### Assignment: Real Life Derivatives

Now that we have explored the meaning of the derivative of a function, let's think about some things in our worlds that can be modeled using rates of change. My example is my 1969 Volkswagen Westfalia. It is obviously old and, because I live at the beach, it has to endure the harsh elements. So, what does this have to do with functions, you ask. Well, I am constantly worrying about rust. My function is the amount of rust on my VW as a function of time, R(t). As much as I wish it were not the case, I think this is probably an increasing function. The more rust there is, the more there is going to be. Perhaps a never ending battle with the elements. Oh to own a garage.

What is the meaning of the derivative of my rust function? Liebniz would consider dR/dt...the change in the amount of rust with respect to time. In other words, the amount of rust that is added at any given moment. Argh, continuous function? Does that mean my beloved Math-Mobile is rusting away as I write this? Maybe I should get out of here and start sanding and grinding. Ah but back to the Calculus...

In this blog, I want you to consider some function in your world and discuss the meaning of the derivative of your function.

What is the meaning of the derivative of my rust function? Liebniz would consider dR/dt...the change in the amount of rust with respect to time. In other words, the amount of rust that is added at any given moment. Argh, continuous function? Does that mean my beloved Math-Mobile is rusting away as I write this? Maybe I should get out of here and start sanding and grinding. Ah but back to the Calculus...

In this blog, I want you to consider some function in your world and discuss the meaning of the derivative of your function.

## 21 Comments:

A function that came to mind upon reading the story of Mr. Mayo’s VW Bus, I will call K(t). This represents the amount of knowledge gained as a function of time since we started school almost thirteen-years-ago. Leibniz would write the derivative of my function as dK/dt , which is always positive. I would love to say that the derivative function of a school-aged person is positive. At least I hope school is not making us all less intelligent. That would sure put a new spin on the No Child Left Behind Policy. However, I do feel that the second derivative of the knowledge function is negative because we all learn something new everyday, but as I’ve heard before, “Everything I needed to know I learned in first grade.” Therefore, as you gain more and more knowledge it becomes harder and harder to learn something new each day. Also, as much as we all hate to admit it, we do forget some things from time to time. Anyway, with all of this being said, let’s go out and raise our K(t) values.

Hmmmm...the function H(t) came to mind when prompted with this blog. The function of H(t) represents height as a function of time. If you have ever had a growth chart on your wall you know that you have a couple of periods when you grew faster than at other times in your life. If the marks were made at regular intervals, they'd be more spread out in certain periods and more clustered together in others. So, how fast you grew is the derivative with respect to time of how tall you were. Leibniz would write the derivative of the function as dH/dt. The derivative would be big sometimes, small sometimes (the derivative is positive) and once you reach a certain age you begin to shrink and your derivative then becomes negative.

Huh..Okayy..

Hrmm..Things in our world that can be modeled using rates of change. Okay, Lets model age as a function of time. A(t). Lebniz would write this as dA/dt. A(t) is more than likely an increasing function, well, I hope it is. Nobel prize for the guy who makes it negitive..Not to mention a fair amount of mola. Hrm.. Age decreasing as time increases, that would be an interesting phnomenom. However, unless someone discovers the perverbial fountain of youth or such, dA/dt will always be increasing and positive.

I have a function about running that the Prewannbe didnt post before me. Yeah! My function is T(m). T represents time with respect to mile number in a race. I hate to say it but for me this is an increasing function outside of saturday's race. I hope to make the fuction constant one day but anyway. The way Leibnizs would have written it would be dT/dm. Since the function is increasing, the derivitive will always be positive. Wow it would be awesome if the deritive was negative, maybe one day.

A derivative function in my everyday life could be the function I(t) where I(t) is the income in dollars as a function of age (t) in years. Leibniz would express this in dI/dt. In other words, change in dollars per year. Hopefully, this function will continue to be an increasing function because it would be impossible to live on my own out in the world with my current income and not receive government food stamps! Generally speaking, this function is increasing for most people. Sure there are exceptions for whatever reason but as a whole, American people make more money the closer they get to retirement. Therefore, the derivative would be positive. The second derivative would be both positive and negative. It would be negative because once you retire, I(t) develops a horizontal asymptote known as social security. This asymptote makes the graph concave down. The second derivative would be positive because the graph of I(t) would be concave up thanks to lemonade stands, birthday checks, and pennies found on the ground while you’re younger (horizontal asymptote y = 0).

One thing that could be a function in my world is memorization with respect to repetitions M(r). Obviously to memorize something very well you must study it with a lot of repetitions. (Well for me at least) Also the same would be true that if you barely look at something, then you probably aren’t going to know it very well. The derivative of my function would be written dM/dr…. which is the change in the amount of memorization or knowledge in respect to repetitions. The function is increasing with repetitions and the derivative will always be positive.

In my world, a function that deals with my everyday life is M(h). This function is the time I spend listening to music as a function of the time I spend on homework. As Liebniz would say, the derivative of my function, dM/dh is positive, since the more time I spend doing homework, the more time I spend listening to music. Similarly, if I do not have much homework, I spend less time listening to music and more time doing things that are more entertaining than just sitting idly and listening to music.

Judging by the last 17 years of my life, it has become clear that R(y) is a function related to my life. R(y) expresses the number of responsibilities that I must uphold as a function of the years I have lived. As I become older, my responsibilities have increased as a result of school, a job, and more freedom, thus making dR/dy of my function positive. Hopefully in the stressful times of my life the derivative will decrease a little bit.

My function would be S(t). This represents the degree of senoritis as a function of time. The derivative would be dS/dt. The derivative would also be positive. Everyday my senoritis gets worse and worse.

a function for me would be H(w). the amount of time i have to do homework at night with the respect f how late i have to work. leibniz would write the derivative as dH/dw. the more i work, the less time i have to do homework before my parents make me go to bed or before i fall asleep. i do my homework though.

i have chosen to use the function of my growth. when i first came into the world, i had a lot of room to grow, but as i get older, i cant grow as much. This is represented by the function g(j). Leibniz would write the derivative of my functions as dg/dj which means, change in growth over the change in time. The function would be increasing at a decreasing rate because i eventually will not be able to grow anymore.

take time and meditate the meaning of this in your mind, and you will find, sight of the blind, and you will not be left behind...

Then take me disappearin' through the smoke rings of my mind,

Down the foggy ruins of time, far past the frozen leaves,

The haunted, frightened trees, out to the windy beach,

Far from the twisted reach of crazy sorrow.

Yes, to dance beneath the diamond sky with one hand waving free,

Silhouetted by the sea, circled by the circus sands,

With all memory and fate driven deep beneath the waves,

Let me forget about today until tomorrow.

(bobby boy)

A function that I thought of would be N(t). This function represents the number of times I got my name on the board as a function of time (years). Liebniz would write the derivative of this function as dN/dt. The number of times my name was on the board per year. Over time the number of times my name has been on the board has accumlated. Every year I would get my name on the board less that year. Therefore, the derivative of my function is positive and the second derivative is negative. My function is increasing at a decreasing rate.

The function that I thought of was F (a). F (a) represents the amount of freedoms with respect to age. The older one becomes the more freedoms one should have. So, the function is increasing at an increasing rate. Leibniz would have written the derivative of this function as follows: dF/da. The derivative of F (a) will positive considering the older one becomes the more they are allowed to do and just the opposite, the younger one is the less they are allowed to do!

A function in my present day life that has a derivative is the amount of pollution in our air, with respect to time P(t). Since the population is increasing daily as well as the pollution producing agents, the function is increasing at an increasing rate. The derivative of this function is positive and will hopefully start to decrease. Liebniz notation would be dP/dt. Thanks to hybrids and our environmental club hopefully this decrease in pollution will happen sooner than later.

Dr. Mayo's story of his VW Van and his nonterminating battle with rust got me thinking about my car as well. My car, a '99 Mitsu Eclipse, is only six years old but I am already having to funnel money into repairing my car, almost as often as I fill up the gas tank. If it's not a new set of tires one week, it's a bum EGR valve or O2 sensor. It seems like the cost of owning and maintaining my car is always increasing with each passing month. The function C(t) represents the cost of maintaining my car over time. This is most definately an increasing function. The derivative function of this one (in Leibniz terms: dC/dt) is the change in cost with respect to time. This derivative fxn. is certainly increasing and will continue to increase at an increasing rate more than likely.

I enjoy music. When I hear a good song- a really good song- I listen to it over and over again. Each time I play it though, it looses some impact. It is still enjoyable, but not as enjoyable. So my function is J(t). "J" is the total joy I have derived from a song after listening to it "t" times. The derivative, in Leibniz Notation, would be dJ/dt- the additional satisfaction per one listening. The first derivative is always positive for good songs, while the second derivative is always negative, as it increases at a decreasing rate.

It seems like the older you get the less sleep you end up getting because of homework, activities, etc. So, a possible function in my life would be S(t), the amount of sleep you get as a function of time. In Leibniz notation this would be dS/dt. Over the years the number of hours you have slept increases, but the number of hours per year decreases, therefore the function is increasing at a decreasing rate. So the derivative is positive and the second derivative is negative.

Cool stuff here. It seems like most of you have this concept nailed. What if the beloved math-mo-beel is 20% rusted now (underneath the cover of equations) and has an R(t)=5 %/year. How long until it is nothing but dust?

CORRECTION TO LAST COMMENT:

What if the beloved math-mo-beel is 20% rusted now (underneath the cover of equations) and has a dR/dt=5 %/year. How long until it is nothing but dust?

Actually Mr. Scientist, if we lose 5% per year we will always have some Math-Mobile left. It is kind of like that old dead guy Zeno. If we take away 5% of 80% we end up with 76%. We then take away 5% of 76% and end up with 72.2%. If we continue this we will never get to 0%. However, as much as I hate to admit it, the ol' Math-Mobile will be far from healthy. Speaking of which, I picked up an odd sound this morning driving in. Sounds like it may be a valve out of adjustment. Oh the woes of Math-Mobile ownership.

Well now, I guess it depends on how you interpret the problem. On the one hand, dR/dt could be referenced to the total surface area of the car (i.e., 5% of the car surface rusts each year). In this case dR/dt is a constant, R(t) is linear, and the math mo-beel's days are definitely numbered. On the other hand dR/dt could refer to that part of the car that is unrusted (i.e., 5% of the unrusted part of the car rusts each year). In this case dR/dt is proportional to (1-R), R(t) is exponential, and the math mo-beel will outlast us all, albeit with only a VW bug's worth of metal before long and only a real bug's worth of metal eventually.

So is the moral to the story that math in general, and calculus in particular, isn't just plug and chug? Rather, it requires understanding, assumptions, decisions, etc, that all impact the final result? I hope so, since this is what makes it most interesting!

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