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Related Rates—Position, Velocity, and Acceleration over time (Jan. 29, 2010)
Position, Velocity, and Acceleration (Related Rates)
Consider a baseball thrown at 10 feet per second, straight down from a building that is 1000 feet tall.
The position function for falling objects is:
...and we're counting time in seconds, velocity in feet per second, and position in feet.
So at time zero, position is 1000 feet, since -16(02) + (-10)(0) + 1000 = 1000 feet. Mathematically, this is expressed:
...which can be read: "At time zero, position is 1000 feet."
At time = 2 seconds:
At what time (t) will it hit the ground? (i.e. What value of t makes p(t) zero?):
What a mess, but it turns out you can discard the negative numerator (since time will always be positive), and it turns out to be (-5 + 126.6)/16 ≈ 7.6 seconds.
Well that's fine and dandy, but that's got nothing to do with derivatives. That's algebra. The position formula becomes more useful when you take derivatives of it.
Mathematically lazy folks describe derivatives of f(x) as f'(x), like this:
So how to take a derivative...? Easy stuff. The “Generalized Power Rule” is the secret weapon they don't show you until you've learned to figure derivatives the way they did it in the 1600's. I'm told it built character, but every time I've built character I've regretted it.
Some rules to know:
For example, it looks syntactically like this:
The d on top says we're taking a derivative, the dx on the bottom says we're taking it with respect to x, which is convenient because we have no other variables here. The stuff in the brackets is the function we're taking the derivative of.
The Power Rule says we take any argument with an x in it, multiply the exponent by the coefficient, and then decrement the exponent, and the result is the derivative.
So our first argument, 3x2, becomes (2)(3)x(2-1) = 6x1 or just plain 6x.
The derivative of that big bundle of crap is 6x + 6.
What if they throw another variable in? Well, then it's a constant and its derivative is zero.
First argument 9x4 = (4)(9)x(4-1) = 36x3
If position is described by:
...and velocity over time is described by the derivative of this function [d/dt], then:
...and acceleration over time is described by the derivative of velocity function, then:
This is a simplified acceleration (-32t feet per second), but it roughly approximates the 9.8 meters per second2 we're used to on planet Earth. You can plug in any positive number for time t and find instantaneous velocity or acceleration or position.
Some examples of derivatives with respect to x (denoted d/dx):
So you can see if you keep taking derivatives, you will eventually reduce the powers all to zero and then be left with a constant whose derivative is zero, and further derivatives are zero. This is the case unless you have a fractional power, and in that case, derivatives go on forever, but their utility does not necessary correspond with the amount of work required to figure them. For instance, the second derivative of the position function is acceleration, but what is the fifth derivative useful for? (I don't know, and I'm not convinced anyone else does.)
That's derivatives in a nutshell. Everything else is some variation. There are rules for trigonometric functions and their inverses, logarithms, e, and a bunch of other stuff, but basically the challenge of any actuarial (or biomedical, etc.) problem is creating an accurate mathematical model... some function that approximates the rates of accidents (or growth, or death (e.g. cancer-killing), etc.), and then making predictions based on the models. There's lots of software to create models with, so it's not just guesswork... the computer does the big stuff.