	$635$	$585$	$535$	$490$	$440$

Let a new function $m$ , that also depends on $h$ , (say $y = m(h)$ ) measure the number of minutes required for a plane at altitude $h$ to climb the next foot of altitude.

Determine a similar table of values for $m(h)$ and explain how it is related to the table above. Be sure to discuss the units on $m$ .
Give a careful interpretation of a function whose derivative is $m(h)$ . Describe what the input is and what the output is. Also, explain in plain English what the function tells us.
Determine a definite integral whose value tells us exactly the number of minutes required for the airplane to ascend to 10,000 feet of altitude. Clearly explain why the value of this integral has the required meaning.
Determine a formula for a function $M(h)$ whose value tells us the exact number of minutes required for the airplane to ascend to $h$ feet of altitude.
Estimate the values of $M(6000)$ and $M(10000)$ as accurately as you can. Include units on your results.

Boelkins, Active Calculus, §5.2 The Second Fundamental Theorem of Calculus, Exercise 7

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