MHB Creating a function for the total change theorem

[zimmertr]

Hello everyone, I'm having some trouble getting started with this problem because I think I misunderstand the meaning. Can anyone help me? I'm confident I'll be able to complete the problem once I finish step 1, I just don't understand what it's asking? How can you design a function to measure the amount of time required to move a single foot when the variable of the function that you provide is the number of feet traveled? It doesn't make sense to me.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. Given below is a table showing 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.

1) Define a new function m(h) that measures the number of minutes required for a plane at altitude h to climb to the next foot of altitude.

2) Construct a table similar to the one above for the values of m(h) and explain how it is related to the table above. Be sure to explain the units.

3) 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.

4) 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.

5) Use the Riemann sum M5 to estimate the value of the integral you found in (c). Include units on your result.

 

[Evgeny.Makarov]

If the climbing speed at height $h$ is $c(h)$, then the time it takes to climb 1 foot at height $h$ is $1/c(h)$ (distance over speed).

 

[zimmertr]

Thanks so much. That really helped. The way that the question is worded semantically was messing with my brain. I've since finished the assignment. I would really appreciate it if you would check it if you have the time:

https://www.dropbox.com/s/zwu0bzmvl7416io/Zimmerman Miniproject8.pdf?dl=0

 

[Evgeny.Makarov]

Next time, please try to post your solution in the body of the message. For help with LaTeX (the language used for typing formulas), check out sticky threads in http://mathhelpboards.com/latex-tips-tutorials-56/.

I don't think that regression analysis is required in this problem unless that's what you've been studying. The definition of $m(h)$ is simply $m(h)=1/c(h)$.

Your problem numbering is off by 1.

In problem 2, you should probably say explicitly that the units of $m(h)$ are min/foot. I first thought that the units are simply minutes (since the question asks about the time to travel one foot), but min/foot is probably what's intended.

Regarding problem 3, my recommendation is: fewer words, more formulas. It is impossible to give the right answer unless you reason with formulas (unless you are an ancient Greek who developed the theory of quadratic curves using only natural language (Smile)).

Suppose that the plane is climbing as rapidly as possible. It's height $h$ is a function of time $t$, say, $h=f(t)$. Each height is passed only once during the ascent, so the function is injective, so there exists the inverse function $t=f^{-1}(h)$. Next, $c(h)$ looks like a derivative of $f$, but it depends on $h$ rather than $t$. This means that $c(h)$ equals $f'(t)$ where $t$ corresponds to the given $h$, i.e., $c(h)=f'(f^{-1}(h))$. Finally,
\[
m(h)=\frac{1}{c(h)}=\frac{1}{f'(f^{-1}(h))}.
\]
The right-hand side is in the derivative of $f^{-1}(h)$ according to the formula in Wikipedia. Therefore, $m(h)=(f^{-1})'(h)$. The antiderivative of this function is $f^{-1}(h)$. So its input is height, and the output is the time when this height is reached.

For problem 4, you correctly write that the time to reach $H=10,000$ feet is $\int_0^{H}m(h)\,dh$, but it is not clear from where it follows. By the fundamental theorem of calculus,
\[
\int_0^{H}m(h)\,dh=\int_0^{H}(f^{-1})'(h)\,dh=f^{-1}(H)-f^{-1}(0),
\]
and, assuming $f^{-1}(0)=0$, the integral equals $f^{-1}(H)$, i.e., the time required to reach 10,000 feet.

If you need to use the Riemann sum with 5 terms only, then you don't need to interpolate the function since the value of $m(h)$ is known in 5 equally-spaced points.

I should note that calculus is not my specialty and I may have missed something, such as an easier way to answer a question.