Difference: SniperZero (3 vs. 4)

Revision 42008-01-28 - DickFurnas

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META TOPICPARENT name="SeasonOne"

Numb3rs 109: Sniper Zero

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  approximate the actual trajectory is to ignore the effects of drag and wind, instead looking only at gravity.
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Figure ##f#.:

 
bullet trajectory, freefall
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  Since we disregard drag, the horizontal component of the velocity does not change.
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Figure ##f#.:

 
velocity vector decomposition
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  follow them and do the exercises, but if you can't, just use the equations mentioned in order to do activity 2.
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  • Tip, idea Technically, the term velocity means the vector pointing in the direction of motion with magnitude equal to the speed of the object. However, in everyday usage and even in many
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  the dean's office and has also incurred substantial long-term debt in the form of lavish salary increases and exponential growth in new, highly compensated faculty and staff directly
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reporting to him." ( _UCSF dean is fired, cites whistle-blowing_, Los Angeles Times, Dec 15,
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reporting to him." ( UCSF dean is fired, cites whistle-blowing, Los Angeles Times, Dec 15,
  2007)

While the above excerpts describe growth in entirely different areas,

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the one thing they have in common is the use of the term _exponential growth._ In mathematics, we say that quantity x grows exponentially
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the one thing they have in common is the use of the term exponential growth. In mathematics, we say that quantity x grows exponentially
  with respect to time t if x satisfies the following differential equation:dx/dt=kx, where k is a constant and dx/dt is either a derivative, when t is continuous, or the change in x in a given
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  where C is some constant, and exponentiating both sides, we finally get x=Dekt, where D is a constant. We can solve for D by plugging in t=0, the starting
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time, to arrive at the general solution x(t)=x(0)e^kt. Exponential growth is much faster than polynomial, as the example below illustrates in case of et versus t3.
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time, to arrive at the general solution x(t)=x(0)e^kt.

Figure ##f#.:

 exponential vs polynomial growth
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Exponential growth is much faster than polynomial, as the figure illustrates in case of et versus t3.
 

Activity ##a#.:

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  1. You open a savings account which earns 2% interest with a deposit of $1000. Would you rather the interest compound daily or monthly? Write down the formula for the amount of money in the account after a year in both cases. (Hint: write down the expression for the amount of money after one period of compounding, now after two periods (don't simplify!), then three... See the pattern?)
  2. Suppose we decide to compound not once a month or a day, but once every split second. In fact, we can let the number of compounding periods go to infinity, thus letting the length of each period approach zero. Use the fact that e^y=lim(1+y/n)^n as n->inf0.02*t dollars. This is continuous compounding.

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Tangent ##t#.:

  In popular usage, the expression "exponential growth" is often used as a synonym for "very fast growth". There's no good reason to describe faculty hiring practices, as the third quote
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  exponential function, sigmoidal growth reflects the fact that at some point growth must slow down due to lack of resources.
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