Difference between revisions of "Piece-wise defined functions"
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Below we try to give a bit more motivation to piece-wise functions (which is not just a nuance for students, but rather indispensable tool in mathematics and applications). | Below we try to give a bit more motivation to piece-wise functions (which is not just a nuance for students, but rather indispensable tool in mathematics and applications). | ||
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''' Pool table ''' | ''' Pool table ''' | ||
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Determine how many positive even numbers are there thad do not exceed a given positive integer. | Determine how many positive even numbers are there thad do not exceed a given positive integer. | ||
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''' Ball trajectory''' | ''' Ball trajectory''' | ||
A person throws a ball forward at 3 m/s. The ball bounces off the ground, reaches some maximal height, then falls and bounces again, and so on. If the initial altitude of the ball is 1.5 m, find the trajectory of the ball if a) no energy is lost b) about 20% of energy is lost on each bounce. | A person throws a ball forward at 3 m/s. The ball bounces off the ground, reaches some maximal height, then falls and bounces again, and so on. If the initial altitude of the ball is 1.5 m, find the trajectory of the ball if a) no energy is lost b) about 20% of energy is lost on each bounce. | ||
Revision as of 15:20, 2 December 2021
A typical introduction of piece-wise defined functions in a calculus textbook looks something like this: there are functions that take on different formulas depending on the intervals (pieces) of their domains, for example: [math]\displaystyle{ f(n) = \begin{cases} x^2, & \mbox{if }x\le 1 \\ 3-2x, & \mbox{if }x \gt 1 \end{cases} }[/math]
(examples may vary).
Below we try to give a bit more motivation to piece-wise functions (which is not just a nuance for students, but rather indispensable tool in mathematics and applications).
Pool table
A pool table has dimensions 10 ft by 5 ft. A ball is hit from a corner towards the longer side at the angle 30 degrees (with respect to the shorter side). Describe the trajectory of the ball before it hits the side second time.
Even numbers
Determine how many positive even numbers are there thad do not exceed a given positive integer.
Ball trajectory
A person throws a ball forward at 3 m/s. The ball bounces off the ground, reaches some maximal height, then falls and bounces again, and so on. If the initial altitude of the ball is 1.5 m, find the trajectory of the ball if a) no energy is lost b) about 20% of energy is lost on each bounce.