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.
posted by mathman at 10:19 AM
7 Comments:
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).
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.
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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