standard_description: Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
problem_activity: Task
Many computer applications use very complex mathematical algorithms. The faster the algorithm, the more smoothly the programs run. The running time of an algorithm depends on the total number of steps needed to complete the algorithm. For image processing, the running time of an algorithm increases as the size of the image increases.
For an $n$-by-$n$ image, algorithm 1 has running time given by $p(n) = n^3+3n+1$ and algorithm 2 has running time given by $q(n)=15n^2+5n+4$ (measured in nanoseconds, or $10^{-9}$ seconds ).
Compute the running time for both algorithms for images of size 10-by-10 pixels and 100-by-100 pixels.
Graph both running time polynomials in an appropriate window (or several windows if necessary).
Which algorithm is more efficient?


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